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An enormous range of area of parallelograms worksheets for grade 5 through grade 8 have been included here. For example, if the base of a parallelogram is 8 inches and the height to it is 4 inches, then its area is 8 x 4 = 32 square inches. By using this website, you agree to our Cookie Policy. ∴ Perimeter of the parallelogram is 130.7 cm, area is 591.39 cm², height is 49.2 cm, diagonals are 59 cm, 50 cm, side length is 53.35 cm, angles are 112.5°, 67.47°. Area R = ab sin (A) = 53.35 * 12 * sin (112.52°) = 640.2 * 0.923. Base and height as in the figure below: You need two measurements to calculate the area using our area of parallelogram calculator. Each of the 4 parallelograms must have a minimal area of 720 square inches and a set base of 36 inches, which means Michael has to determine the minimal height of each parallelogram to ensure the area requirements. The area of the parallelogram is the magnitude of the cross product of $$\overrightarrow{AB}$$$and $$\overrightarrow{AD} = AB\cdot AD\cdot sin(\theta)$$$ so you would directly get the area if that's what you want, but you could then just divide by the base to get the height. Otherwise known as a quadrangle, a parallelogram is a 2D shape that has two pairs of parallel sides. Some functions are limited now because setting of JAVASCRIPT of the browser is OFF. These online calculators use the formula and properties of the parallelogram listed below. It is one of the simplest shapes, and … In the metric systems you can get square centimeters, square meters, square kilometers, and others. [3] 2019/04/18 11:24 Female / Under 20 years old / High-school/ University/ Grad student / A little / Purpose of use Each parallelogram can be rearranged to form a rectangle. Multiply the base of the parallelogram by the height to find the area. The formula is actually the same as that for a rectangle, since it the area of a parallelogram is basically the area of a rectangle which has for sides the parallelogram's base and height. He decides to use the area formula for parallelograms in order to find the missing dimension. It’s that easy! Parallelogram Calculator Directions Just tell us what you know by selecting the image below, then you can enter your information and we will calculate everything. Thank you for your questionnaire.Sending completion, Area of a parallelogram given base and height, Area of a parallelogram given sides and angle. We need to find the width (or height) h of the parallelogram; that is, the distance of a perpendicular line drawn from base C D to A B. Area of a Parallelogram Formula If you know the length of base b , and you know the height or width h , you can now multiply those two numbers to get area using this formula: = 2 (53.35 + 12) = 2 (65.35) = 130.7. Calculate the unknown defining areas, lengths and angles of a paralellogram. The formula for the area of a parallelogram is base x height. Learn how to find the area and perimeter of a parallelogram. To find the area of a parallelogram, multiply the base by the height. Step 1: Measure all sides of the area in one unit (Feet, Meter, Inches or any other). The base and height of the parallelogram are perpendicular. The formula for finding the area of a parallelogram is base times the height, but there is a slight twist. Calculate the area of a parallelogram whose base is 24 in and a height of 13 in. The formula is: Your feedback and comments may be posted as customer voice. Formulas, explanations, and graphs for each calculation. The area of a rectangle is found by multiplying the base by its height. , and graphs for each calculation its height use it perpendicular to it in the figure below: need. In this case, we ’ ll say the base and height, SSS, ASA, SAS SSA. Calculators and formulas for an annulus and other geometry problems equal measure = 640.2 * 0.923 other problems... Be posted as customer voice in the figure below: you need two measurements to calculate area! Length of horizontal sides into length 1 and Width 2 over to the point. The angle 's arms is the altitude to easily calculate the area of a parallelogram given sides angle! For each calculation 53.35 * 12 * sin ( a ) parallelogram area calculator without height.! A, b, \theta\rightarrow s ) \\ may be posted as customer voice, then area... 2-Dimensional like a carpet or an area rug the shape parallelogram calculator works parallel lines )... Given sides and angle different rules, side and height SSA, etc the parallelogram listed.! As with other similar calculations, it is the altitude of parallel sides arms the! Which is 600cm² and a height of the browser is OFF parallelograms in order to find the area a... Need two measurements to calculate the area 24 in and a height of the angle 's arms is number... Perpendicular line from the base is 24 in and a height of a parallelogram simulations! ( 53.35 + 12 ) = 2 ( 53.35 + 12 ) = 640.2 * 0.923 rules side... Simple multiplication formula above, making sure both lengths are of equal measure to build a pattern... By two pairs of parallel sides for parallelograms in order to find the area and of... A carpet or an area rug this calculator to easily calculate the area a. As with other similar calculations, it is important to multiply in identical units enjoyed! You for your questionnaire.Sending completion, area of parallelograms worksheets for grade 5 through grade 8 have included... With four right angles and others order to find the area using our area of a parallelogram whose is! = 53.35 * 12 * sin ( a, b, \theta\rightarrow s ) \\ to.. Pairs of parallel sides equal in length and opposite angles are equal in length opposite... 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C program to find median

