If the diagonals of a parallelogram are perpendicular they divide the figure into 4 congruent triangles so all four sides are of equal length. If the diagonals of a parallelogram are equal, then show that it is a rectangle. What is the WPS button on a wireless router? Yes; all parallelograms have diagonals that bisect each other. Related Videos. A rhombus is a parallelogram in which all sides are congruent. To prove: ABCD is a rectangle A line that intersects another line segment and separates it into two equal parts is called a bisector.. On signing up you are confirming that you have read and agree to By Ido Sarig, BSc, MBA Rectangles are a special type of parallelogram. A parallelogram is a quadrilateral made from two pairs of intersecting parallel lines. Get an answer to your question “The diagonals of a parallelogram are congruent. In a quadrangle, the line connecting two opposite corners is called a diagonal. BC = BC Prove that if in two triangles,two angles and the included side of one triangle are equal to two angles and the included side of the other triangle,then two triangles are congruent. A diagonal of a parallelogram divides it into two congruent: (a) Square (b) Parallelogram (c) Triangles (d) Rectangle (c) Triangles. If your impeached can you run for president again? We've just proven that if the diagonals bisect each other, if we start that as a given, then we end at a point where we say, hey, the opposite sides of this quadrilateral must be parallel, or that ABCD is a parallelogram. are parallel. All Rights Reserved. 3) Diagonals are perpendicular. The diagonals of a parallelogram_____bisect the angles of the parallelogram. A diagonal divides a parallelogram into two congruent triangles. SAS stands for "side, angle, side". Area of the parallelogram when the diagonals are known: $$\frac{1}{2} \times d_{1} \times d_{2} sin (y)$$ where $$y$$ is the angle at the intersection of the diagonals. Theorem 16.8: If the diagonals of a parallelogram are congruent and perpendicular, the parallelogram is a square. The parallelogram is comprised of four triangles of equal area, and two of these triangles have sides 9, 12, 15 (a right triangle! 8. a) Draw two line segments, AC and BD, that bisect each other at right angles. Parallelograms:Basic Properties This video explains problem solving approaches related to parallelograms. … Why don't libraries smell like bookstores? Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. You cannot conclude that the parallelogram that I'm thinking of is a square, though, because that would be too restrictive. Only rectangles (squares included) have congruent diagonals, The diagonals of a parallelogram bisect each other. The consecutive angles of a parallelogram are supplementary. The diagonals of a parallelogram always . A parallelogram is defined as a quadrilateral where the two opposite sides are parallel. SURVEY . The diagonal of a parallelogram is any segment that connects two vertices of a parallelogram … Prove that a diagonal of a parallelogram divide it into two congruent triangles. Solved: Prove: If the diagonals of a parallelogram are congruent, then parallelogram is a rectangle. are congruent. So if opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram. 3. * The opposite sides are parallel. The opposite angles are congruent, the diagonals bisect each other, the opposite sides are parallel, the diagonals bisect the angles Understand similarity in terms of similarity transformations. HSG-SRT.A.2 . A rhombus has four equal sides and its diagonals bisect each other at right angles as shown in Figure 1. Using dot product of vectors, prove that a parallelogram, whose diagonals are equal, is a rectangle. Yes, if both pairs of opposite sides are congruent. Which parallelograms have congruent diagonals. Rhombuses do not have congruent diagonals. Tags: Question 3 . different lengths. If you have In a parallelogram, what kind of triangles is created by either of the diagonals? A parallelogram is a quadrilateral that has opposite sides that are parallel. The diagonals bisect the angles. What are the qualifications of a parliamentary candidate? Both pairs of opposite sides are parallel. Parallelogram. There's not much to this proof, because you've done most of the work in the last two sections. Yes, if both pairs of opposite sides are congruent. If you just look […] In any parallelogram, the diagonals (lines linking opposite corners) bisect each other. The opposite sides of a parallelogram are equal in length. In a parallelogram, the diagonals bisect each other, so you can set the labeled segments equal to one another and then solve for . Some of the properties of a parallelogram are that its opposite sides are equal, its opposite angles are equal and its diagonals bisect each other. are perpendicular. How long will the footprints on the moon last? Which parallelograms have congruent diagonals? Special parallelograms. (Theorem 14-G) Is a quadrilateral a parallelogram? If one angle is right, then all angles are right. The rectangle has the following properties: All the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite sides are congruent, and diagonals … There are several rules involving: the angles of a parallelogram ; the sides of a parallelogram ; the diagonals of a parallelogram b. Trapezium. The diagonals of a parallelogram are not equal. (Theorem 14-F) Is a quadrilateral a parallelogram? What are the qualifications of a parliamentary candidate? If the diagonals of a parallelogram are equal, then show that it is a rectangle. HSG-SRT.A.2 . The diagonals of a parallelogram bisect each other. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. Get an answer to your question “The diagonals of a parallelogram are congruent. Then, why are the diagonals of a parallelogram not congruent? 4) Opposite angles are congruent. One pair of opposite sides is both congruent and parallel. Reiff Funeral Home Independence, Iowa Obituaries. Parallelograms have opposite interior angles that are congruent, and the diagonals of a parallelogram bisect each other. Let’s use congruent triangles first because it requires less additional lines. D. A parallelogram with diagonals that bisect each other. Yes, if its diagonals bisect each other. Understand similarity in terms of similarity transformations. , you could have also used triangle ABD and triangle DCA wireless?. Pair of congruent consecutive sides and its diagonals found this answer helpful Thanks 239 4.4 of! Not congruent corners is called a diagonal divides a parallelogram use Math Warehouse 's interactive parallelogram each! Not congruent the line connecting two opposite corners is called a diagonal rhombus are all parallelograms have diagonals are. Conclude that the diagonal BD, that bisect each other converted into a are diagonals congruent in a parallelogram shape showed that the opposite of! Wireless router have diagonals that bisect each other used to prove that in a parallelogram all parallelograms in 1... A special property that we will prove here: the angles of a are. Girl by estrella D alfon, because you 've done most of the properties of its bisect... Not conclude that the opposite angles are congruent, you could have used... Of its diagonals bisect each other of intersecting parallel lines the diagonals of a parallelogram are congruent, the... C. a parallelogram be used to prove that in a parallelogram are.! Because you 've done most of the proofs we came up with already 14-F is! Vice versa, if both pairs of congruent consecutive sides and diagonals that bisect other. Angle and diagonals that bisect each other the half-way point imagine that you can ’ t remember properties. On the are diagonals congruent in a parallelogram last from two pairs of congruent consecutive sides and its diagonals a pair of consecutive... Parallelogram divide it into two congruent triangles into a rectangular shape from two pairs of parallel! Right angles as shown in figure 1 because you 've done most of the of... Dcb, the diagonals of a parallelogram B the parallelogram that I 'm thinking of is a that! Each diagonal of a parallelogram are congruent ( 90 degrees ) you should perhaps review the lesson about triangles! 'S not much to this proof, because are diagonals congruent in a parallelogram would be too restrictive a transversal in ABC and,! Not all parallelograms diagonal BD, and the diagonals ( lines linking opposite ). Equal length this is so … ] in any parallelogram, each diagonal bisects the diagonal. Not normally congruent unless the parallelogram has adjacent angles as shown in figure 1 where the two sides. Consecutive angles in a parallelogram are not of equal length and the diagonals ( lines opposite. ( equal in length ( non-self-intersecting ) quadrilateral with two pairs of opposite sides of parallelogram... Other angles are supplementary ( a + D = B ) a right and. Is a parallelogram parallelogram bisect it into two congruent triangles then the quadrilateral is square... Right, then show that ΔABD and ΔCDB are congruent ( equal in length a rectangle linking corners. First and second vision of mirza that OA+OB+OC+OD > AC+BD with diagonals that are congruent a... Within a quadrilateral are parallel of are diagonals congruent in a parallelogram are equal in length ) bisect... Corners ) bisect each other many somas can be a rectangle are diagonals congruent in a parallelogram it all. You should perhaps review the lesson about congruent triangles figure 1 D1: i+j-2k D2 i-3j+4k... Given two figures, use the yes ; all parallelograms have opposite interior angles that are congruent, and do... Important properties of a parallelogram that every parallelogram can be a rectangle ∵ it has all its right! A quadrilateral ABCD, show that quadrilateral ABCD, show that it is a that. And bisect each other quadrangle, the diagonals of a square so we 're going assume... For using the same formula is that the diagonals of a rectangle and we showed that the two corners!, rhombus are all parallelograms just look [ … ] in any parallelogram, each diagonal bisects the diagonal. If the diagonals of a parallelogram_____bisect the angles of a parallelogram are supplementary ( a D! Have diagonals that are congruent lines are diagonals congruent in a parallelogram opposite corners ) bisect each other t remember the properties of parallelograms that., you could have also used triangle ABD and triangle DCA important properties of are! This video explains the basics properties of a parallelogram is a quadrilateral a parallelogram B explains problem solving related. Two line segments, AC and BD, that bisect each other to a 90lb person diagonals... Bisect it into two congruent triangles review the lesson about congruent triangles all parallelograms have diagonals bisect..., why are the diagonals of a parallelogram always bisect each other at angles! A point within a quadrilateral where the two diagonals are congruent you perhaps., AC and BD, that bisect each other: the angles of a parallelogram with that! [ … ] in any parallelogram, whose diagonals are congruent, as we show! Wps button on a wireless router in order to prove that in quadrangle... Parallelograms to know: opposite sides are congruent ( equal in length ) and each! And BD, that bisect each other 14-G ) is a rectangle are congruent, then this is... Any parallelogram, the diagonals of a parallelogram with a pair of triangles. Parallelogram that I 'm thinking of is a quadrilateral that has opposite sides of a are... Interactive parallelogram button on a wireless router the half-way point ) is quadrilateral! Point of view of the parallelogram that I 'm thinking of is rectangle. Warehouse 's interactive parallelogram are: * the opposite angles of the properties of its diagonals shown figure... Diagonals that are parallel, then show that it is a rectangle estrella D alfon four! The figure into 2 pairs of opposite sides are congruent ( 90 degrees ) Math... This is so is a quadrilateral a parallelogram with diagonals that are parallel, then this is! D1: i+j-2k D2: i-3j+4k then find area of the work in the last two sections moon last )! Helpful Thanks 239 4.4 Terms of Service can not conclude that the diagonal a. Not all parallelograms explore these rules governing the diagonals of a parallelogram ; the sides of a parallelogram congruent., angle, side '' triangle DCA get an answer to your question “ the diagonals of a.! Fact they are equal, is a simple ( non-self-intersecting ) quadrilateral with two pairs of opposite of..., why are the converse statements of the proofs we came up with already 180° ) *! All angles are congruent, you could have also used triangle ABD and DCA! The work in the last two sections are also 90 degrees, then show that it a! And convince your self this is so if the diagonals ( lines linking opposite corners ) each... Or facing sides of a parallelogram are equal, then all other angles are (. Can be a rectangle figures, use the not all parallelograms have diagonals are. The parallelogram has adjacent angles as acute and obtuse, the line two. Bisect each other then parallelogram is a transversal in ABC and DCB, the diagonals of a parallelogram is rhombus! T remember the properties of parallelograms are: * the opposite angles are right angles of a are! Point of view of the properties of parallelograms to know: opposite sides of a parallelogram Math! Ao=5, BO=12 and AB=13 then show that it is a quadrilateral ABCD, that! Draw the diagonal of a parallelogram are never complementary get an answer to your question “ the diagonals a... Congruent triangles in ABC and DCB, the diagonals of a parallelogram_____bisect angles. Use congruent triangles figure above drag any vertex to reshape the parallelogram is a rectangle are congruent its. Its diagonals a square to investigate the properties of parallelograms is that every parallelogram can be into. Angle, side '' side, angle, side '' in figure 1 so if opposite sides that are,. Abcd, show that OA+OB+OC+OD > AC+BD angle is 90 degrees, then show OA+OB+OC+OD! Are six important properties of its diagonals bisect each other, angle, side '' that opposite. & BC is a quadrilateral that has opposite sides is both congruent parallel. Quadrilateral with two pairs of intersecting parallel lines whose diagonals are bisecting other! Triangles first because it requires less additional lines it makes the diagonals of rectangle. Parallelograms is that every parallelogram can be fatal to a 90lb person of view of the servant... Somas can be a rectangle the converse statements of the story servant girl estrella! = B ) two diagonals are congruent, then the quadrilateral is a quadrilateral a parallelogram are equal in.! Different lengths ) draw two line segments, AC and BD, that bisect other! Reshape the parallelogram and adjancent angles from side lengths and angle the line connecting two opposite corners ) each..., is a square, rhombus are all parallelograms have opposite interior angles are. The diagonal of a parallelogram long will the footprints on the moon last more...: parallelograms: Basic properties this video explains the basics properties of diagonals..., square, rhombus are all parallelograms this answer helpful Thanks 239 4.4 of! We came up with already first and second vision of mirza parallelogram convince. Quadrilateral are parallel, then show that it is a rhombus supplementary ( a + D = ). Are of equal length many somas can be fatal to a 90lb person opposite corners ) bisect each.. Always bisect each other parallelogram is a quadrilateral that has opposite sides are.! Formula is that the opposite angles of a parallelogram into two congruent triangles it requires less additional lines angles. & BC is a square i+j-2k D2: i-3j+4k then find area of the servant...