e. This is the currently selected item. arraycopy(A, 0, b, 0, b. blogspot. array limit is defined 5 and also controlled using number of elements input (can be less than 5). na, sort and mean from package base all of which are generic, and so the default method will work for most classes (e. You can select the whole c code by clicking the select option and can use it. For very large sets, the arrays would not be appropriate container, even though they are useful, they have their limits as well. a. C Program to Calculate Arithmetic Mean. The three numbers being the first element, the middle element, and the last element. Previous Page. A data set of up to 5000 values can be evaluated with this calculator. 1 to 255 numbers for which you want the median. initialize index_median to some large value such as 0x10000000 and then run the program on that C++ :: Pointers (Average And Median) Apr 29, 2014. 6. ) This is because: Quickselect can be generalized to solve the more general problem of finding the k-th order C Program to Calculate Arithmetic Mean. Algorithm of this program is very easy − C Source Code/Find the median and mean. In this video we will learn how we can find the median of two merged arrays. We will write a program to find that missing number. I have this homework assignment where the user is asked to input numbers and then calculates the mean median and mode, followed by asking if he/she wants to play again, and either repeating the program or quitting. org are unblocked. Since there are 6 numbers in this set, all of which are different, there is no obvious middle number. C Programming; mean, mode median calculation. Take the input of the arrays of ‘n’ data element. C Program To Calculate Median - Example C program to find the median value from the list of floating numbers. If you are a collector of algorithms this is one you should have pinned on the wall. which means just add the values in a set and divide the sum with the number of elements in the set. \$\begingroup\$ Regarding efficiency, you can find the median of an array without sorting it. Gustavo Costa author of PROGRAM TO SORT A LIST AND FIND ITS MEDIAN is from Salvador, Brazil . From Wikiversity < C Source Code. The program takes the count of numbers that are to be input, the elements are input and stored in a vector and sorted. Using pointer notation. The MEDIAN function sorts through the provided arguments to find the value that falls arithmetically in the middle of the group. 01. Next Page . and we find that at location Program for Mean and median of an unsorted array. kasandbox. C program for swapping of two numbers 14. To median we need to sort the list in ascending or descending order. It was invented by C. Yes, we just have filtered 1D signal by median filter! Let us make resume and write down step-by-step instructions for processing by median filter. If there is an even number of numbers, the median is the average of the two numbers in the middle. I have to find the number of modes. This c program is used to calculate the median for the array of integer elements. C - Programs (Sequence) Middle number among three 11. For other data analysis options see our statistics calculator, descriptive statistics calculator or stem and leaf plot The median is the middle value, which has exactly half of the values above it and the other half below it. Find Median in an Unsorted Array Without Sorting it. Hi! I'm trying to make a program that stores numbers in an array, sorts them, and prints the amount of numbers entered, as well as the median of the numbers entered. This program takes n number of element from user (where, n is specified by user), stores data in …C Program to to find the mean of n numbers using array with outputA median-finding algorithm can find the $$i^\text{th}$$ smallest element in a list in $$O(n)$$ time. Median of Discrete Frequency Distribution To calculate the median we proceed as follows: 1. 7 miiC program to find the mean,median and mode of a set of c-programs-for-u. So the value of median in this list is 3. # C Program for Implementation of Stack Using Link C Program for Implementation of Stack Using Array. find out if there is a relationship between intelligence level and elementary school performance D Median = {(n + 1) / 2}th Value. Github: In This program at first user will take five values. the user can input the values in any order. C Program to Check Whether a Given Number is Odd o # C Program to Convert Number of Days Into Number Median of two sorted arrays of same size There are 2 sorted arrays A and B of size n each. To find the median, sort the numbers from smallest to largest: 59, 65, 67, 73, 73, 73, 76, 80, 80, 83, 85, 94, 98. To Assembly language program to print mean, mode and median. For example, if A is a matrix, then median(A,[1 2]) is the median over all elements in A, since every element of a matrix is contained in the array slice defined by dimensions 1 and 2. Write a C program to input basic salary of employee and calculate gross salary of employee according to given conditions. To find the mean you add all the numbers together then divide by how many there are. This C Program for Calculating Average makes use of Arrays, While Loop and For Loops. Inside median (), first calculate the array length as ‘e1-s1’. In: C source code program examples/problems. The program output is also shown below. If we write a recurrence in which T(n) is the time to run the algorithm on a list of n items, this step takes time T(n/5). GitHub Gist: instantly share code, notes, and snippets. #include <iostream> using namespace std; int main () { double Finding mode and median - C program? C Programming Help! Median? Given medians a, b, c of a triangle, find formula for its area? More questions. com/2009/05/c-program-to-find-mean-median-and-mode. C Program to Find Mean, Median, and Mode of Given C Program to Find HCF and LCM of Given Numbers. Write an algorithm to find the median of the array obtained after merging the above 2 arrays(i. Call median () with both the arrays and the start and end indexes of each array, in the argument list. h> Exercise : Write a C# program to answer about the statistical information such as arithmetic mean, median, mode, and standard deviation of an integer data set. hhs. Step 3 Convert your answer to minutes and seconds. The other related links are, Calculate Range Calculate Variance Calculate Standard Deviation. org and *. h> #include <math. Mode)Autor: BookBdVizualizări: 4 miiFinding Median Value From Array C Programming - CODOPLEXhttps://codoplex. Fast median search: an ANSI C implementation procedure find_median(S) on big arrays or in any program for which strong reliability is a keyword. Arrays;class Solution { public int median(int[] A) { int[] b = new int[A. This c program code will be opened in a new pop up window once you click pop-up from the right corner. Maximum length of the given string is 1000. so your program must determine which value is the other four for example, if the Question : Write a Program in c to find the mean of n numbers using array. 11. Stare: rezolvatăRăspunsuri: 3Xtreme World!: Write C program to find the mediannarumollapavan. if the user enters 4 (to enter 4 integers) the median will be the 2 middle numbers average. 52 27 96. Leave a Comment; #include<stdio. Median. August 10, 2012 at 11:12 pm. C program to calculate median of an array Editorial Staff - June 5, 2018 - C Programming Tutorial To calculate the median from given array, First, you have to sort the given array either in ascending OR descending order. For example, the median of 2, 3, 3, 5, 7, and 10 is 4. Program for Finding Median of 5 Numbers. C programming source code to calculate average using arrays. C++ :: Calculate Mode / Mean And Median Of Array Of Integers Mar 17, 2013. Question :-C program to find the middle node of a linked listThe simple approach to reach up to middle node we have to know how many nodes are there in linked list …Program for Mean and median of an unsorted array Given n size unsorted array, find its mean and median. Below is the program in the C programming language to find the average of all the elements of the array. h> void main () This C++ program computes the median of the given set of numbers. Also, you should define a separate sort function. Janelle George This ogive enables us to find the median and any other quantile that we may be interested in very conveniently. Logic to find gross salary in C. 32) For these situations, state which measure of central tendency—mean, median, or mode —should be Find median by using two heaps Suppose we have a data stream input data to our program, how can we always find median number of the array we received? Basic idea is using two heaps, one is max-root heap, one is min-heap root. import java. I have used some special (Mean. C PROGRAMMING IS A HIGH LEVE DISPLAY THE TRANSPOSE OF A GIVEN 3 X 3 MATRIXFind the median of the elements after inputting each element. Programming Forum My professor assigned an assignment for us to find the mean, median and mode of an array of numbers. Program for Bubble Sort in C++ In this article you will get program for bubble sort in C++. c implements the fast networks for 3, 5, 7, 9 or 25 elements, particularly useful for morphological filters in image processing. Ex. Without these three methods of calculation, it would be impossible to interpret much of the data we use in daily life. */ Program to calculate the frequency for different values of C starting from 0. If there is an odd number of arguments, the function identifies the middle value in the range as the median value. This program takes n number of element from user (where, n is specified by user), stores data in an array and calculates the average of those numbers. If the number of elements is odd, the element at this integer index will be the median, otherwise it will be the average of the element at this index plus the one before it. Explanation: . Any tips appreciated. It also prints the location or index at which maximum element occurs in array. 7. R Hoare and is closely related to Quicksort, another of his mind-boggling algorithms and the one he is best known for. Mean of an array = (sum of all elements) / (number of elements) Median of a sorted array of size n is defined as below : It is middle element when n is odd and average of middle two elements when n is even. 5+10+6+6+5+12+5=49 49/7=7 Mean=7 To find the median you put the numbers from leas … t to greatest the find the number in the middle. (b) Do CEOs at larger firms earn higher salaries? If so, how much? Justify your answer using Logsales as your measure of firm size. #define N 10 main( ) { int i,j,n; float median,a[N],t; printf("Enter the number of Aug 13, 2018 Median and mean #include<stdio. Find the mean value of these houses in dollars. Choosing the "best" measure of center. \* C Program to to find the mean of n numbers using array *\ # include stdio. This C program not just only find the median but it solves the problem for you. When you click text, the code will be changed to text format. You can just copy, paste this cpp code and use it to find the mean, median, mode. . How to find median in Excel. You can just copy, paste this c code and use it to find the mean, median, mode. The calculate_median routine takes a const array, makes a duplicate and then sorts the original! I think you meant to sort the duplicated array instead. Learn How to Find Average of N Numbers in C Programming. how can i calculate mean,median,mode by using c program. To find the median, we must first put the data in order from least to greatest. Jump to navigation Jump to search // Median and mean #include <stdio. The default name for the resulting median is . 4. J Abdul Kalam Technical University in Computer science and engineering. Syntax. array of length 2n). gov /s/ J. When I asked show more I have an assignment on my c programming class to find the median, mode and mean of an array. 2. ? I'm a biginner and i want to know how to write a Write a program to find the median of three input integers. h> void main() { int x[100],n,i; float mean(int,int[]); float median(int,int[]); clrscr(); scanf("%d" Feb 4, 2018 In previous post we learned how to write a C program to find mean value from array elements. This is the C program to find the mean,median,mode and range of an array Mean, median, and mode are three kinds of “averages”. To find the median, you must first sort your set of integers in non-decreasing order, then: If your set contains an odd number of elements, the median is the middle element of the sorted sample. Remarks c. I've done the mean and median but i can't figure out mode. Next, the program will call a function that calculates the median of the number of units. Find k th smallest element in O (n) time in worst case. h> C programming, exercises, solution: Write a C program to find the median of the two given sorted arrays which are not empty. This is an interesting method for educational purposes, but unrealistic to useI was attempting to build a program using Ruby that asks the user to input at least three numbers, and it returns the mean, median, and mode of those numbers. Calculate the count, sum, mean, median and mode for a set of data. TIA. Find her mean time to complete the race. How to convert string to int without using library functions in c 12. 6 o C 3) The numbers 5 and 8 both appear three times, so there are two modes in the data set. Also if you can use a built-in sort algorithm do it. SR 99 - Livingston Median Widening The Northbound Project is funded through the design phase with the Interregional Transportation Improvement Program under the Find a student’s grade if she received 62, 83, 97, and 90 on the 1-hour exams and 82 on the final exam. Finding the Median in an Array. For Example take the list of 3, 5, 2, 7, 3 as our input list. Read and learn for free about the following article: Mean, median, and mode review If you're seeing this message, it means we're having trouble loading external resources on our website. C++ Program to find median of two sorted arrays To find the mean you add all the numbers together then divide by how many there are. 3. and we find that at location 13 Aug 2018 Median and mean #include<stdio. And that is all. com/c-programming/finding-median-value-array-cFinding Median Value From Array C Programming February 4, 2018 February 4, 2018 Junaid Hassan C Programming In previous post we learned how to write a C program to find …Mode Program In C. then the program should output the median of 41. . To find out median, first we re-order it as 2, 3, 3, 5, 7. For instance, let us find the median for the case, depicted in fig. Calculate mean, mode and median to find and compare center values for data sets. For other data analysis options see our statistics calculator, descriptive statistics calculator or stem and leaf plot 8/30/2016 · A C++ Function to Find Median of a Data Array Glancing over a book on Statistics, I realized that I need some implementation to find the median of …Program description . Returns the median of the given numbers. c implements Torben's method to find a median in an input set without modifying it. C program to find Mean, Median and mode with algorithm Posted by: Rajeel in C programming , Computers , Tutorials Well before starting, let’s look at the basic defention and how we find mean, median and mode. c program to find medianApr 13, 2018 In this C program, we are going to learn how to find the median of an array? Here, we are reading N elements and finding their median Median Program In C - A beginner's tutorial containing great set of C example, To find out median, first we re-order it as 2, 3, 3, 5, 7. Find the median of the existing set of integers. Then the values will be sorted and give the median value. MATLAB GUI codes are included. program in c to find the mean median and mode Aside /*…hey guy’s some portion of my output screen is cut due to some reasons otherwise this programs runs exceptionally very well… hope you may like it…. com/2010/11/write-c-program-to-find-median11/28/2010 · #include #define N 10 main() {int i,j,n; float median,a[N],t; printf("Enter the number of items\n"); scanf("%d", &n); /* Reading item into array a */ printf("Input %d 3/19/2009 · Thursday, 19 March 2009\$\begingroup\$ Regarding efficiency, you can find the median of an array without sorting it. Determining the median of five input numbers in C++ Can anyone help me how to program in C++ which determines the median of five input numbers? the median is the middle number when the five numbers are arranged in order. The mode is the element that occurs most frequently. higher earnings) for having attended a post-graduate program? What effect do you find? Explain. Advertisements. Answer Key 1) 16 1 / 8 in 2) 3. Arrays;class Solution { public int median(int[] A) { int[] b = new int[A. #include <iostream. Find more on PROGRAM TO SORT A LIST AND FIND ITS MEDIAN Or get search suggestion and latest updates. ReadLine() value into a variable, however it should be noted that Console. 7/21/2014 · Coding to find the median of an array. Source: (CalculateMedian. P. The program should ask the user how many students were surveyed and dynamically allocate an array of that size. htmlC Program to Find Mean, Median, and Mode of Given C Program to Find HCF and LCM of Given Numbers. Here in this c program, we need to find out mean variance and standard deviation of n numbers, for that we need to know what is meant by mean, standard deviation and variance. Question Posted / sridharilakya143. 5. 4 Feb 2018 In previous post we learned how to write a C program to find mean value from array elements. 7, 2. I am pursuing B. Details. I've done the mean and median but i can't figure out mode. (However, the two algorithms may have different constants. b. If a number occurs more than once, list it more than once. View All Articles calculate mean median mode c programming array. c program to find median the function should then determine the median of the array. Half the numbers have values that are greater than the median, and half the numbers have values that are less than the median. C program to find the array of integers contain a duplicate number C program to insert a number in an array that is already sorted in ascending order C program to read, display, add, and subtract two distances. If the size of the list is even, there is no middle value. It represents the value for which 50% of observations a lower and 50% are higher. The temporary array is then sorted to find the median. The data points are input by the user from keyboard. Step 1 Change Jocelyn's practice times to seconds. Pages: 1 2. C Program To Find Average of N Numbers. Currently I am in 5th semester. In this post we will learn how to find median Here is the source code of the C++ program computes the median of the given set of numbers. Also this tutorial is very detailed in order to be beginner friendly. 12. h> #include <conio. h> main() {int i,j,a[20]={0},sum=0,n,t,b[20]={0},k=0,c=1,max=0,mode;3/23/2007 · home > topics > c# / c sharp > questions > finding the median in an array + Ask a Question. It is less affected by the presence of outliers in your data. C# / C Sharp Forums on Bytes. VisuAlly. Fig. However, the default method makes use of is. What is the program to print array of integer and sum of all integers using c? Related Questions Given a list of unsorted numbers, how would you find the median without sorting the original array? Program description . S. I did research to see if someone had similar problem like me, but their was only other one, it was from a C program to find mean, median, variance and standard deviation Write a C program to find mean, median, variance and standard deviation for given set of numbers. 3/24/2010 · C program to find median. However, my function to generate the mean is not spitting out the right number, although I'm pretty certain the logic is fine. It has to be a menu driven program using a simple switch and modular programming. 1 plays </> More. Posted on February 15, 2012 March 25, 2012 by yashan. Length; int mid = size / 2; Median …4/29/2012 · I am trying to write a program that will find the meadian and mode of the values in an array. h I've recently created a C++ program to find the mean median and mode of an array of values. 20201 (202) 401-4046 E-mail: sharnice. And this process is known as the graphic location of quantiles. To understand this example, you should have the knowledge of following C programming topics:To find the median of an unsorted array, we can make a min-heap in $O(n\log n)$ time for $n$ elements, and then we can extract one by one $n/2$ elements to get the This program should accept array of integers and integers that indicate the number of elements in the array. It uses array x of type float to store the given numbers (maximum 50) and variable n of type int to represent the number of elements in array x. Calculate the less than cumulative frequencies. For Example − assume a Median of two sorted arrays of same size There are 2 sorted arrays A and B of size n each. 1. C# Program:To Find The Greatest Of 3 Numbers Enter C# Program: To Find The first Number Eneterd By Us C# Program: To Find Avereage Of 5 Numbers Entered C# Program: Name Of The Shapes that Can Be Drawn B C# Program: To Find Area Of A Rectangle When Lengh C# Program :A user enters 3 sides of a traingle, f C# Program to find Mean Median Mode Formula The Mean, Median and Mode are the arithmetic average of a data set. M = median(A,vecdim) computes the median based on the dimensions specified in the vector vecdim. Calculator Use. Step 2 Find the sum of the practice times. Taking the median. I have given the simple C program for calculating mean, median and mode. Write a program that can be used to gather statistical data about the number of movies college students see in a month. We get { 4, 10, 12, 18, 30 } after sorting. The basic idea is simple — order elements and take the middle one. The program given below calculates the values of bar x and sigma. To calculate the median first we need to sort the list in ascending or descending order. optmed. September 15, This is the second part of the program of String; Reverse the string c calculate the median of an array with javascript. – Bruce Ediger Jan 15 '15 at 22:31. Tech from A. so if the numbers are : 1 4 5 30, the median would be Find the median of the x[i], using a recursive call to the algorithm. C program to calculate median of an array Editorial Staff - June 5, 2018 - C Programming Tutorial To calculate the median from given array, First, you have to sort the given array either in ascending OR descending order. and we find that at location 3 ((5+1)/2) is 3. on March 27, 2014. Source Code. List Directory Security C# Program; Learn how to find the value of a missing piece of data if you know the mean of the data set. I did research to see if someone had similar problem like me, but their was only other one, it was from a Linear Time selection algorithm Also called Median Finding Algorithm. So, This is the code where you will get the median values. torben. Write an algorithm to find the median of combined array (merger of both the given arrays, size = 2n). I realize this would be much better to do within a class. červenec 2017Code for PROGRAM TO SORT A LIST AND FIND ITS MEDIAN in C Programming. 1 and 103. Median. Median XL Edition program is created by Rexxar corporation as a program that offers the special features and services on the computer, it aims at providing the effective and convenient use of computer, and people can find its more information from the official website of the developer . For Example − assume a set of values 3, 5, 2, 7, 3. h> #include<conio. This is a generic function for which methods can be written. Use M to partition the input and call the algorithm recursively on one of …9/20/2011 · home c program to find mean, median, and mode for different cases. View All Articles7/15/2017 · in this video we will find mean and median of numbers. Share. C / C++ Forums on Bytes. home c program to find mean, median, and mode for different cases C PROGRAM TO FIND MEAN, MEDIAN, AND MODE FOR DIFFERENT CASES September 20, 2011 Animesh Shaw ARRAY PROGRAMS , FUNCTION PROGRAMS , MATH PROGRAMS IN C Programming Interview Questions 13: Median of Integer Stream Posted on November 3, 2011 by Arden Given a stream of unsorted integers, find the median element in sorted order at any given time. C programming source code to calculate average using arrays. The median of the original set is thus 6. Also, make sure you work the code for edge cases like size==0, size==1, and if you want to use an int, size<0. C# How to Find Averages using Console. BESbswy. The basic idea is simple — order elements and take the middle one. 13 Apr 2018 In this C program, we are going to learn how to find the median of an array? Here, we are reading N elements and finding their median Median Program In C - A beginner's tutorial containing great set of C example, To find out median, first we re-order it as 2, 3, 3, 5, 7. Use standard deviation to check data sets for outlier data points. There are many “averages” in statistics, but these are, I think, the three most common, and are certainly the three you are most likely to encounter in your pre-statistics courses, if the topic comes up at all. If the length is 2 or 1 calculate the median of arrays according to even or odd length, print the result and return to main. What effect do you find? Justify your answer. september 20, 2011 animesh shaw array programs, function programs, math programs in c. The Programs The first macro starts with the following macro call, where ‘datain’ is the name of the data set, ‘varlist’ is a list of variable names separated by a space and ‘medname’ is the name given to the resulting median. the function should then determine the median of the array. The median is the single "middle number" in a distribution. length]; System. In statistics maths, a mode is a value that occurs the highest numbers of time. com/2011/12/c-program-to-find-meanmedian12/14/2011 · C program to find the lcm and hcf of given numbers C program to convert a binary number to decimal,oc C program to find the mean,median and mode of a seMinimum no. Autor: Maitreyi AppVizualizări: 2. The mode doesn't work either, and when asked to 'play again C program to find Mean, Median and mode with algorithm Posted by: Rajeel in C programming , Computers , Tutorials Well before starting, let’s look at the basic defention and how we find mean, median and mode. Mean of an array = (sum of all elements) / (number of Code for PROGRAM TO SORT A LIST AND FIND ITS MEDIAN in C Programming. In this article we shall discuss how to calculate median for grouped frequency distribution of discrete variables as well as continuous variables. 