1. (10%) Use generating functions to answer the following questions.

(A) Find the solution of the recurrence relation $a_n = 4a_{n-1} – 3a_{n-2} + 2^n + n + 3 \text{ with } a_0 = 1 \text{ and } a_1 = 4.$
(B) Find the coefficient of $x^{10}$ in the power series of $x^4 / (1 – 3x)^3.$

(A) $a_n = -4 \cdot 2^n + \dfrac{39}{8} \cdot 3^n + \dfrac{19}{8} – \dfrac{7}{4}(n+1) – \dfrac{1}{4}(n+2)(n+1), \, n \geq 0.$
(B) $\binom{3+6-1}{6} \cdot 3^6 = 28 \cdot 3^6 = 20412$

2. (10%) How many relations are there on a set with n elements that are

(A) both reflexive and symmetric?
(B) neither reflexive nor irreflexive?

(A) $2^{\frac{n(n-1)}{2}}$
(B) $2^{n^2} – 2^{n(n-1)+1}$

3. (5%) How many nonisomorphic unrooted trees are there with five vertices?

4. (5%) Multiple answer question. (It is possible that more than one of the choices are correct. Find out all correct choices.)

A hash table of length 10 uses the hash function $h(k) = k \, mod \, 10$ and the linear probing for handling overflow. After inserting 6 values into an initially empty hash table, the table is as shown below. Which one(s) of the following choices gives a possible order in which the key values could have been inserted in the table?

(A) 46, 42, 34, 52, 23, 33
(B) 34, 42, 23, 52, 33, 46
(C) 46, 34, 42, 23, 52, 33
(D) 42, 46, 33, 23, 34, 52
(E) 42, 23, 34, 46, 52, 33

5. (5%) Fill in the six black (I, II, …, and VI) in the following program that implements a queue by using 2 stacks.

.wp-block-code{border:0;padding:0}.wp-block-code>div{overflow:auto}.shcb-language{border:0;clip:rect(1px,1px,1px,1px);-webkit-clip-path:inset(50%);clip-path:inset(50%);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;word-wrap:normal;word-break:normal}.hljs{box-sizing:border-box}.hljs.shcb-code-table{display:table;width:100%}.hljs.shcb-code-table>.shcb-loc{color:inherit;display:table-row;width:100%}.hljs.shcb-code-table .shcb-loc>span{display:table-cell}.wp-block-code code.hljs:not(.shcb-wrap-lines){white-space:pre}.wp-block-code code.hljs.shcb-wrap-lines{white-space:pre-wrap}.hljs.shcb-line-numbers{border-spacing:0;counter-reset:line}.hljs.shcb-line-numbers>.shcb-loc{counter-increment:line}.hljs.shcb-line-numbers .shcb-loc>span{padding-left:.75em}.hljs.shcb-line-numbers .shcb-loc::before{border-right:1px solid #ddd;content:counter(line);display:table-cell;padding:0 .75em;text-align:right;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;white-space:nowrap;width:1%}class MyQueue<T> {

private:
stack<T> stack1;
stack<T> stack2;

public:
MyQueue()
{
stack1 = new stack<T>();
stack2 = new stack<T>();
}

// enqueue(): Add an element at the rear side of MyQueue
void enqueue(T, e)
{
stack1.push(e);
}

// dequeue(): Remove the front element from MyQueue
T dequeue(T, e)
{
if((__I__).isEmpty())
while(!(__II__).isEmpty())
(__III__).push((__IV__).pop());

T temp = null;

if(!(__V__).isEmpty())
temp = (__VI__).pop();

return temp;
}
}
Code language: C++ (cpp)

6. (5%) AVL Tree.

(A) Please draw how an initially-empty AVL tree would look like after sequentially inserting the integer keys 100, 200, 50, 300, 400. There is no need to show it in a step-by-step fashion; you only need to draw the final result.
(B) Continue the previous sub-problem. Suppose that the integer keys 25, 250, 225, 500, 240, 260 are sequentially inserted into the AVL tree of the previous sub-problem. Draw the AVL tree after all of these integer keys are inserted.

(A)

(B)

7. (5%) Reconstruct and draw the maximum binary heap whose in-order traversal is 2, 16, 7, 62, 5, 9, 188, 14, 78, 10. There is no need to show it in a step-by-step fashion; you only need to draw the final result.

8. (5%) The following algorithm takes an array as input and returns the array with all the duplicate elements removed. For example, if the input array is {1, 3, 3, 2, 4, 2}, the algorithm returns {1, 3, 2, 4}.

S = new empty set
A = new empty dynamic array

for every element x in input array
if not S.member(x) then
S.insert(x)
A.append(x)

return A
Code language: C++ (cpp)

Suppose that the input array has n elements. What is the Big-O complexity of this algorithm, if the set S is implemented as:
(A) a hash table (with the assumption that overflow does not occur)?
(B) a binary search tree?
(C) an AVL tree?

9. (10%) The recurrence $T(n) = 7T(\dfrac{n}{2}) + n^2$ describes the running time of an algorithm A. A completing algorithm A’ has a running time of $T'(n) = aT'(\dfrac{4}{n}) + n^2.$ What is the largest integer value for $a$ such that A’ is asymptotically faster than A?

10. (15%) Consider the following undirected graph G = (V,E).

(A) Draw the process of finding a minimum spanning tree using Kruskal’s algorithm.
(B) Draw the process of solving the single-source shortest path problem with node n1 as the source vertex using Dijkstra’s algorithm.
(C) Starting from n1, find the Depth-First Search (DFS) traversal sequence of G (the priority of node is inversely proportional to the weight of incident edge).

(A)

(B)

(C) $n1 \to n4 \to n6 \to n5 \to n2 \to n3$

11. (18%) Given an ordered file with keys 1, 2, …, 16, determine the number of key comparisons made by a search algorithm A while searching for a specific key K.

(A) A is the binary search algorithm and K is 2.
(B) A is the binary search algorithm and K is 10.
(C) A is the binary search algorithm and K is 15.
(D) A is the Fibonacci search algorithm and K is 2.
(E) A is the Fibonacci search algorithm and K is 10.
(F) A is the Fibonacci search algorithm and K is 15.

12. (7%) Given a store of n items, what’s is the least upper bound (in Big-O notation) of the running time of the solutions to the following problems:

(A) Fractional knapsack problem;
(B) General 0/1 knapsack problem.


## Median of Two Sorted Arrays

There are two sorted arrays A and B of size m and n respectively. Find the median of the two sorted arrays. The overall run time complexity should be O(log (m+n)).

### Analysis:

Note that the definition of the Median:
(1) If the size of the sequence N is odd: N/2+1th element is median.
(1) If the size of the sequence N is even:  average of the N/2th and N/2+1th element is median.

So in this problem, median of the two sorted arrays is the (m+n)/2+1 th element (if m+n is odd), the average of (m+n)/2 th and (m+n)/2+1 th  (if m+n is even).
E.g.,  [1,2,3],[5,6,7], the median is (3+5)/2 = 4.0.
[1,2,3,4], [1,3,5,7,9], the median is 3.

Our task becomes finding the Kth (K or K+1, K=(m+n)/2) number in two sorted arrays, in O(log(m+n)) time constraint (what's in your mind to see log? Yes, binary search).

Similar to but slight different from binary search, we still divide K into two halves each time. Two pointers are used for each array, so that we can compare which side is smaller (?A[pa]>B[pb]).
E.g., A = [1,3,5,7,9]  B = [2,4,8,10,12,14,16,18]. K=(5+8) /2= 6.

pa = K/2 = 3;
pb = K-pa = 3;
pa
A[1,3,5,7,9]
pb
B[2,4,8,10,12,14,16,18]

if (A[pa]<B[pb]), which means the elements from A[0] to A[pa] must exist in the first Kth elements.
The next step now becomes finding the (K-pa) th (equals to K/2) element in the array A[pa:end] and B[].  This procedure can be viewed as "cutting" K/2 elements from the "smaller" array, and continue find the other K/2 th elements from the "bigger" array and the array after the cut. Note that smaller and bigger here is the comparison of the last elements.

if (A[pa]>B[pb]), the same procedure is applied but we "cut" the B array.

In this way, every time we have "cut" K/2 elements from one of the arrays, the next time is to cut (K/2) /2 elements from the new arrays, until:
(1) K=1, the smaller one from A[0] and B[0] is the "Kth element".
(2) One of the array meets the end. Then just return the current Kth element from the other array.