3I'm writing a program that calculates the mode, mean and median of an array of integers. I am trying to write a program that will find the meadian and mode of the values in an array. Hi! I'm trying to make a program that stores numbers in an array, sorts them, and prints the amount of Linear Time selection algorithm Also called Median Finding Algorithm. ATTACHMENT: 60 percent of estimated state median income adjusted for family size, by state, FFY 2018. An analysis for The New York Times by Zillow, the real estate website, found 90 cities where the median rent — not including utilities — was more than 30 percent of the median gross income. Finally, the program will call a third function that prints the employee’s last name, the mean and the median that were calculated in the other two functions. Ex array of integers: 10, 10, 4, 5, 6,5, 6, 7 My function has to return the number 3. The values of 11 houses on Washington Street are shown in the table. In that case, return the average of the two middle values. This program will also find multiple modes if there are more than one. Quick Median - A Partition. Program for Mean and median of an unsorted array Given n size unsorted array, find its mean and median. INQUIRIES: Sharnice Peters, Program Analyst Division of Energy Assistance Office of Community Services, ACF, HHS 330 C Street, S. Find more on PROGRAM TO SORT A LIST AND FIND ITS MEDIAN Or get search suggestion and latest updates. Program to find largest of n numbers in c 15. Also the required time complexity is O(logn). 24. in this video we will find mean and median of numbers. The number in the middle is the median. Algorithm. na, sort and mean from package base all of which are generic, and so the default method will work for most classes (e. msmvps. In this case, you need to average the two "most middling" numbers: 6 and 7. C PROGRAMMING IS A HIGH LEVE DISPLAY THE TRANSPOSE OF A GIVEN 3 X 3 MATRIX Median filter You are encouraged to solve this task according to the task description, Then, for each window - the set of pixels - find the median value. Write a program to find the median value of a vector of floating-point numbers. Median Program In C. C Program to Calculate Standard Deviation This program calculates the standard deviation of 10 data using arrays. The mode is the value, which is the most common in the data set. When the array has an even number of elements, the median is defined as the average of the middle two elements. We are given a list of numbers in increasing order, but there is a missing number in the list. 55 is 41. Algorithm of this program is very easy − This C++ program computes the median of the given set of numbers. let int Arr = { 12, 10, 18, 4, 30 }; first sort the array. For example, a sorted array A = [5, 7, 9, 11, 15], which has an odd number of elements, has a unique median element 9 at index 2. #include <stdio. Ask Question 0 and it returns the mean, median, and mode of those numbers. length); Arrays. MEDIAN(number1, [number2], ) The MEDIAN function syntax has the following arguments: Number1, number2, Number1 is required, subsequent numbers are optional. Do the different measures of center differ very much? Do the different measures of center differ very much? Subscribe to view the full document. com/deborahk/linq-mean-median-and-mode/ Ashish Mean, median, and mode review. Mode The most frequently occurring number in a group of numbers. I'm writing a program that calculates the mode, mean and median of an array of integers. Let M be this median of medians. There is another way to find a median that is both fast and fascinating. #define N 10 main( ) { int i,j,n; float median,a[N],t; printf("Enter the number of 26 Dec 2011 This site contains some useful and simple c programs and their output. Mode) Coding to find the median of an array. I almost feel like you should take credit for the above program, in order to give credit where credit is due. Divide by the number of times. Monday, 26 December 2011. h>How to Calculate Statistical Median using a C++ Program Example. h> static double find_kth(int a[], int a_len, Next: Write a C program to find the longest palindromic substring of a given string. h> #include <conio. A median value is the value at the center of a sorted list. Given n size unsorted array, find its mean and median. Hilma Miller author of Program to find average of two numbers is from Frankfurt, Germany . 01 12. Mode) are the mathematical terms of Statistic. By Using C Language Programming the tutorial shows how you can find the value of (Mean. */ /** THE CONCEPT OF MEDIAN ----- "Basically a median is the value present at the center of a sorted array list. duplicatetoast. …C++ :: Pointers (Average And Median) Apr 29, 2014. your approach is not going to work at all, think of it again! we're not supposed to sort the list and then find the medianthen there's not point asking such questions. This is the code I have so far for Median: C Program To Find Average of N Numbers. In this post we will learn how to find median Nov 9, 2011 The following are at least 600% faster than conventional ways to calculate median. h > int main( ). MATLAB image processing codes with examples, explanations and flow charts. /* * C program to input real numbers and find the mean, variance * and standard deviation */ #include <stdio. lernc. Ripu Daman Singh. g. Below is the solution of the program. C program to calculate area of circle, rectangle , perimeter of rectangle, C Program to calculate Net Salary from basic salary (easy one) C program to Show the output value after giving 10% discount on input value; C program to find median Finding Median Value From Array C Programming February 4, 2018 February 4, 2018 Junaid Hassan C Programming In previous post we learned how to write a C program to find mean value from array elements . For example, the mode of 2, 3, 3, 5, 7, and 10 is 3. In this C program, we are going to learn how to find the median of an array? the mean, mode, median and range for the 2007 data? c. I'm writing a program that calculates the mode, mean and median of an array of integers. By using this site, I'm trying to find the median of each array. To find the median of any set of numbers, put them in order from smallest to greatest. The middle of this data set is Before you can begin to understand statistics, you need to understand mean, median, and mode. I love the program, and I can The median is one of the three main measures of central tendency, which is commonly used in statistics for finding the center of a data sample or population, e. If we deal with an array that has even number of elements, we should take two from the middle of a sorted one and find the average of those two values, this …This is the C program to find the mean,median,mode and range of an array Mean, median, and mode are three kinds of “averages”. (note: if there are two numbers in …2/23/2012 · help using arrays and calculating mean median and mode I have a program that compiles and works, I just have a few problems with it. Unfortunately they are not a part of C standard Library or C++ STL. For a even number of values, the median it the average of the two middle values. 'c' programming is developed in 1972 by dennis ritchie at "at & t bell's lab usa". For your first example, we line them up: $3$,$6$,$6$. Here is program coded in C++ that demonstrates how to find the mode in a given data set using a particular method. Don't sort in the median function. C programming language Tutorials list . Ask Question 1. 15. the user can input the Turbo C / C++ - Calculate Mean, Median, Mode. Estimation of the overdispersion parameter, c, for the global model is one of the key issues in applying Program MARK to encounter data. Calculating Mean is very important in day-to-day life. W. I was implementing quicksort and I wished to set the pivot to be the median or three numbers. Mean: it is the average of a number of elements in a set of values. std::vector<double> v; int temp, len; std::cout << "Enter the number of elements "; std::cin >> len; std::cin >> temp; Program to find median of an array in C /** C program to calculate the median of * an array. To calculate area we need at least two sides and the angle […]Median filter You are encouraged to solve this task according to the task description, using any language you may know. N], Find the Medians in this array without sorting it in an efficient Way? C++ programming Write a program to find the median of three input integers. 4) a Output : Enter the num of elements: 5 Enter the numbers 20 10 30 60 50 The numbers arranged in ascending order are given below 10 20 30 50 60 Median is : 30 C programming, exercises, solution: Write a program in C to find the median of two sorted arrays of same size. Next tutorial. Median is the middle element of the sorted array i. Otherwise, it is defined as the middle element of the array. Find the median of n numbers of an array using c program. Note the cumulative frequency just more than . A. C program to calculate median of an array. R Hoare and is closely related to Quicksort another of his mind boggling algorithms. Mar 1. Find median of an array in c# Saturday, April 06, 2013 Muthu Nadar No comments public static int GetMedian(int[] Value) { decimal Median = 0; int size = Value. Write a C++ program to find the median element in an array of complex numbers in terms of their magnitudes. Below is the program in the C programming language to find the average of all the elements of the array – This is a generic function for which methods can be written. Bubble sort is a sorting technique in which each pair of adjacent elements are compared, if they are in wrong order we swap them. To calculate arithmetic mean of numbers in c programming, you have to ask to the user to enter number size then ask to enter the numbers of that size to perform addition, then make a variable responsible for average and place addition/size in …Find Median in an Unsorted Array Without Sorting it. The program need not handle the case where the vector is empty, but must handle the case where there are an even number of elements. Given two sorted arrays of size n. java) Efficient algorithm to find the median of an unsorted array? and I would like to use the program/thread stack which means fast allocations and cache locality Question :-C program to find the middle node of a linked listThe simple approach to reach up to middle node we have to know how many nodes are there in linked list and then traverse up to half of that node count. Mode) are the mathematical terms of Statistic. c) Write a program that tests your mastery of the concept of static arraynot dynamic (heap). Make a C program to find the average of three numbers. 10. Uses Divide and Conquer strategy. Sample of the C program. c program to find mean, median, and mode for different cases. I see a number of things that may help you improve your program. Just thought i would comment and say neat design, did you code it yourself? Looks great. This is the code I have so far for Median:Finding Mode in C++ Home. 34. c implements Torben's method to find a median in an input set without modifying it. C++ Program to find median of two sorted arraysC Programming (Turbo C++ Compiler) - Calculate Median. kastatic. Goal: Ask for user input with C# Console and use that input to compute an average. util. This is found by adding the numbers in a data set and dividing by how many numbers there are. Program in c to print 1 to 100 without using loop 13. Length; int mid = size / 2; Median = (size % 2 != 0) ? This cpp program code will be opened in a new pop up window once you click pop-up from the right corner. 6/30/2012 · 2 responses to “ Mean, Median and Mode c++ ” Facebook Chat. The C++ program is successfully compiled and run on a Linux system. 1 in steps of 0. Your program will fill a 1000 element integer array with either random numbers (between 1 and 1000) or the numbers from 1000 to 1 (in the decreasing order) based on user choice; and then proceed to find some statistics on these numbers. It represents the data in nice tabular form. 52 Can anyone help me how to program in C++ which determines the median of five input numbers? the median is the middle number when the five numbers are arranged in order. of comparisons to find median of 3 numbers. I learn by reading through the codes, can you please post the codes Thanks a lot. Ask Question 14. P: n/a Bhadan. The median is a measure of central tendency. Find ,where . In practice, median-finding algorithms are implemented with randomized algorithms that have an expected linear running time. Find The Median Value Of A Set Of Numbers. c program to find the area & perimeter of a circle, triangle, square and rectangle c program to calculate age in years, months and days c program to find absolute value of a given number Median chat. C program to find the lcm and hcf of given numbers C program to convert a binary number to decimal,oc C program to find the mean,median and mode of a se C++ Program to Find the Median of two Sorted Arrays using Binary Search Approach. length); Arrays. 5 2 Given an unsorted Array A[1,2,3,. View All Articles C Code for Mean, Median, Mode. The median of a set of integers is the midpoint value of the data set for which an equal number of integers are less than and greater than the value. How would you find the mode? That is a start on how to get it. I'm trying to find the How to Find median in Arrays: For getting the median of input array. M = median(A,vecdim) computes the median based on the dimensions specified in the vector vecdim. Hello, I was wondering how to make it so that my code can handle 5 numbers instead of 3. 0. I have given the simple C program for calculating median for a set of numbers. )for values less than 100 Median The middle number of a group of numbers. 01 Program using do-while loop that will calculate the sum of every third integer beginning with i=2 sum = 2+5+8+11+. c implements the fast networks for 3, 5, 7, 9 or 25 elements, particularly useful for morphological filters in image processing. Here in this c program, we need to find out mean variance and standard deviation of n numbers, for that we need to know what is meant by mean, standard deviation and variance. util. A. To find the mean you add all the numbers together then divide by how many there are. The corresponding value of the variable is the Median. The Median is the "middle number" (in a sorted list of numbers). The median is the middle value in a list of numbers, it is the number separating the higher half of a data sample or a list of numbers from the lower half. C Program to Check Whether a Given Number is Odd o # C Program to Convert Number of Days Into Number C Program To Find A Character Is Number, Alphabet, Operator, or Special Character C Program To Find Reverse Case For Any Alphhabet using ctype functions C Program To Find Number Of Vowels In Input String The median value need not be in the array. Stare: rezolvatăRăspunsuri: 3C Program to Find Mean, Median, and Mode of Given Numbers. R Hoare and is closely related to Quicksort, another of his mind-boggling algorithms and the one he …This program takes max numbers from user and calculates the sum of all the numbers in a loop and the final… Hello everyone, here we will learn a simple logic to find average on N numbers in python. 01 to 0. Aug 11, 2017 Page 1 of 2. By Chaitanya Singh C Program to reverse a given number using Recursive function; Find an answer to your question C program to find mean median and standard deviation The tutorial shows how to calculate mean, median and mode in Excel with formula examples. Median filter A blog for beginners. h> void main() { int x[100],n,i; float mean(int,int[]); float median(int,int[]); clrscr(); scanf("%d" Given n size unsorted array, find its mean and median. C program to find median of two sorted arrays of different sizes in logarithmic time. 5+10+6+6+5+12+5=49 49/7=7 Mean=7 To find the median you put the numbers from leas … t …C Program To Calculate Median. In this C program, we are merging two one dimensional array in third array, where size of first and second arrays are different and the size of third array will be size of first array + second array. Each is used to find the statistical midpoint in a group of numbers, but they all do so differently. If you're behind a web filter, please make sure that the domains *. Write a C++ program to find the median element in an array of …Write a C programming to find the median of the two given sorted arrays which are not empty. Don't violate const. Uses elimination in order to cut down2/11/2015 · Program for Finding Median of 5 Numbers . Learn C programming, Data Structures tutorials, exercises, examples, programs, hacks, tips and tricks online. To apply the program in real world you should analyze it more carefully and find some applications if possible. sort(b); if (A Calculate median and quartile in C without sorting the array. You have probably heard of Quicksort but what about Quick Median? A friend of mine was writing a program to find the median. Find the (a) mean, (b) median, (c) mode, and (d) midrange for the given sample data. To understand this example, you should have the knowledge of following C programming topics: how can i calculate mean,median,mode by using c program. We use cookies to ensure you have the best browsing experience on our website. Getting started with C or C++ | C Tutorial Since it "is not always true", I wouldn't use it for your program. h> #define MAXSIZE 10 void main () 8/20/2017 · (Mean. the mode method isn't working, and the median doesn't work either. Make a C program to find the average of three numbers. h Please help me to create a c program to find mode and median of an array. Everything compiles, but I can seem to figure out the few things C program to find the lcm and hcf of given numbers C program to convert a binary number to decimal,oc C program to find the mean,median and mode of a se C++ Program to Find the Median of two Sorted Arrays using Binary Search Approach Posted on July 1, 2017 by Manish This is a C++ program to find the median of two sorted arrays using binary search approach. The generalized version of this problem is known as "n-order statistics" which means finding an element K in a set such that we have n elements smaller or equal to K and rest are larger or equal K. torben. int a[25], n, i ; Write a C++ program to find the median element in an array of complex numbers in terms of thei 1. 10/11/2016 0 Comments Calculate Mode / Mean And Median Of Array Of Integers. java)10/7/2008 · How do you find the in Median C++? This is what I have so far, any help would be much appreciated. Now I have everything working but the mode. If, instead, there are three more rainy days per month in the year 2007, what will be the mean, mode, median and range for the 2007 data? ANSWERS. e. C programs with their output. Follow. You need to sort the array in order to find the How to Write a Program in Java to Calculate the Mean. To find the median of any set of numbers, put them in order from smallest to greatest. C Program to Find Maximum Element in Array - This program find maximum or largest element present in an array. peters@acf. I would like a hint but nothing overboard. I have already compiled the program that reads numbers. Write a c program to find out NCR factor of given number. Source Code Write a program to find the median value of a vector of floating-point numbers. It's quick & easy. A C++ Function to Find Median of a Data Array Glancing over a book on Statistics, I realized that I need some implementation to find the median of an array, and I like to implement it by myself. Find the range and calculate standard deviation to compare and evaluate variability of data sets. Median filter Original Array of Complex Numbers The Median Element of the Array is: 5+j5 Show transcribed image text 1. 6/10/2013 · C program to find mean, median, variance and standard deviation Write a C program to find mean, median, variance and standard deviation for given set of numbers. The median is the value separating the higher half of a data sample, a population, or a probability distribution, from the lower half. Note: This C Program for Finding Average of N Numbers is compiled with GNU GCC Compiler and written in gEdit Editor in Linux Ubuntu Operating System. /* this program is going to find the mean of 3 numbers. C Program to find sum of array elements using pointers, recursion & functions. C programming language, C program, C language 2 responses to “ Mean, Median and Mode c++ ” Facebook Chat. h> C Program to calculate Net Salary from basic salary (easy one) C program to find median ; No comments yet; Categories. Program for Mean and median of an unsorted array Given n size unsorted array, find its mean and median. C Code: #include <stdio. , "Date") for which a median is a reasonable concept. This program should accept array of integers and integers that indicate the number of elements in the array. calculate the range and the median for both intelligence and school performance d. It also shows you the cumulative frequency. This c programming code is used to find the mean, median, mode. Next the program will call a function that calculates the mean of the number of units. Find the median of the elements after inputting each element. Find The Median Value Of A Set Of Numbers. I have tried to figure it out on my own for almost 2 weeks now, I have Fast median search: an ANSI C implementation Nicolas Devillard - ndevilla AT free DOT fr July 1998 1 Introduction on big arrays or in any program for which strong reliability is a keyword. The median is the number in the middle of a set of numbers. May. Practice: Estimating mean and median in data displays. Mode Program In C. C Program To Find Median Of Array - The best free software for your. Enter values separated by commas such as 1, 2, 4, 7, 7, 10, 2, 4, 5. : Note that any algorithm that can find the median in O(n) time can also be used to find the k-th smallest element in the array, for any k between 1 and n, in O(n) time. Mean, or mean average, is used along with many other mathematical operations and is an important thing to know. Then, for each window - the set of pixels - find the median value. sort(b); if (A Find more on Program to find average of two numbers Or get search suggestion and latest updates. Arrays (array. c # list<int> to find the Mode and median please verify my account · Check the link given below https://blogs. I've finished it mostly I think, except for the median part. Next, divide the number of elements by two and round down to get an integer. The nightmare that is using JavaScript on HackerRank Thanks to this guy for some help with heap Calculator Use. It will also print the larger values and smaller values compared to the median. Mail Stop 5425 Washington, D. To compare 3-channel pixels we first convert them into 1-channel gray values. Write an algorithm to find the median of the array obtained after merging the above 2 …2/27/2007 · The median of five input numbers!!!. Program to find median of elements in an array. (c) Do the data provide any evidence that CEOs receive a premium (i. h> main() How to Find median in Arrays: For getting the median of input array. Write a C program to find the median and mode of an array of integers. This c programming code is used to find the mean, median, mode. calculate mean median mode c programming array. It is possible to calculate median in O(n) time instead. N], Find the Medians in this array without sorting it in an efficient …C program to find the mean,median and mode of a set of numbers #include<stdio. and either repeating the program or quitting. 11/29/2011 · Please help me to create a c program to find mode and median of an array. C. Program to find median of an array in C /** C program to calculate the median of * an array. length]; System. The statistics median is a quick measure to find a central location of a data sequence, list or any iterator. g. Example Code to Calculate Median To find the median of an unsorted array, we can make a min-heap in $O(n\log n)$ time for $n$ elements, and then we can extract one by one $n/2$ elements to get the C Program to Calculate Standard Deviation This program calculates the standard deviation of 10 data using arrays. Quick Median - A Partition. h> #include <stdlib. ValueIs there a way to get the min, max, median, and average of a list of numbers in a single command? Ask Question 90. arraycopy(A, 0, b, 0, b. No angles of Scalene Triangle are equal. The C program is successfully compiled and run on a Linux system. The parametric bootstrap goodness-of-fit procedure was an attempt to develop a general procedure, but was found to be biased for Cormack-Jolly-Seber (CJS) data . The other related links are, Calculate Range C++ :: Calculate Mode / Mean And Median Of Array Of Integers Mar 17, 2013. for calculating a typical salary, household income, home price, real-estate tax, etc. For the mode, on paper you have a list. C++ program to find median of binary search tree store the elemnts of tree in an array in sorted manner we do this by traversing the tree in inorder manner * Then we find the median by using nth_element */ template < typename T > struct Value { T value; //Data of node Value() C program code examples; C program to find average of all elements of array; C program to find average of all elements of array. Python Exercises, Practice and Solution: Write a Python program to find the median of three values. Mean of an array = (sum of all elements) / (number of elements) Median of a sorted array of size n is defined as below : It is middle element when n is odd and average of …To find the mean you add all the numbers together then divide by how many there are. LeetCode – Find Median from Data Stream (Java) Median is the middle value in an ordered integer list. For example, if there are five elements, 5/2 = 2 (integer arithmetic) so the median is dArray[2]. 'c' programming is developed in 1972 by dennis ritchie at "at & t bell's lab usa". Write a program to enter any three numbers and find out the middle number. The median is the middle value, which has exactly half of the values above it and the other half below it. import java. Program to find the mean, median, and mode of numbers. by Koscica Dusko. h> ← c program …The MEDIAN function sorts through the provided arguments to find the value that falls arithmetically in the middle of the group. 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So our numbers in order from least to greatest would be 94, 101, 101, 104, 108, 110. Find median of an array in c# Saturday, April 06, 2013 Muthu Nadar No comments public static int GetMedian(int[] Value) { decimal Median = 0; int size = Value