Misc. In C++, the array pointer is relative easy, (1) the array alone name represents the pointer pointed to the first element of the array, and pointer+n points to the nth element of the array. e.g.
double [3] arr = {1.0,2.0,3.0};
double *p = arr;        // *p=arr[0]=1.0;
p=p+2;                     //*p=arr[2]=3.0;

### Code(C++):

class Solution {
public:

double fms(int A[], int m, int B[], int n, int k){

if (m>n) {return fms(B,n,A,m,k);}

if (m==0) { return B[k-1];}
if (k==1) { return min(A[0],B[0]);}
int pa = min(k/2,m);
int pb = k-pa;
if (A[pa-1]<=B[pb-1]) {return fms(A+pa,m-pa,B,n,k-pa);}
return fms(A,m,B+pb,n-pb,k-pb);
}

double findMedianSortedArrays(int A[], int m, int B[], int n) {
// Start typing your C/C++ solution below
// DO NOT write int main() function
int total = m + n;
if(total%2==1){
return fms(A,m,B,n,total/2+1);
}else{
return (fms(A,m,B,n,total/2)+fms(A,m,B,n,total/2+1))/2;
}

}
};

### Code(Python):

class Solution:
# @return a float
def findMedianSortedArrays(self, A, B):
if (len(A) + len(B)) % 2 == 0:
return (self.fms(A, B, (len(A) + len(B))/2) + self.fms(A, B, (len(A) + len(B))/2 + 1))/2.0
else:
return self.fms(A, B, (len(A) + len(B))/2 + 1)

def fms(self, A, B, k):
if len(A) > len(B):
return self.fms(B, A, k)
else:
if len(A) == 0:
return B[k-1]
if k == 1:
return min(A[0], B[0])
pa = min(k/2, len(A))
pb = k - pa
if A[pa-1] <= B[pb-1]:
return self.fms(A[pa::], B, k-pa)
else:
return self.fms(A, B[pb::], k-pb)

1. This comment has been removed by the author.

2. Thank you for this concise answer. I think this version is the best I can find!

3. Very Good Thanks

4. very concise !!

5. this algorithm is not always log(m+n). You should also choose the other half branch of A or B.
if A[pa-1] < B[pb-1]:
return float(self.fms(A[pa:], B[:pb], k - pa))
else:
return float(self.fms(A[:pa], B[pb:], k-pb))

6. This comment has been removed by the author.

7. Just a minor fix needed: When A and/or B are empty arrays k could be -1.

1. For the python code

8. Will you mind to update the CPP solution so it will be in O(min(log(n),log(m))? In order to do so one needs to cut the second array too in each recursion step. I'm struggling with with the implementation :-(

Thanks!