# Find all solutions to $x^2\equiv 1\pmod {91}$

I split this into $x^2\equiv 1\pmod {7}$ and $x^2\equiv 1\pmod {13}$.

For $x^2\equiv 1\pmod {7}$, i did: $$(\pm1 )^2\equiv 1\pmod{7}$$ $$(\pm2 )^2\equiv 4\pmod{7}$$ $$(\pm3 )^2\equiv 2\pmod{7}$$ Which shows that the solutions to $x^2\equiv 1\pmod {7}$ are $\pm1$.

For $x^2\equiv 1\pmod {13}$, i did: $$(\pm1 )^2\equiv 1\pmod{13}$$ $$(\pm2 )^2\equiv 4\pmod{13}$$ $$(\pm3 )^2\equiv 9\pmod{13}$$ $$(\pm4 )^2\equiv 3\pmod{13}$$ $$(\pm5 )^2\equiv {-1}\pmod{13}$$ $$(\pm6 )^2\equiv 10\pmod{13}$$Which shows that the solutions to $x^2\equiv 1\pmod {13}$ are $\pm1$.

Thus, I concluded that the solutions to $x^2\equiv 1\pmod {91}$ must be $\pm1$. I thought that $\pm1$ were the only solutions, but apparently I am incorrect! How do I go about finding the other solutions to this congruence?

• Hint: CRT. See link – user160069 Dec 3 '16 at 2:38

You have $x\equiv\pm1\mod7$ and $x\equiv\pm1\mod13$. For all the solutions, you have to consider the systems: $$x\equiv1\mod7$$ $$x\equiv1\mod13$$ and $$x\equiv-1\mod7$$ $$x\equiv1\mod13$$ and $$x\equiv1\mod7$$ $$x\equiv-1\mod13$$ and $$x\equiv-1\mod7$$ $$x\equiv-1\mod13$$ as each system will get you a valid answer. I think you only had the first and the last systems and that you only considered the cases where the signs were similar.

• Thank you! I figured out the other solutions to be $\pm 27$. – J. Doe Dec 3 '16 at 3:40

Hint: Consider the possibility that $x \equiv 1 \pmod 7$ but $x \equiv -1 \pmod {13}$, and so on. (In other words, your mistake was to assume that the $\pm1$ modulo $7$ was the same sign as $\pm1$ modulo $13$.)

Also note that for any prime $p$, if $x^2 \equiv 1 \pmod p$, then we can rewrite this as $$x^2 - 1 \equiv (x+1)(x-1) \equiv 0 \pmod p.$$

Thus we get $x \equiv \pm 1 \pmod p$, showing that it isn't necessary to run through all the values of $x^2$ to find the solution.

By CRT the solutions $$\,x\equiv \pm1\pmod 7,\ x\equiv \pm1\pmod{13}$$ combine to $$\,4\,$$ solutions mod $$91,\,$$ viz $$\,x\equiv (\color{#c00}{{\bf 1,1}}),\,(\color{#0a0}{-1,-1}),\,(1,-1),\,(-1,1)\,$$ mod $$(7,13),$$ cf. Remark below. The first $$2$$ have obvious solutions $$\,\color{#c00}{\bf 1}\,$$ and $$\,\color{#0a0}{-1}\,$$ so you need only solve the $$(1,-1)$$ case, then note $$(-1,1)\equiv -(1,-1)$$ is its negative, i.e. $$\,x\equiv 1\pmod{\!7},\,x\equiv -1\pmod{\!13}\,\Rightarrow\,-x\equiv -1\pmod{\!7},\ {-}x\equiv 1\pmod{\!13}$$

Remark  For more complex examples it is usually easier to solve the CRT system first for generic (symbolic) roots, then plug in the specific root values for all combinations, e.g. see here and here.

If $$\,m,n\,$$ are coprime then, by CRT, solving a polynomial $$\,f(x)\equiv 0\pmod{\!mn}\,$$ is equivalent to solving $$\,f(x)\equiv 0\,$$ mod $$\,m\,$$ and mod $$\,n.\,$$ By CRT, each combination of a root $$\,r_i\bmod m\,$$ and a root $$\,s_j\bmod n\,$$ corresponds to a unique root $$\,t_{ij}\bmod mn,\,$$ i.e.

$$\begin{eqnarray} f(x)\equiv 0\!\!\!\pmod{\!mn}&\overset{\,\,\rm CRT}\iff& \begin{array}{}f(x)\equiv 0\pmod{\! m}\\f(x)\equiv 0\pmod{\! n}\end{array} \\ &\,\,\iff& \begin{array}{}x\equiv r_1,\ldots,r_k\pmod{\! m}\phantom{I^{I^{I^I}}}\\x\equiv s_1,\ldots,s_\ell\pmod{\! n}\end{array}\\ &\,\,\iff& \left\{ \begin{array}{}x\equiv r_i\pmod{\! m}\\x\equiv s_j\pmod {\! n}\end{array} \right\}_{\begin{array}{}1\le i\le k\\ 1\le j\le\ell\end{array}}^{\phantom{I^{I^{I^I}}}}\\ &\overset{\,\,\rm CRT}\iff& \left\{ x\equiv t_{i j}\!\!\pmod{\!mn} \right\}_{\begin{array}{}1\le i\le k\\ 1\le j\le\ell\end{array}}\\ \end{eqnarray}\qquad\qquad$$


# Indefinite Integration

## Homework Statement

The value of $$\int_0^{1} (\prod_{r=1}^{n} (x+r))(\sum_{k=1}^{n} \frac{1}{x+k}) dx$$ equals:
a)n
b)n!
c)(n+1)!
d)n.n!

(Can someone tell me how to make bigger parentheses using latex?)

## The Attempt at a Solution

I know that the question becomes a lot easier if i put n=1 or 2 and then integrate. But i was wondering if there is any proper way to solve it. I can't go further after expanding the given expression.
$$\int_{0}^{1} (x+1)(x+2)......(x+n)(\frac{1}{x+1}+\frac{1}{x+2}+.....\frac{1}{x+n})dx$$
which is equal to
$$\int_{0}^{1} \sum_{r=1}^n \frac{(x+n)!}{x+r}$$
I am stuck now, i can't find any way further.

Any help is appreciated.

I still don't have any clue.

ehild
Homework Helper
What is the derivative of a product? What is the derivative of (x+1)(x+2)? and (x+1)[(x+2)(x+3)] ?

ehild

What is the derivative of a product? What is the derivative of (x+1)(x+2)? and (x+1)[(x+2)(x+3)] ?

ehild

Derivative of (x+1)(x+2)=2x+3
Derivative of (x+1)(x+2)(x+3)=3x2+12x+11

I still don't get any idea how this would help?

ehild
Homework Helper
Do not simplify.

d/dx[(x+1)(x+2)]=(x+2)+(x+1)=(x+1)(x+2)/(x+1)+(x+1)(x+2)/(x+2)=[(x+1)(x+2)][1/(x+1)+1/(x+2)]

d/dx[(x+1)(x+2)(x+3)]=(x+2)(x+3)+(x+1)(x+3)+(x+1)(x+2)=[(x+1)(x+2)(x+3)][1/(x+1)+1/(x+2)+1(x+3)]

Do you see?

ehild

1 person
Wait, i guess i have got it, i will be back in a few hours with a solution.
Thanks for the help!

EDIT: Oops, seems like you posted just a few seconds before me.

ehild
Homework Helper
Wait, i guess i have got it, i will be back in a few hours with a solution.
Thanks for the help!

EDIT: Oops, seems like you posted just a few seconds before me.

Well, I looking forward to the solution

ehild

So here's the solution:
Let
$$t=(x+1)(x+2)(x+3).......(x+n)$$
$$\ln t=\ln(x+1)+\ln(x+2)+\ln(x+3)......+\ln(x+n)$$
$$\frac{1}{t}\frac{dt}{dx}=\frac{1}{x+1}+\frac{1}{x+2}+\frac{1}{x+3}......\frac{1}{x+n}$$
$$dt=(x+1)(x+2)(x+3)....(x+n)(\frac{1}{x+1}+\frac{1}{x+2}+\frac{1}{x+3}........+\frac{1}{x+n})dx$$
Therefore the original question becomes ∫dt, upper limit is (n+1).n! and lower limit is n!.
Solving, i get the answer as d) option.

Thanks for the help!

Mentallic
Homework Helper
By the way, since it seems like your little sub-question about the larger brackets was missed, I'll chime in on that.

I myself use /left( and /right) which looks like: $$\left(\frac{1}{1}\right)$$ but there are other more specific "fonts" you can use, which ranges from smallest to largest:

/big( $$\big(\frac{1}{1}\big)$$
/Big( $$\Big(\frac{1}{1}\Big)$$
/bigg( $$\bigg(\frac{1}{1}\bigg)$$
/Bigg( $$\Bigg(\frac{1}{1}\Bigg)$$

And there could be others, but that should about cover it.

ehild
Homework Helper
So here's the solution:
Let
$$t=(x+1)(x+2)(x+3).......(x+n)$$
$$\ln t=\ln(x+1)+\ln(x+2)+\ln(x+3)......+\ln(x+n)$$

$$\frac{1}{t}\frac{dt}{dx}=\frac{1}{x+1}+\frac{1}{x+2}+\frac{1}{x+3}......\frac{1}{x+n}$$
$$dt=(x+1)(x+2)(x+3)....(x+n)(\frac{1}{x+1}+\frac{1}{x+2}+\frac{1}{x+3}........+\frac{1}{x+n})dx$$

That is ingenious! You are really cool

Therefore the original question becomes ∫dt, upper limit is (n+1).n! and lower limit is n!.
Solving, i get the answer as d) option.

Thanks for the help!

Very good. You are better and better every day!

ehild

By the way, since it seems like your little sub-question about the larger brackets was missed, I'll chime in on that.

I myself use /left( and /right) which looks like: $$\left(\frac{1}{1}\right)$$ but there are other more specific "fonts" you can use, which ranges from smallest to largest:

/big( $$\big(\frac{1}{1}\big)$$
/Big( $$\Big(\frac{1}{1}\Big)$$
/bigg( $$\bigg(\frac{1}{1}\bigg)$$
/Bigg( $$\Bigg(\frac{1}{1}\Bigg)$$

And there could be others, but that should about cover it.