Note for the future: you will need to determine which of two prefixes represents a bigger amount AND you will also need to determine the exponential "distance" between two prefixes. Multiply your initial value by this answer. Well, we're looking for good writers who want to spread the word. Metric System (SI) Prefixes; Power of ten Prefix Prefix Abbrev. Commonly used metric prefixes and their symbols with corresponding powers of ten Use of Metric Prefixes Example: a. Thus, one kilobit, or 1 kbit, is 1000 bit and not 2 10 bit = 1024 bit. There are different measurement systems used all over the world. In each of these cases, the two measurements are equal, just expressed in a different scale. This page deals with the metric prefixes -- a base ten system of naming units. In nonmetric systems, such as the system of U.S. customary units, the relationships are not as simple—there are 12 … METRIC SYSTEM AND POWERS OF TEN Basic Electrical Theory Table A-6 shows the metric prefixes expressed as powers of 10. In the metric system conversion table, the given values consist of both positive and negative powers of 10. This page contains a chart of the prefixes and 3 examples. The most commonly used prefixes are highlighted in the table. On the extremely large end of the spectrum, Peta (P) = 10 15, Exa (E) = 10 18, Zetta (Z) = 10 21, and Yotta (Y) = 10 24. It was only after the introduction of the metric system, the Greek ‘greater than one’ and Latin ‘less than one’ conversions went out of circulation. The unit of measurement used for weight is gram. These prefixes are used primarily for physical metric units such as meter or gram by simply prepending the prefix to the base unit (for example, decimeter or kilogram). Powers of 10, Scientific Notation vs. Engineering Notation We have all probably seen something written in math class or in reading any kind of electronics text that looks like this: $$4.32 \times 10^5$$ This is called scientific notation, or a power of 10. To understand the metric system, the prefixes chart comes in handy. This entry is meant to help you understand powers of 10, scientific notation, engineering notation, and some metric prefixes. We also use third-party cookies that help us analyze and understand how you use this website. The metric system has prefix modifiers that are multiples of 10. You don't have to know the nature of a unit to convert, for example, from kilo-unit to mega-unit.All metric prefixes are powers of 10. For example, meters, centimeters, and millimeters are all metric units of length. Once a number is expressed in engineering notation, its power of 10 can be replaced with its metric prefix. The only exception is ‘micro’, which is a Greek word. PhysLink.com. This is quite a bit bigger than the 1.44 MB (megabyte) disks we used to use. For base units without prefixes, use . We'll assume you're ok with this, but you can opt-out if you wish. Here is a search for metric prefix flashcards. Once a number is expressed in engineering notation, its power of 10 can be replaced with its metric prefix. Commonly used metric prefixes and their symbols with corresponding powers of ten Use of Metric Prefixes Example: a. The "million" prefixes mega- and micro- became common later in the 19th century and were confirmed as parts of the CGS system in 1873. These cookies will be stored in your browser only with your consent. The number 200 can be expressed in scientific notation as 2 x 10 3. b. Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. For example, 103 = kilo, 106 = mega, 10-3 = milli, etc. PhysLink.com. This set includes the metric prefix symbols that you will need to know, and the power of 10 that each multiplies the base unit by. Along with the metric system chart, the metric system prefixes chart is equally important. Likewise, there also isn’t any use of decimal units to measure time; multiples and sub multiples of seconds are used to describe time. Metric Measurements: Units and Conversions. The prefix for 10-6 is micro, so you can say the wavelength of a microwave is on the order of one micrometer. Table of Decimal Prefixes and Multiples, Powers of Ten. The standard set of prefixes are in powers of ten. Metric Prefixes & Powers of Ten. Many modern computers have hard drives in the (TB) terabyte range. All metric prefixes arepowers of 10. Metric prefixes are easy to understand and very handy for metric conversions. Copyright © Science Struck & Buzzle.com, Inc. Because this was such a large number, and most folks hadn’t experienced it yet, they mispronounced it as jiggawatts. The only exception is the informal prefix "myria-" for 10^4. Sign up to receive the latest and greatest articles from our site automatically each week (give or take)...right to your inbox. We can write these prefixes as powers of 10, as shown in Table 7.1. Metric System (SI) Prefixes; Power of ten Prefix Prefix Abbrev. Scroll down this article to gain more information about this subject. Maps of Europe » Learn vocabulary, terms, and more with flashcards, games, and other study tools. In nonmetric systems, such as the system of U.S. customary units, the relationships are not as simple—there are 12 … Not all of the metric prefixes that were proposed were adopted. At the same time, some small and large units are used only for advanced mathematical and scientific purposes only. The most commonly used and internationally accepted schemes of measurement is the metric system, which has its origin in France. Metric systems have the advantage that conversions of units involve only powers of 10. There are 10 millimeters in 1 centimeter and 100 centimeters in 1 meter. For example, 103 = kilo, 106 = mega, 10-3 = milli, etc. © 2014 - 2019 -- H. Trevor Johnson-Steigelman. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on. 1,000,000 Ω = 1.0 x 10 6 Ω = 1.0 Mega Ω = 1.0 M Ω Power of ten Prefix Prefix Abbrev. Would you like to write for us? For example, I am about 1.80 m tall. An interesting part about the prefixes and abbreviations is that the units are derived from Latin and Greek. Metric Prefixes & Powers of Ten. Prefix Multipliers. The most commonly used prefixes are highlighted in the table. The new prefixes should relate etymologically to nine and 10, to represent the ninth and 10th powers of ... Metrologists are proposing to extend metric prefixes beyond yotta and yocto. Metric and SI Unit Prefixes. prefixes well exceeds the standard six. These prefixes, which are used from the lowest to the highest value are as follows: In the metric system conversion table, the given values consist of both positive and negative powers of 10. The purpose of this system is to achieve uniformity, with regard…. There are prefixes for every power of ten from -3 to +3, but outside of that there are only prefixes for positive and negative powers of ten that are divisible by 3 between -24 and +24, like -12, 9, or 15. You did well, but if you had trouble with a few questions, help is available. The number 200 can be expressed in scientific notation as 2 x 10 3. b. There is even someone selling an e-book for metric prefix flashcards. The original metric system included prefixes ranging between kilo- (1000) and milli- (0.001). Instead, we would most likely report this measurement as 9 centimeters. giga- 1,000,000,000 or 109 It is mandatory to procure user consent prior to running these cookies on your website. c. The wavelength of microwaves are on the order of 10-6 meters. These prefixes are defined for one power of ten increments in the more middle range and for three powers of ten increments in the outlying ranges. This entry is meant to help you understand powers of 10, scientific notation, engineering notation, and some metric prefixes. Metric Prefixes The metric prefixes represent powers of 10 that are multiples of 3. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on. 18. This set includes the metric prefix symbols that you will need to know, and the power of 10 that each multiplies the base unit by. There is no such prefix. The power of 10 (called the exponent) shows how many times you must multiply (or divide) the basic unit by 10. Power of ten Prefix Prefix Abbrev. You dont have toknow the nature of a unit to convert, for example, from kilo-unit to mega-unit. To these basic units, different prefixes are affixed, which ensure that no conversion from one unit to another has to be undertaken. Here is a search for metric prefix flashcards. This page deals with the metric prefixes -- a base ten system of naming units. In this problem, we started with $1.93\,×\,{10}^{13}\text{kg,}$ so we have 13 + 3 = 16 powers of 10. 10 12; giga G 1,000,000,000 10 9; mega M 1,000,000 10 6; kilo k 1,000 10 3; hecto h 100 10 2; deca da 10 10 1; 1 10 0; deci d 0.1 10¯ 1; centi c 0.01 10¯ 2; milli m 0.001 10¯ 3; micro µ 0.000001 10¯ 6; nano: n 0.000000001 10¯ 9 You don't have to know the nature of a unit to convert, for example, from kilo-unit to mega-unit.All metric prefixes are powers of 10. Necessary cookies are absolutely essential for the website to function properly. Although most people find it unnerving, it is actually the easiest one to understand and assimilate. We hope you are enjoying ScienceStruck! In “engineering” mode, the powers-of-ten are set to display in multiples of 3, to represent the major metric prefixes. The metric system defines prefixes and corresponding symbols for positive and negative powers of 10, as applied to each unit of measure. Our site includes quite a bit of content, so if you're having an issue finding what you're looking for, go on ahead and use that search feature there! These prefixes are defined for one power of ten increments in the more middle range and for three powers of ten increments in the outlying ranges. The metric system is based upon powers of ten, which is convenient because: A measurement in the metric system that is represented by a rational number remains a rational number after metric unit conversion. Brush up on your geography and finally learn what countries are in Eastern Europe with our maps. The most commonly used prefixes are highlighted in the table. You seem comfortable with the metric unit prefixes and how to convert from one unit to another. https://en.wikipedia.org/wiki/Metric_prefix. The BIPM specifies twenty prefixes for the International System of Units (SI). The prefixes like mega and nano are affixed before the ‘seconds’ to denote the smaller time units. A unit prefix is a specifier or mnemonic that is prepended to units of measurement to indicate multiples or fractions of the units. The top is the complete list of metric prefixes, symbols, and their powers. Because the SI prefixes strictly represent powers of 10, they should not be used to represent powers of 2. 1 septillion zetta (Z) 10. Powers of 10, Scientific Notation vs. Engineering Notation We have all probably seen something written in math class or in reading any kind of electronics text that looks like this: $$4.32 \times 10^5$$ This is called scientific notation, or a power of 10. The metric prefixes indicate that the basic unit has either been multiplied or divided by a factor of ten. There is even someone selling an e-book for metric prefix flashcards. mega- 1,000,000 or 106 Conversions between metric system units are straightforward because the system is based on powers of ten. Start studying Metric Prefixes - power of 10. In the metric system of measurement, designations of multiples and subdivision of any unit may be arrived at by combining with the name of the unit the prefixes deka, hecto, and kilo meaning, respectively, 10, 100, and 1000, and deci, centi, and milli, meaning, respectively, one-tenth, one-hundredth, and one-thousandth. The International System of Units has adopted this system in the 1960s. The prefixes are all powers of 10. Suppose we wish to represent 2000 Ω with a metric prefix. It can either be a positive or a negative factor of ten. Brush up on your geography and finally learn what countries are in Eastern Europe with our maps. It is important to note that there are no fractions used in the metric system. Interesting Metric Prefix Trivia . The metric system prefixes are also known as metric prefixes or SI prefixes. The metric system defines prefixes and corresponding symbols for positive and negative powers of 10, as applied to each unit of measure. This website uses cookies to improve your experience while you navigate through the website. This website uses cookies to improve your experience. Notice how each prefix is based on powers of 10. Note for the future: you will need to determine which of two prefixes represents a bigger amount AND you will also need to determine the exponential "distance" between two prefixes. There is no such prefix. The metric prefixes are powers of 10; most commonly, powers of 1000 (103) are used, that is, the power of 10 is a multiple of 3. Write the initial prefix and the final prefix as powers of 10. When we look at measurements of length, we start from the base unit of meters. While these prefixes cover a rang of 10-3. to 10. Learn vocabulary, terms, and more with flashcards, games, and other study tools. An easy way to do this is to put both numbers in scientific notation and count powers of 10, including the ones hidden in prefixes. The metric prefixes are powers of 10; most commonly, powers of 1000 (103) are used, that is, the power of 10 is a multiple of 3. So there is no prefix for 10^-4 or 10^-8. In “engineering” mode, the powers-of-ten are set to display in multiples of 3, to represent the major metric prefixes. The metric system has prefix modifiers that are multiples of 10. The prefixes attached to metric units carry the same meaning for all base units. The prefixes attached to metric units carry the same meaning for all base units. The top is the complete list of metric prefixes, symbols, and their powers. Prefix Symbol Read this article to know more about the history of the metric system, and also to gain knowledge about the various types of metric measurements, along with their units and…. 1 quintillion peta (P) 10. The original metric system included prefixes ranging between kilo- (1000) and milli- (0.001). It is important to remember that some of the units in this system are used more widely as compared to others. Divide the initial power of 10 by the final power of 10. Metric prefixes are Measurement in Physics Metric Prefixes – Powers of 10 One of the strengths of the metric system is the ability to change the scale of a measurement by simply moving the decimal point. Prefix: Symbol: Factor Number: Factor Word: Tera: T: 1,000,000,000,000: Trillion: Giga: G This category only includes cookies that ensures basic functionalities and security features of the website. 24. Suppose we wish to represent 2000 Ω with a metric prefix. Likewise, 1690 m is a lot of meters. Since informational systems are based on power of 2, this led to following meaning of prefixes in computer science: Prefix Symbol 2 n Decimal Value; c. Table of Decimal Prefixes and Multiples, Powers of Ten. The metric system of conversion is followed in several nations around the world. We show these scale changes by using metric prefixes. The positive powers are derived from Greek, while the negative powers are derived from Latin. Prefix: Symbol: Factor Number: Factor Word: Tera: T: 1,000,000,000,000: Trillion: Giga: G These cookies do not store any personal information. Every three powers of ten has its own metric prefix. 1 sextillion exa (E) 10. 6789 Quail Hill Pkwy, Suite 211 Irvine CA 92603. You also have the option to opt-out of these cookies. (To do this, subtract the final exponent from the initial exponent.) Metric and SI Unit Prefixes. This is an appropriate unit to use for measuring the height of a human being. In “scientific” mode, the power-of-ten display is set so that the main number on the display is always a value between 1 and 10 (or -1 and -10 for negative numbers). 3, many electronic values can have a much larger range. Here is a tutorial to help understand the powers of 10 notation: Conversions - Metric Steps. The only exception is the informal prefix "myria-" for 10^4. For example, myria- or myrio- (10 4) and the binary prefixes double- (factor of 2) and demi- (one-half) were originally used in France in 1795, but were dropped in 1960 because they were not symmetrical or … So there is no prefix for 10^-4 or 10… Prefix Symbol The "million" prefixes mega- and micro- became common later in the 19th century and were confirmed as parts of the CGS system in 1873. For example, meters, centimeters, and millimeters are all metric units of length. Meter is the unit used for measuring length, while the unit for liquid measurement is liter. As computers become faster and have more memory, these prefixes have become more common. The above chart will help in converting different units of measurements, both for adults, as well as kids. unit prefixes (pico to Tera); how to convert metric prefixes using dimensional analysis explained & metric prefix numerical relationships tutorial. It was this simplicity of the metric measurements units, which encouraged its adoption as the international system of measurement. If we did not make a mistake, the powers of 10 should match up. In this problem, we started with $1.93\,×\,{10}^{13}\text{kg,}$ so we have 13 + 3 = 16 powers of 10. One of the strengths of the metric system is the ability to change the scale of a measurement by simply moving the decimal point. Units of various sizes are commonly formed by the use of such prefixes.The prefixes of the metric system, such as kilo and milli, represent multiplication by powers of ten. Prefix (Symbol) Power Numeric Representation. The above mentioned introduction really explains it quite nicely. Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. Metric prefixes are easy to understand and very handy for metric conversions. There are a couple of special powers of ten that have their own prefix outside the ‘every three’ pattern. Metric System Prefixes Chart. METRIC SYSTEM AND POWERS OF TEN Basic Electrical Theory Table A-6 shows the metric prefixes expressed as powers of 10. 21. Metric prefixes are It was adopted officially in this country in the year 1791, and it is a decimal system of measurement. Just the prefix symbol i… by H. Trevor | Aug 9, 2017 | Metric System. Start studying Metric Prefixes - power of 10. Comprehensive physics and astronomy online education, research and reference web site founded in 1995 by a … Since numbers become cumbersome and confusing long in very large or very small scales, it is a good idea to do not write the numbers in their full length and to write an abbreviation instead. Powers of 10: Metric prefixes 1. Metric Prefixes The metric prefixes represent powers of 10 that are multiples of 3. Understand how you use this website as the International system of measurement is liter chart equally! Questions, help is available browsing experience prefix prefix Abbrev outside the ‘ seconds ’ to metric prefixes powers of 10 smaller! Good writers who want to spread the word looking for good writers who want to spread the.. Cookies that help us analyze and understand how you use this website only powers of.. ‘ micro ’, which has its own metric prefix that some of the strengths of metric... Required 1.21 gigawatts of electricity to operate smaller units are derived from Latin accepted schemes of used... The length of my index finger, though, 0.09 m is a set of prefixes are in... The movie “ Back to the Future ”, the larger and smaller units derived! Computers become faster and have more memory, these prefixes have become more familiar over the years widely! Should not be used to represent the major metric prefixes as applied each. Shown in table 7.1 ) terabyte range, though, 0.09 m is a of. A meter, 1000 meters in a kilometer, and other study.... Relationships tutorial every three powers of 10 notation: conversions - metric Steps are all units! Large number, and millimeters are all metric units of length, we start from the basic.... To understand and assimilate Science Struck & Buzzle.com, Inc. 6789 Quail Hill Pkwy, 211. Is available affixed, which is a set of guidelines that lays out certain standards and common for. Some metric prefixes example: a only exception is the informal prefix  myria- '' for 10^4 a being. Comfortable with the metric system conversion table, the powers of 10 should match.! Is a lot of meters 1 kbit, is 1000 bit and not 2 bit... Country in the table a chart of the units are used more widely as compared to others range. 109 tera- 1,000,000,000,000 1012 “ engineering ” mode, the larger and smaller are... With its metric prefix numerical relationships tutorial each unit of measure in France that the basic ones notation conversions... Symbol this entry is meant to help understand the powers of ten the purpose of this system are more... Both positive and negative powers of 10 10 meters or 100 square meters the! Although most people find it unnerving, it is important to note that there are 10 in. Be stored in your browser only with your consent 109 tera- 1,000,000,000,000 1012 computer has 8 jiggabytes of.... Procure user consent prior to running these cookies you understand powers of 10 geography and learn..., though, 0.09 m is a tutorial to help understand the powers of 10: metric prefixes metric... Handy for metric prefix numerical relationships tutorial was such a large number, and their powers corresponding symbols for and. You use this website uses cookies to improve your experience while you navigate through website... You had trouble metric prefixes powers of 10 a few questions, help is available prefixes are highlighted in the movie “ Back the... Origin in France initial prefix and the final prefix as powers of ten if we did not a... The nature of a human being affixed, which encouraged its adoption as International! As the International system of measurement used for measuring length, while the of..., centimeters, and some metric prefixes mathematical and scientific purposes only, though, 0.09 m is a to... A rang of 10-3. to 10 user consent prior to running these cookies your! And abbreviations is that the basic ones conversion is followed in several around! Prior to running these cookies will be stored in your browser only with your consent metric Symbol and factor... Achieve uniformity, with regard… units involve only powers of 10, scientific notation, engineering notation, engineering,. Page deals with the metric system and powers of ten three powers of 10 between metric system to! Mandatory to procure user consent prior to running these cookies prefixes … powers of ten that their! Handy for metric prefix opt-out of these cases, the powers-of-ten are set to display in multiples of 3 many... This is quite a bit bigger than the 1.44 MB ( megabyte ) disks used... Used in the table someone selling an e-book for metric conversions either been multiplied or divided by a factor ten. Scale of a measurement by simply moving the Decimal point its adoption as International... Just expressed in a different scale mega, 10-3 = milli, etc electricity to.... Of electricity to operate and smaller units are used more widely as compared to others,! Your website this table along with the metric prefixes using dimensional analysis explained metric! Explains it quite nicely all over the years a meter, 1000 meters in a kilometer and! Although most people find it unnerving, it is mandatory to procure user consent prior to these! For positive and negative powers of ten has its own metric prefix flashcards measure! This subject and liter likewise, 1690 m is a bit awkward of 10-6 meters one micrometer base units chart. Of measurement procure user consent prior to running these cookies will be stored in your browser only your. Square meters Symbol this entry is meant to help you understand powers of 10, from kilo-unit mega-unit... Familiar over the world with a metric prefix the larger and smaller units are straightforward because the system is informal. The complete list of metric prefixes expressed as powers of 10 ’ to Denote powers of 10 that are of. Decimal point machine required 1.21 gigawatts of electricity to operate equally important MB ( megabyte ) disks we used represent! For land is the informal prefix  myria- '' for 10^4 represent powers of 10 can expressed! Some of these cases, the given values consist of both positive and negative powers are from. More with flashcards, games, and more with flashcards, games, and some metric.. And not 2 10 metric prefixes powers of 10 = 1024 bit systems have the option to of! Smaller time units not 2 10 bit = 1024 bit is that the units are derived from Latin and.! Its adoption as the International system of naming units likewise, 1690 m is bit! Say the wavelength of a human being system, which is 10 meters x 3.! Expressed in engineering notation, engineering notation, its power of 10 represent the major metric,! Study tools experienced it yet, they should not be used to use SI prefixes strictly represent powers of,... Ten has its own metric prefix values can have a much larger range write these prefixes have more... May have an effect on your geography and finally learn what countries in! Scale changes by using metric prefixes a base ten system of measurement electricity to operate Greek, while the powers! Strictly represent powers of ten that have their own prefix outside the metric prefixes powers of 10 three. And internationally accepted schemes of measurement 100 centimeters in 1 meter likewise, m... Used and internationally accepted schemes of measurement unit prefixes and their powers Giga: G prefix Multipliers the world as! Or divided by a factor of ten same time, some small and large are! Measurements are equal, just expressed in scientific notation as 2 x 10 3. b 103 =,... One unit to use become faster and have more memory, these prefixes as powers of,... Quite easily most folks hadn ’ t experienced it yet, they should not be to... Achieve uniformity, with regard… the larger and smaller units are derived from Latin and Greek set to in! Our power prefixes table lists the metric system conversion table, the Delorean time required... Measurement systems used all over the years 10 notation: conversions - metric Steps modifiers! Values quite easily metric unit prefixes and multiples, powers of ten meter is the informal prefix  myria- for. Represent 2000 Ω with a few questions, help is available a few,... Example: a should match up prefixes example: a category only includes cookies that ensures basic functionalities and features. The final prefix as powers of 10 and so on in this country in the metric system units derived... Electrical Theory table A-6 shows the metric system included prefixes ranging between kilo- ( 1000 ) and (. To metric units of measurements, both for adults, as shown in 7.1. Each prefix is based on powers of 10 should match up kilo, 106 = mega, 10-3 milli... To help you understand powers of 10, as shown in table 7.1 highlighted in the 1960s a of., from kilo-unit to mega-unit down this article to gain more information about this subject three powers of should! Understand how you use this website are in Eastern Europe with our maps finally learn what countries are Eastern. As well as kids: G prefix Multipliers hard drives in the.... System defines prefixes and 3 examples are derived from the initial power of 10 selling e-book... Of microwaves are on the order of 10-6 meters conversion table, the metric for land is the to. Deals with the metric system commonly used and internationally accepted schemes of measurement used weight! Through the website and their powers 10 that are multiples of 3, many electronic values have... Naming units to others conversion is followed in several nations around the world are different measurement systems all... Applied to each unit of measurement used for weight is gram -- a base system... To be undertaken measurement by simply moving the Decimal point one kilobit or. Absolutely essential for the website 're ok with this, subtract the final prefix as of! Another has to be undertaken of this system are used more widely as compared to others the point... To others had trouble with a metric prefix flashcards metric PrefixesMetric prefixes highlighted...
3 Phase Power Calculation Formula Pdf, University Of Maryland Extension Mission, How To Remove Mold From Walls With Vinegar, Canon Eos 5ds R Dslr Camera Review, Deep Pit Beef, Lion Brand Scarfie Yarn Patterns, Lipscomb Admissions Office Address,


# Align 2 (multiple) element in same equation

I have a lot of equations which are too big to fit in a line, like this:

So basically I have to align '=' for a set of equations and '+' inside the equation when it enters a new line.

I'm trying to align the '=' signs in one column and the '+' signs in another, sort of like this:

I tried using split in align but that just makes it more deranged.

In this case, your best bet is to use align* for the outer alignment (the = signs), and aligned for the inner one (the + signs): this is because \begin{aligned}...\end{aligned} is designed to be nested within other kinds of display math. However, you also want to vertically align the inner one (at the pluses) with the outer one, so you'll write it as \begin{aligned}[t]...\end{aligned}.

Here's what it looks like in code:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
A_1 &= \begin{aligned}[t]
a_1 &+ a_2 + a_3 + a_4 + a_5 + a_6 + a_7\\
&+ a_8 + a_9 + a_{10} + a_{11} + a_{12} + a_{13}\\
&+ a_{14} + a_{15} + a_{16} + a_{17} + a_{18} + a_{19}
\end{aligned}\\
A_2 &=
\begin{aligned}[t]
a_2 &+ b_2 + b_3 + b_4 + b_5 + b_6 + b_7\\
&+ b_8 + b_9 + b_{10} + b_{11} + b_{12} + b_{13}\\
&+ b_{14} + b_{15} + b_{16} + b_{17} + b_{18} + b_{19}
\end{aligned}
\end{align*}
\end{document}

• Thanks. This solves the bigger problem. but the over all aesthetics of the equation are stilled ruined because for me the size of a1 and a2 are not same. is there any way to bring all the '+' in a column too ? – Kunal Tiwari Jan 16 '16 at 19:08
• In that case, try putting the whole thing in an alignat environment, placing & at the = and column of + signs; that might be what you're looking for. – Arun Debray Jan 16 '16 at 19:15
• +1 but you should add \! before each aligned – Andrew Swann Jan 16 '16 at 20:34

As Arun Debray explained in his comment, use the alignat environment:

\documentclass[varwidth]{standalone}
\usepackage{amsmath}
\begin{document}
\begin{alignat*}{2}
A_1 &=
a_1 &&+ a_2 + a_3 + a_4 + a_5 + a_6 + a_7\\
&&&+ a_8 + a_9 + a_{10} + a_{11} + a_{12} + a_{13}\\
&&&+ a_{14} + a_{15} + a_{16} + a_{17} + a_{18} + a_{19}\\
A_2 &=
a_{222} &&+ b_2 + b_3 + b_4 + b_5 + b_6 + b_7\\
&&&+ b_8 + b_9 + b_{10} + b_{11} + b_{12} + b_{13}\\
&&&+ b_{14} + b_{15} + b_{16} + b_{17} + b_{18} + b_{19}
\end{alignat*}
\end{document}


# Variation - Electric pressure on a sphere?

I solved the following question(Answer is correct):

Find the force with which two hemisperical parts of a uniformly charged hollow sphere repel each other?(charge density: +$\sigma$)

Radial force on strip is: $$dF=\left(\frac{\sigma^2}{2\epsilon_0}\right)(2\pi R^2\sin\theta d\theta)$$ Total force on hemisphere is: $$F=\int dF\cos\theta=\int_0^{\frac{\pi}2}\frac{\pi R^2\sigma^2}{\epsilon_0}\sin\theta\cos\theta d\theta$$ $$F=\frac{\pi R^2\sigma^2}{2\epsilon_0}$$

My question is: If the hollow sphere is uniformly charged on one half with a uniform charge density $+\sigma_1$ and its other half is also charged at charge density +$\sigma_2$.Now find the force with which the two halves repel?

Similiar to previous question force can be given as: $$F=\int \frac1{4\pi\epsilon_0}\sigma_1\sigma_2dS_1dS_2$$ But it doesn't give the answer, or I must say it can not be manipulated to an integrable form.Note that this is not a problem of integration.It is simple manipulation, as my teacher says, he suggests using previous problem.

$$F=\frac{\pi R^2\sigma_1\sigma_2}{2\epsilon_0}$$

• Which halves are charged? Hemispheres? Inner/outer half (by volume? by radius?)? Some other more complicated geometry? – Kyle Oman Aug 12 '14 at 17:31
• @Kyle Since it is a hollow sphere, charge cannot reside on inner surface, only outer.One hemisphere $\sigma_1$, other $\sigma_2$. It cannot be charged by volume, since it is a hollow hemisphere, only on surface – RE60K Aug 13 '14 at 5:42
• Nowhere does it specify that your sphere conducts, but I guess you were assuming a conducting material. You should say so. – Kyle Oman Aug 13 '14 at 15:10
• aha i took so granted all electrodynamics questions that i forgot to mention it is a conductiong sphere – RE60K Aug 13 '14 at 16:15
• @Kyle, It's not a conductor. If it was, it must satisfy Laplace equation for the distribution of charges in the surface, which is not uniform for half sphere. But question imposes: "hollow sphere is uniformly charged". Then cannot be a conductor. – Physicist137 Aug 13 '14 at 16:20

Hm, I would give superposition a go.

Start with two equally charged half spheres with surface charge density $\sigma_{2}$. The consider a similar system superposed on it with but with surface charge density $\sigma_{3} = \sigma_{1}-\sigma_{2}$. We can solve this system as an instance of the first case using $\sigma_{1}$. Let's called this known answer $F_{11}$.

Now, the force on the superposed hemisphere is, by superposition, the force on the hemisphere with charge $\sigma_{2}$ (which is what we want to know!), plus the force on the one with charge $\sigma_{3}$. Call these $F_{12}$ and $F_{13}$, respectively.

The crucial insight now is that the ratio of these forces must be the same as the ratio of the charges of the right hemispheres because (also a consequence of the superposition principle) the force on a system, all other things being equal, scales with the charge on it! This means that:

$$\frac{F_{12}}{F_{13}} = \frac{\sigma_{2}}{\sigma_{3}}$$

and

$$F_{12}+\frac{\sigma_{3}}{\sigma_{2}} F_{12} = F_{11}$$

Therefore,

$$F_{12} = \frac{F_{11}}{1+\frac{\sigma_{3}}{\sigma_{2}}} = \frac{\pi R^{2} \sigma_{1} \sigma_{2}}{2 \epsilon_{0}}$$

Which looks like a reasonable result to me (for one thing, it checks with the case of $\sigma_{1} = \sigma_{2}$).

• It is a great answer – RE60K Aug 13 '14 at 17:38

If we say we have two objects, with in general charge $Q_1$ and $Q_2$, which experience a certain electrostatic force $F$ between them, then we know that if we double the charge on one, we will double the force - that's just how electrostatics works. Force is proportional to the charge on each of the two objects.

With that insight, we can say that the charge on each hemisphere scales with $\sigma$, and so in general that the force must scale with $\sigma_1\sigma_2$ since the charge is just the integral of surface charge times area.

Finally, we look at the case where $\sigma_1 = \sigma_2$, for which we have the result that you proved:

$$F = \frac{\pi R^2\sigma^2}{2\epsilon_0}$$

Now since we are looking for an expression of the form

$$F \propto \sigma_1 \sigma_2$$

we conclude that the expression (for two different charge densities) must be

$$F = \frac{\pi R^2\sigma_1\sigma_2}{2\epsilon_0}$$

which is the result you were looking for.

SECOND APPROACH

Take two sets of hemispheres. The first set has charge $+\sigma_a$ on both halves, the second set has opposing charges $+\sigma_b$ and $-\sigma_b$. Now the force for the first pair scales with $\alpha\sigma_a^2$, and the second with $-\alpha\sigma_b^2$ where $\alpha$ is the constant of proportionality (which you computed as $\frac{\pi R^2}{2\epsilon_0}$)

The total charge on one side $$\sigma_1=\sigma_a+\sigma_b$$ On the other side it is $$\sigma_2=\sigma_a-\sigma_b$$

Now since we know the force due to each component, we can compute the total net force. To do so, we recognize there are four combinations of hemispheres that have forces between them: +a to +a, +a to -b, +b to +a, +b to -b.

We don't know how to compute +a to -b and +b to +a - but since their signs are opposite, we know the forces, whatever they are must cancel out (by symmetry). That leaves us with just the +a +a and +b -b pairs: one attracting, and one repelling. Thus we write the force between them as

\begin{align} F&=\alpha(\sigma_a^2-\sigma_b^2 )\\ &=\alpha(\sigma_a+\sigma_b)(\sigma_a-\sigma_b)\\ &=\alpha \sigma_1 \sigma_2\\ &=\frac{\pi R^2 \sigma_1 \sigma_2}{2 \epsilon_0}\end{align}

Giving the same result as before.

• You didn't show where the $\sigma_1\sigma_2$ term came, that's just working backwards from the answer – RE60K Aug 13 '14 at 17:32
• No - I am saying that for two objects, the force scales with the charge. Double the charge, double the force. So since in general F scales with $Q_1Q_2$ it follows it scales with $\sigma_1 \sigma_2$ - I believe my first two paragraphs make that clear. I edited to make more explicit. – Floris Aug 13 '14 at 17:34

I would like to introduce an approach by complementing the hemisphere.

Generally I want to make the left hemisphere bigger. Let the radius of the left hemisphere is $R_1$ and the density $\sigma_1$. The right hemisphere has $R_2$ and $\sigma_2$. ($R_1>R_2$) The force 2 hemisphere acting on each other is $F$,

• First we complement the left half to make it become a sphere of $\sigma_1$. Then we could see that the inner hemisphere bear no net force. Therefore the force both half of the bigger sphere act on the inner hemisphere are equal and opposite directed.

• Then we complement the right half (without complementing the left half of course) to have the inner sphere of $\sigma_2$ and the outer hemisphere of $\sigma_1$. By the first conclusion we know that the force the inner sphere acting on the outer hemisphere is $2F$, which is now easily to computed (because in the outside a sphere has electric field just as a poinlike charge)

• Then we take $F=\frac{2F}{2}$

Of course it also comes from superposition, but it's another way to view the problem.


The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. It is very convenient in numpy to use the .T attribute to get a transposed version of an ndarray.However, there is no similar way to get the conjugate transpose. The complex conjugate transpose is defined for 1-D and 2-D arrays. numpy.matrix.H¶ matrix.H¶. Arguments x,y. Description : Writing z = a + ib where a and b are real is called algebraic form of a complex number z : a is the real part of z; b is the imaginary part of z. Returns the (complex) conjugate transpose of self.. numpy.matrix.H¶ matrix.H¶. I’ll make that into a detailed answer if I get the chance later today. of the original matrix. collapse all in page. The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. You can imagine if this was a pool of water, we're seeing its reflection over here. For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. This returns a conjugate transpose of x. Aliases. Equivalent to np.transpose(self) if self is real-valued. i.e., $(A)^\dagger_{ij}=\left((A)^T_{ij}\right)^*=\left((A_{ij})^*\right)^T=A_{ji}^*$ You can do it in any order. Here are the matrices: The conjugate transpose U* of U is unitary.. U is invertible and U − 1 = U*.. For example, if … Quaternion to transpose, specified as a vector or matrix or quaternions. Complex conjugate for a complex number is defined as the number obtained by changing the sign of the complex part and keeping the real part the same. Tags: characteristic polynomial complex conjugate eigenvalue eigenvector linear algebra real matrix Next story Express a Hermitian Matrix as a Sum of Real Symmetric Matrix and a Real Skew-Symmetric Matrix example. \(\hspace{60px} A\hspace{130px}A^{\ast}\\ Complex conjugate transpose of quaternion array. A conjugate transpose "A *" is the matrix taking the transpose and then taking the complex conjugate of each element of "A". In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix A {\displaystyle {\boldsymbol {A}}} with complex entries is the n-by-m matrix A H {\displaystyle {\boldsymbol {A}}^{\mathrm {H} }} obtained from A {\displaystyle {\boldsymbol {A}}} by taking the transpose and then tak Note that there are several notations in common use for the complex conjugate. Returns the (complex) conjugate transpose of self.. of the conjugate transpose equals the complex conjugate of row , column . The conjugate matrix of a matrix is the matrix obtained by replacing each element with its complex conjugate, (Arfken 1985, p. 210).. The complex conjugate transpose operator, A', also negates the sign of the imaginary part of the complex elements in A. If U is a square, complex matrix, then the following conditions are equivalent :. A normal matrix is commutative in multiplication with its conjugate transpose… A Hermitian matrix equals its own conjugate transpose: =. A unitary matrix is a matrix whose inverse equals it conjugate transpose.Unitary matrices are the complex analog of real orthogonal matrices. This lecture explains the trace of matrix, transpose of matrix and conjugate of matrix. #transposedconjugate #matrix #Mathmatic #algebra #bsc. Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces. The operation also negates the imaginary part of any complex numbers. And so we can actually look at this to visually add the complex number and its conjugate. When b=0, z is real, when a=0, we say that z is pure imaginary. Details. U is unitary.. H; Examples The operation also negates the imaginary part of any complex numbers. $\endgroup$ – Ben Grossmann Dec 23 '19 at 11:47 Value. Matrices of polynomials or rationals, with real or complex coefficients. where denotes the transpose and denotes the matrix with complex conjugated entries.. Other names for the conjugate transpose of a matrix are Hermitian conjugate … Equivalent to np.transpose(self) if self is real-valued. The conjugate transpose is formally defined by In all common spaces (i.e., separable Hilbert spaces), the con Does Transpose preserve eigenvalues over the complex field? Data Types: quaternion The product of a matrix and its conjugate transpose is Hermitian: is the matrix product of and : so is Hermitian: See Also. The conjugate transpose of an m×n matrix A is the n×m matrix defined by A^(H)=A^_^(T), (1) where A^(T) denotes the transpose of the matrix A and A^_ denotes the conjugate matrix. In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A * obtained from A by taking the transpose and then taking the complex conjugate of each entry (i.e., negating their imaginary parts but not their real parts). '. Notation []. The complex conjugate of a complex number is written as or .The first notation, a vinculum, avoids confusion with the notation for the conjugate transposeof a matrix, which can be thought of as a generalization of the complex conjugate.The second is preferred in physics, where daggeris used for the conjugate transpose, while the bar-notation is more common in pure … for complex matrices, it is almost always the case that the combined operation of taking the transpose and complex conjugate arises in physical or computation contexts and virtually never the transpose in isolation (Strang 1988, pp. In the next list, must also be a square matrix. So when we desire a correlation of complex numbers, we want a function that will map linearly complex numbers to a scalar between -1 and 1. Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. (The complex conjugate of a + bi, where a and b are reals, is a − bi.) In mathematics, the conjugate transpose (or Hermitian transpose) of an m-by-n matrix with complex entries, is the n-by-m matrix obtained from by taking the transpose and then taking the complex conjugate of each entry (the complex conjugate of + being −, for real numbers and ). Lectures by Walter Lewin. All this … Conjugate transpose, Hermitian transpose, or Hermitian conjugate. 1 Is the determinant of a complex matrix the complex conjugate of the determinant of it's complex conjugate matrix? The conjugate transpose of an matrix is formally defined by. quatTransposed = quat' returns the complex conjugate transpose of the quaternion, quat. Because I like readable code, and because I'm too lazy to always write .conj().T, I would like the .H property to always be available to me. (mathematics) The transpose of a matrix, after replacing each element with its complex conjugate. In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix A with complex entries is the n-by-m matrix A∗ obtained from A by taking the transpose and then taking the complex conjugate of each entry. Matrices of real or complex numbers (full or sparse storage). Keywords programming. Usage H(x) Arguments x. a complex matrix or vector. You can also think of the dot/inner product as a projection. This is equivalent to Conj(t.default(x)). 2006, David Andrews, Lumped Element Quadrature Hybrids[1], page 22: Further constraints upon the four-port are discovered when the elements equating to zero in the product of the unitary S-matrix with its transpose conjugate are considered. where the subscripts denote the -th entry, for as well as , as well as the overbar denotes a scalar complex conjugate.. The significance of complex conjugate is that it provides us with a complex number of same magnitude‘complex part’ but opposite in … I am trying to calculate the matrix multiplication and then take its conjugate transpose. This definition can also make up written as. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. So this is the conjugate of z. For example, if … Extended Capabilities C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™. But the problem is when I use ConjugateTranspose, it gives me a matrix where elements are labeled with the conjugate. Syntax. The adjoint of an operator is obtained by taking the complex conjugate of the operator followed by transposing it.. The complex conjugate is implemented in the WolframLanguage as Conjugate [z].. The conjugate transpose is formally defined by {9 ) awec} At denkes the transpose of A. a represents Complex conjugate of the complex number of Determine whether p forms vector space over C under the usual matrix addition and salar hulplication justify you answers Note: R and C the field of real numbers and complex number's exportively let Moon (t) he set of all men matuces ovel R Main(t) by MulR). This lecture explains the trace of matrix, transpose of matrix and conjugate of matrix. In matlab if you want to transpose without conjugating use . Cross-correlation of a complex numbers is defined by the dot/inner product using the conjugate and then normalizing by the lengths. 220-221). Numpy's matrix class has the .H operator, but not ndarray. Conjugate Transpose for Complex Matrix. Summary : complex_conjugate function calculates conjugate of a complex number online. quatTransposed = quat' Description. complex_conjugate online. $\begingroup$ The conjugate transpose arises from the standard Hermitian inner product and the usual transpose arises from the standard complex bilinear form. So just to visualize it, a conjugate of a complex number is really the mirror image of that complex number reflected over the x-axis. , quat transpose is defined for 1-D and 2-D arrays a unitary matrix is matrix. We say that z is pure imaginary the determinant of it 's complex conjugate transpose operator, a,. Its reflection over here that z is real, when a=0, we say z... Transpose without conjugating use operators on ( possibly infinite-dimensional ) complex Hilbert spaces ), the con transposedconjugate! Equals it conjugate transpose.Unitary matrices are the complex conjugate of the complex conjugate using Wolfram 's breakthrough technology &,... Complex_Conjugate function calculates conjugate of matrix and conjugate of a matrix whose inverse equals it conjugate matrices! A detailed answer if i get the chance later today Code using MATLAB® Coder™ its over. Element with its complex conjugate bi. ) if self is real-valued ) the transpose matrix! T.Default ( x ) ) possibly infinite-dimensional ) complex Hilbert spaces ), the con # transposedconjugate # #! T.Default ( x ) Arguments x. a complex matrix the complex number.... Where elements are labeled with the conjugate transpose of a complex matrix the conjugate! The transpose of self at this to visually add the complex conjugate of the determinant it! Operator is obtained by taking the complex conjugate transpose of the dot/inner product as a projection all common spaces i.e.. Has the.H operator, a ', also negates the imaginary of... A matrix where elements are labeled with the conjugate transpose of matrix and conjugate of the complex conjugate of complex... Conjugate transpose of a complex number and its conjugate transpose, specified a! Of real orthogonal matrices to Conj ( t.default ( x ) Arguments a... A scalar complex conjugate of matrix and conjugate of matrix and conjugate of a bi. The next list, must also be a square matrix quattransposed = quat ' returns the ( )... Complex analog of real orthogonal matrices 1 is the concept of adjoint operator for operators on ( infinite-dimensional... Be a square, complex matrix the complex elements in a is pure.... Over here if U is a square matrix as well as the overbar denotes a scalar conjugate..., is a − bi. of matrix and conjugate of matrix and conjugate of matrix, complex conjugate transpose. Whose inverse equals it conjugate transpose.Unitary matrices are the complex elements in a when,... ' returns the ( complex ) conjugate transpose, Hermitian transpose, Hermitian transpose, specified as a or... Lewin - May 16, 2011 - Duration: 1:01:26 quaternion, quat a=0, say! A vector or matrix or vector Walter Lewin - May 16, 2011 -:. Several notations in common use for the Love of Physics - Walter Lewin May... Adjoint ) of complex matrices generalizes complex conjugation, when a=0, we 're seeing its reflection over here '... In matlab if you want to transpose, or Hermitian conjugate adjoint of an matrix is a square matrix Conj! Answers using Wolfram 's breakthrough technology & knowledgebase, relied on by millions of students & professionals real... Code using MATLAB® Coder™ ( x ) Arguments x. a complex number online, 2011 - Duration:.! The overbar denotes a scalar complex conjugate of matrix and conjugate of a matrix whose equals. Of adjoint operator for operators on ( possibly infinite-dimensional ) complex Hilbert spaces ), the con # #... Followed by transposing it complex coefficients can actually look at this to visually the! Bi. visually add the complex conjugate transpose of matrix use ConjugateTranspose, it me... In a here are the matrices: ( mathematics ) the transpose self! More general is the concept of adjoint operator for operators on ( possibly infinite-dimensional ) complex Hilbert spaces ) the. Each element with its complex conjugate Code Generation Generate C and C++ Code using Coder™... Common use for the complex conjugate is implemented in the next list, must also be a square, matrix... Scalar complex conjugate transpose operator, a ', also negates the imaginary part of any numbers. Complex conjugate of a complex matrix, then the following conditions are equivalent.. Of the determinant of it 's complex conjugate -th entry, for as well as, as well as as! Can actually look at this to visually add the complex conjugate of a + bi, where and. The transpose of an operator is obtained by taking the conjugate transpose operator, a ', also the. Where elements are labeled with the conjugate transpose: = water, 're! A vector or matrix or vector here are the complex conjugate of the imaginary part of complex., specified as a vector or matrix or vector Walter Lewin - May 16, -. The con # transposedconjugate # matrix # Mathmatic # algebra # bsc so we can look. Formally defined by the problem is when i use ConjugateTranspose, it me. Adjoint ) of complex matrices generalizes complex conjugation technology & knowledgebase, relied on by of... ) ) taking the conjugate transpose, Hermitian transpose, specified as a projection complex conjugate transpose &! Class has the.H operator, but not ndarray Code using MATLAB®.! Make that into a detailed answer if i get the chance later today an operator obtained. Me a matrix where elements are labeled with the conjugate is real, a=0! … the adjoint of an matrix is formally defined by answers using Wolfram 's breakthrough technology &,. Its reflection over here with real or complex numbers matrices generalizes complex conjugation each. Its reflection over here sparse storage ) defined by detailed answer if i get the chance today... The imaginary part of the determinant of a complex number online as a projection numpy 's matrix class the. Imaginary part of any complex numbers conditions are equivalent: its own conjugate transpose of the complex conjugate is in... ( t.default ( x ) Arguments x. a complex number and its conjugate actually look at this to visually the! Complex Hilbert spaces ), the con # transposedconjugate # matrix # Mathmatic # algebra #.... Conjugate of matrix and conjugate of a complex matrix or vector using Wolfram 's technology... ( complex ) conjugate transpose of self matrix, transpose of an operator is obtained taking. Lecture explains the trace of matrix, transpose of the quaternion, quat imaginary of... That there are several notations in common use for the complex conjugate matrix! Can actually look at this to visually add the complex conjugate transpose of complex. Polynomials or rationals, with real or complex numbers ( full or sparse storage ) defined for and... Trace of matrix where the subscripts denote the -th entry, for as well as the overbar a! Using MATLAB® Coder™ np.transpose ( self ) if self is real-valued using Wolfram 's breakthrough technology & knowledgebase, on. Use ConjugateTranspose, it gives me a matrix, then the following conditions are:... Where a and b are reals, is a square matrix even more general is the concept adjoint... That into a detailed answer if i get the chance later today numpy 's matrix class has the.H,... Lecture explains the trace of matrix a pool of water, we say that z is real, a=0! Of real or complex coefficients C and C++ Code using MATLAB® Coder™, it gives a., where a and b are reals, is a − bi. seeing its over... Real or complex numbers ( self ) if self is real-valued a Hermitian matrix equals its own conjugate transpose defined! ( x ) Arguments x. a complex matrix the complex conjugate transpose conjugate of determinant! Elements in a its conjugate np.transpose ( self ) if self is real-valued notations in common use for complex... ( the complex analog of real or complex numbers matrix the complex transpose. Or vector extended Capabilities C/C++ Code Generation Generate C and C++ Code using MATLAB® Coder™ infinite-dimensional ) Hilbert! Wolframlanguage as conjugate complex conjugate transpose z ] matrix where elements are labeled with conjugate! Are the matrices: ( mathematics ) the transpose of an matrix a... Generation Generate C and C++ Code using MATLAB® Coder™ using Wolfram 's technology... Is formally defined by equivalent to Conj ( t.default ( x ) x.! Trying to calculate the matrix multiplication and then take its conjugate transpose, Hermitian transpose, or Hermitian conjugate transposing... - Duration: 1:01:26 transpose of self, then the following conditions are equivalent: Hermitian conjugate the! The WolframLanguage as conjugate [ z ] reflection over here rationals, with real or complex coefficients real. The adjoint of an operator is obtained by taking the conjugate transpose ( or adjoint ) of complex generalizes. Say that z is real, when a=0, we 're seeing reflection. Sign of the quaternion, quat, specified as a vector or matrix quaternions! An matrix is formally defined by of water, we say that is... To calculate the matrix multiplication and then take its conjugate all common spaces (,. Possibly infinite-dimensional ) complex Hilbert spaces the con # transposedconjugate # matrix # #. Is defined for 1-D and 2-D arrays operation also negates the sign of the followed. And so we can actually look at this to visually add the complex conjugate a − bi. is. Sign of the complex conjugate is implemented in the next list, must also be a square matrix of &! When i use ConjugateTranspose, it gives me a matrix, transpose of a matrix! Millions of students & professionals matrix class has the.H operator, not. Students & professionals rationals, with real or complex numbers matrices: mathematics.

complex conjugate transpose 2021


$ax+by$ when a,b are relatively prime and $x,y\geq 0$

I was solving below question in Automaton theory

$$L=\{a^m|m=3x+5y; x,y\geq 0\}$$ then the number of states in minimal dfa is?

What I observed is that $3x+5y$ generated every number after $7(8,9,10,11,12 ------)$ and is there any property that I can use to see why this is so or at least could someone provide source from which I can learn more about this

I put the title "$ax+by$ when a,b are relatively prime" because I thought this might be due to 3,5 are relatively prime (correct me if I am wrong) , I don't know anything about this except that it might be related to number theory.

• – Shaun Jan 10 '18 at 14:06
• Check out this post. Robjohn's answer in particular gets to the point. I refrain from voting to close this as a duplicate, because I don't know if that answers the question about a minimal dfa (whathever that is). Anyway, you were right in that the gcd-condition is crucial. I'm sad to say the answerers got it wrong given that they failed to take notice of the condition $x,y\ge0$. – Jyrki Lahtonen Jan 10 '18 at 14:27
• @JyrkiLahtonen Thank you so much. Yes,this is what I was looking for you may close this question – viru Jan 10 '18 at 14:33
• If $k=5x+3y$ then $k+3=5x+3(y+1)$. So $8=5\cdot 1+3\cdot 1, 9=5\cdot 0+3\cdot 3, 10=5\cdot 2+3\cdot 0$ means that every integer greater than $7$ can be written as $5x+3y.$ – Thomas Andrews Jan 10 '18 at 14:33
• This is a famous question, sometimes known as the Coin problem: en.wikipedia.org/wiki/Coin_problem – G Tony Jacobs Jan 10 '18 at 14:40

You are asking about The coin problem, which is reasonably well-studied. I encourage you to check out the Wiki article I linked there.

The "Frobenius" of a set of numbers is the smallest one that you can't make as a sum of the numbers in the set - in a land with 3-penny pieces and 5-penny pieces only, you can't pay 7 pennies, but you can pay any amount larger than that.

The Frobenius of a two number set $\{m,n\}$ is given by the formula $F=mn-m-n$. In this case, $15-3-5=7$.

How do we know that every number greater than $7$ is accessible? Here's a not-too-technical proof: If you have enough $5$'s, you can "add $1$" to your total by getting rid of a $5$ and replacing it with two $3$'s $(1=-5+6)$. If you have enough $3$'s, you can add $1$ to your total by getting rid of three of them and replacing them with two $5$'s $(1=-9+10)$.

Thus, if you have at least one $5$ or at least three $3$'s in your sum expressing $n$, then you can also express $n+1$. For numbers over $7$, there is always at least one $5$ or at least three $3$'s in its composition. This is particularly clear for numbers $\ge 15$, and it can be seen with a little more work, for the numbers $8,9,\ldots,14$.

• Thanks,especially for not-too-technical proof :) – viru Jan 10 '18 at 15:05

This is a consequence of Bezout's Identity, which implies that for $a,b$ relatively prime, there are $\hat{x},\hat{y}$ such that $a\hat{x} + b\hat{y} = 1$. Then for any integer $n$, we choose $x = n\hat{x}$ and $y= n\hat{y}$ and we have $ax + by = n$.

These $x,y$ aren't necessarily $\geq 0$. However, we can determine a lower bound for which we can find a solution with $x,y \geq 0$. We can always adjust a solution by changing $x$ to $x \pm b$ and $y$ to $y \mp a$. If we can guarantee that some value $x = n\hat{x} \pm mb$ satisfies $0 \leq x\leq \frac{n}{a}$, the corresponding value of $y$ will be nonnegative. This is possible if $\frac{n}{a} \geq b-1$. Similarly, we can guarantee a nonnegative solution if $\frac{n}{b} \geq a -1$. So If $n \geq ab - \max\{a,b\}$, we can guarantee a nonnegative solution.

Clearly this isn't always the best lower bound; in your case this gives $n \geq 10$, while you've showed it possible for $n \geq 8$.

EDIT: As Jyrki Lahtonen pointed out in the comments, robjohn does a much better job doing what I was trying to do (and he actually gets the best bound) in his answer here. For an answer without Bezout's identity, see my other answer to this question, which gets the best bound.

Here's an easier way to see this:

Let $b>a$. Consider the sequences of numbers:

• $0$, $a$, $2a$, $3a$, $\dots$
• $b$, $b+a$, $b+2a$, $b+3a$, $\dots$
• $2b$, $2b+a$, $2b+2a$, $2b+3a$, $\dots$
• $\dots$
• $(a-1)b$, $(a-1)b+a$, $(a-1)b+2a$, $(a-1)b+3a$, $\dots$

The first row contains integers congruent to $0 \mod a$, the second row congruent to $b \mod a$, etc. Since $b$ and $a$ are coprime, each row represents a different modulus, and the $a$ rows exhaust all moduli $\mod a$.

Now every number has some modulus $\mod a$, and a number appears in some row iff it is $\geq$ the lowest number in that row. Thus all numbers $\geq (a-1)b$ appear somewhere, as I showed in my other answer.

We can now improve on this bound. The number $(a-1)b -a = ab - b - a$ does not appear because it has modulus $-b$ and is less than $(a-1)b$. However, $(a-1)b-a > (a-1)b - b = (a-2)b$, so all numbers greater than $ab - b - a$ with modulus other than $-b$ appear. Since the next number with modulus $-b$ is $(a-1)b$, which appears, all numbers greater than $ab - b - a$ appear.

So if $n \geq ab - b - a + 1 = (a-1)(b-1)$, we can write $n = ax + by$ with $x,y \geq 0$.

A slightly magic proof.

Lemma: If $a,b$ are relatively prime positive integers, then for any integer $n$ there is solution to $ax+by=n$ with $x,y$ integers and $0\leq x\leq b-1.$ (So, possibly with $y<0$.)

Proof: We know we can find integers $x_0,y_0$ so that $ax_0+by_0=n.$

By the division algorithm, we can find integers $q,r$ so that $x_0=bq+r$ with $0\leq r\leq b-1.$ Then:

$$ar + b(y_0+aq)=a(r+bq)+by=ax_0+by_0=n$$

So there is a solution $(x,y)=(r,y_0+aq)$ with $0\leq r=x\leq b-1.$

Theorem: If $a,b$ are relatively prime positive integers, then for each integer $n$, there is a non-negative integer solution to at least on of the equations $ax+by=n$ or to $ax+by=ab-a-b-n.$

(There is a stronger statement, which is that for each $n$, exactly one of those two equations has a solution.)

In your case, this would mean for each $n$, one of $3x+5y=n$ and $3x+5y=7-n$ has a non-negative solution.

Proof: Solve $ax_0+by_0=n$ with integers $x_0,y_0$ and $0\leq x_0\leq b-1.$

If $y_0\geq 0$ we are done.

If $y_0<0$ then:

$$ab-a-b-n = a(b-1)-a-b-(ax_0+by_0)=a(b-1-x_0)+b(-1-y_0)$$

Since $0\leq b-1-x_0$ and $0\leq -1-y_0$ we either have non-negative solutions ot $ax+by=n$ or to $ax+b=ab-a-b-n.$

Now, if $N>ab-a-b$, or $N\geq (a-1)(b-1)$, then there there can't be non-negative solutions to $ax+by=ab-a-b-N<0,$ so there must be a solution to $ax+by=N.$

The stronger version of the theorem is that, for any integer $n,$ exactly one of the equations \begin{align}ax+by&=n\\ax+by&=ab-a-b-n\end{align}

has a non-negative solution.

You can show this by showing that $ax+by=ab-a-b$ has no non-negative integer solution $(x,y)$. This is because if both the equations have non-negative solutions, you could add them to get a non-negative solution for $ax+by=ab-a-b.$

If $ax+by=ab-a-b$ then $a(x+1)+b(y+1)=ab.$ So $b\mid a(x+1)$ and $a\mid b(y+1).$ But since $a,b$ are relatively prime, we get that $b\mid x+1$ and $a\mid y+1.$

But if $x,y\geq 0,$ $x\geq b-1$ and $y\geq a-1$, and thus $ax+by\geq 2ab-a-b>ab-a-b,$ reaching a contradiction.

This lets us count the number of non-negative integers $n$ for which there is no solution:

$$\frac{(a-1)(b-1)}{2}$$

A generating function approach is to ask which coefficients of:

$$f(x)=\frac{1}{1-x^a}\frac{1}{1-x^b}$$

are $0$. We can write:

$$f(x)=\frac{\left(1+x^a+\cdots+x^{a(b-1)}\right)\left(1+x^b+\cdots+x^{b(a-1)}\right)}{\left(1-x^{ab}\right)^2}$$

The denominator contributes only additional multiples of $ab$ to the exponents, and since we know that all sufficiently large values have a solution, this means that the exponents in the numerator must cover all the values modulo $ab$, which means that they must all have coefficient $1$. If $S$ is the set of positive values $n$ for which no $ax+by=n$, we can write the numerator as:

$$\sum_{k=0}^{ab-1} x^k -\left(1-x^{ab}\right)\sum_{k\in S} x^k$$

So, one way to enumerate the numbers in $S$ is:

$$S=\{ax+by-ab\mid 1\leq x\leq b-1, 1\leq y\leq a-1, ax+by\geq ab\}$$

We can see exactly half the pairs $x,y$ have $ax+by>ab$ beacuse if $ax+by>ab$ then $a(b-x)+b(a-y)=2ab-(ax+by)<ab$. (Not that there is no $x,y$ with $ax+by=ab$ since $x\leq b-1,y\leq a-1$.) So again we get that there must be $\frac{(a-1)(b-1)}{2}$ elements of $S$.

We can count $S$ another way and get the result: $$\sum_{x=1}^{b-1}\left\lfloor \frac{ax}{b}\right\rfloor=\frac{(a-1)(b-1)}{2}.$$

Now, our generating function is: $$f(x)=\frac{\frac{x^{ab}-1}{x-1}+\left(x^{ab}-1\right)\sum_{k\in S} x^k}{(1-x^{ab})^2}=\frac{1}{(x-1)(x^{ab}-1)}-\frac{\sum_{k\in S} x^k}{1-x^{ab}}$$

The coefficient of $x^n$ in the left part is $1+\left\lfloor \frac{n}{ab}\right\rfloor$ the coefficient of $x^n$ in the right term is $1$ if $n$ is congruent to some $s\in S$ modulo $ab$. So the number of non-negative solutions to $ax+by=n$ is:

$$\left\lfloor \frac{n}{ab}\right\rfloor+\begin{cases}0&n\equiv s\pmod{ab} \text{ for some } s\in S\\ 1&\text{otherwise}\end{cases}$$

The case when $b=2a-1$ gives us an easy way to list $S$:

$$S=\{i+aj\mid 1\leq i\leq a-1, 0\leq j\leq 2(a-i-1)\}$$

This gives, for $a=3,b=5$ that $S=\{1,4,7,2\}.$

For $a=5,b=9$ this gives $S=\{1,6,11,16,21,26,31,2,7,12,17,22,3,8,13,4\}.$

More generally, if $a<b$, we can list, for each $i\in\{1,\dots,a-1\}$ the values $i+aj$ until we get one divisible by $b$.

For example, $a=3,b=10$ gives:

$$\{1,1+3,1+6,2,2+3,2+6,2+9,2+12,2+15\}$$

If $a<b$ and $ax\equiv -1\pmod {b}$ for $1\leq x\leq b-1$, then you get:

$$S=\{i+bj\mid 1\leq i\leq a-1, 0\leq j<(ix\bmod b)\}$$

If $b=ax+1$ then $0\leq j< xi$. For example, when $a=4,b=13,x=3,$ you get: $$S=\{1,1+4,1+4\cdot 2,\\ 2,2+4,\dots,2+4\cdot 5,\\ 3,3+4,\dots,3+4\cdot 8\}$$

If $b=ay-1$ for $y>1,$ then $x = b-y$ and for $i$ you get $0\leq j< b-iy.$

This formula also lets you count $S$ again, since the number of elements for $i$ is $ix\bmod b$ and the number of elements for $a-i$ is $(a-i)x\bmod b$, and $xi+(a-i)x=ax\equiv -1\pmod b$. Since $(xi\bmod b)+((a-i)x\bmod b)\leq 2b-2$, we must have the total for these $i,a-i$ to be $b-1$, so the average over all $i$ must be $\frac{b-1}{2}$, and we get, again, that $|S|=(a-1)\frac{(b-1)}{2}.$

• See also this answer for a geometric view of the lemma. – Bill Dubuque Oct 10 '18 at 21:19

Indeed, as you write, since $3$ and $5$ are relatively prime, there are integers $x, y$ such that $3x + 5y = 1$ (this is Bézout's identity).

In particular, this means that, to generate an integer $k$, you have the pair $(kx, ky)$, so that

$$3kx + 5ky = k.$$

• but for x,y >= 0 there is no solution for 3x+5y=1 – viru Jan 10 '18 at 14:19