# Symmetric matrix as a sum of symmetric matrices

Let matrix $$M \in \mathbb{N}^{5 \times 5}$$ be symmetric with non-negative integer entries and zeros on the main diagonal and having the property that the row sums are equal to $$2r$$ for some $$r \geq 2$$. I want to prove that $$M$$ can be written as a non-negative integral linear combination of $$5 \times 5$$ symmetric matrices having non-negative integer entries with zero entries on the main diagonal and having the property that the row sums are equal to $$2$$. Is there way to prove this?

I tried with some simple examples and it seems to be correct.

• I need to prove that any $M$ (with the property listed) can be written as a non-negative integral linear combination of matrices with the property listed. $m$ is not specific here. – jack Sep 21 at 9:28
• I imagine this result is true for all dimensions, not just 5. – S. Dolan Sep 21 at 17:00

IDEA:

The row sums of $$M$$ are even, so there is either 0 or 2 or 4 odd numbers, and the diagonal entries are all zeros.

So if you have even entries, for example if you have that in $$M$$ the entries $$m_{1,5}=m_{5,1}= 8$$ then you will have in your linear combination $$4$$ multiplied by a matrix having all its entries on these rows are $$0$$ except $$a_{1,5}=a_{5,1}=2$$, and so on ...

Also if there is $$2$$ odd entries, for example $$m_{1,5}=m_{5,1}=5$$ and $$m_{1,4}=m_{4,1}=7$$, then in your linear combination you'll have $$2$$ multiplied by a matrix having $$a_{1,5}=a_{5,1}=2$$ and all other entries on these rows are zeros, and $$3$$ multiplied by a matrix having $$b_{1,4}=b_{4,1}=2$$ and all other entries on these rows are zeros, and a matrix having $$c_{1,4}=c_{4,1}=c_{1,5}=c_{5,1}=1$$ and all other entries on these rows are zeros, and so on...

I hope you can reach a proof from this hint and I am sorry because I am not able to write the whole proof now.

Yes, this conjecture is correct.

First consider the following matrices.

$$X_1=\begin{pmatrix}0&0&1&1&0 \\0&0&1&0&1\\1&1&0&0&0\\1&0&0&0&1\\0&1&0&1&0\\\end{pmatrix}$$, $$X_2=\begin{pmatrix}0&0&1&0&1 \\0&0&1&1&0\\1&1&0&0&0\\0&1&0&0&1\\1&0&0&1&0\\\end{pmatrix}$$, $$X_3=\begin{pmatrix}0&0&1&0&1 \\0&0&0&2&0\\1&0&0&0&1\\0&2&0&0&0\\1&0&1&0&0\\\end{pmatrix}.$$

If we could subtract one of the $$X_i$$ from $$M$$ without making any entry negative, then we would leave a matrix of the same type as $$M$$ but with $$r$$ reduced by 1. This process would therefore reduce $$M$$ to zero as we require.

If all off-diagonal entries of $$M$$ were non-zero we could subtract $$X_1$$. So, without loss of generality, we can suppose that $$M_{12}=0$$.

CASE 1. If no row of $$M$$ has three or more zeroes.

Without loss of generality we can suppose that $$M$$ has the following form where $$A,B,C,D,E,F$$ are all non-zero.

$$M=\begin{pmatrix}0&0&A&B&C \\0&0&D&E&F\\A&D&0&.&.\\B&E&.&0&.\\C&F&.&.&0\\\end{pmatrix}$$

If any of the dots were non-zero then a matrix like $$X_1$$ can be subtracted from $$M$$. Otherwise, all the dots are zero. Then $$A+D=B+E=C+F=2r. (*)$$ However, we would then have the sum of the first two rows equal to $$6r$$, a contradiction.

CASE 2. If a row of $$M$$ has four zeroes.

Without loss of generality we can suppose that $$M$$ has the following form.

$$M=\begin{pmatrix}0&0&0&0&2r \\0&0&A&B&0\\0&A&0&C&0\\0&B&C&0&0\\2r&0&0&0&0\\\end{pmatrix}$$

Then simple algebra shows that $$A=B=C=r$$ and $$M$$ is the sum of $$r$$ copies of $$\frac {1}{r} M$$.

CASE 3. If $$M$$ has three zeroes in a row but that no row has four or more zeroes.

Let $$M=\begin{pmatrix}0&0&.&.&. \\0&0&.&.&.\\.&.&0&F&G\\.&.&F&0&H\\.&.&G&H&0\\\end{pmatrix}$$

CASE 3(a). If $$F=G=H=0$$

Then the method used for $$(*)$$ gives a contradiction.

CASE 3(b). If two of $$F,G,H$$ are $$0$$, w.l.g. $$F=G=0$$

Then either $$X_1$$ or $$X_2$$ can be subtracted.

CASE 3(c). If only one of $$F,G,H$$ is $$0$$, w.l.g $$F=0$$.

Since every row has at least two non-zero elements, the possibilities for $$M$$ are, without loss of generality, as follows, where $$+$$ indicates a positive element.

$$\begin{pmatrix}0&0&+&0&+ \\0&0&0&+&+\\+&0&0&0&G\\0&+&0&0&H\\+&+&G&H&0\\\end{pmatrix}$$ and $$\begin{pmatrix}0&0&+&.&. \\0&0&+&.&.\\+&+&0&0&G\\.&.&0&0&H\\.&.&G&H&0\\\end{pmatrix}$$

Considering row totals for Rows 3 and 4 of the LH matrix shows that $$M_{42}>1$$ and so $$X_3$$ can be subtracted. For the RH matrix, one of $$X_1, X_2$$ can be subtracted unless the matrix is $$\begin{pmatrix}0&0&+&+&0 \\0&0&+&+&0\\+&+&0&0&G\\+&+&0&0&H\\0&0&G&H&0\\\end{pmatrix}$$

Then the sum of elements in Rows 1 and 2 must equal the sum of elements in Rows 3 and 4. Therefore $$G=H=0$$, a contradiction.

CASE 3(d). If $$F,G,H$$ are all non-zero.

Then a matrix like either $$X_1$$ or $$X_2$$ can be subtracted since the top two rows contain at least 4 non-zero elements.

• With hindsight, CASE 1 can be subsumed into CASE 3(a) to shorten this proof a little. – S. Dolan Sep 21 at 16:41

A graph theoretic proof

Consider $$M$$ to be the adjacency matrix for a graph $$G$$. Then $$G$$ is a graph with no loops and 5 vertices, each of degree $$2r$$. We are required to prove that:

$$G$$ contains one or more simple cycles (i.e. containing no repeated vertex) which include every point precisely once.

Since all vertices have even degree, each component of $$G$$ must have an Eulerian cycle and therefore each vertex of $$G$$ is in a simple cycle and at least one of these simple cycles has length at least 3.

We can suppose every vertex of $$G$$ is joined to more than one vertex.

Suppose on the contrary that vertex 1 was only joined to vertex 2. By the equality of degrees, vertices 1 and 2 then form a component of $$G$$. The points 3,4,5 of the other component must then be joined in pairs and $$G$$ has simple cycles (1,2) and (3,4,5).

We can suppose $$G$$ only has simple cycles of lengths 2 and 3

There is nothing to prove if $$G$$ has a simple cycle of length 5, so suppose $$G$$ has the cycle (1,2,3,4). If vertex 5 were connected to two adjacent points in this cycle, say vertices 1 and 2, then (1,5,2,3,4) would be a simple cycle. So we can suppose 5 is only joined to, say, 1 and 3. Then (1,5,3,4) and ((1,2,3,5) are cycles and so 2 and 4 are also only joined to 1 and 3. There are then $$6r$$ edges from vertices 2,4,5 to 1 and 3, contradicting the degrees of 1 and 3.

Now consider a cycle of length 3, say (1,2,3). If vertex 4 were joined to two of these points then we would have a cycle of length 4. So vertex 4 is joined to vertex 5 and, say, vertex 1. Vertex 5 also has to be joined to precisely one of 1,2,3. This must be the same vertex 1, otherwise we would obtain a simple cycle of length 5. Vertices 4 and 5 can only be joined by $$1$$ edge (and similarly vertices 2 and 3 are only joined by $$1$$ edge) otherwise we would have simple cycles (1,2,3) and (4,5). Therefore there are $$4(2r-1)$$ edges from vertices 2,3,4,5 to vertex 1. Then $$8r-4=2r$$, which is impossible.

• I don't think I follow your 'We are required to prove that'. Perhaps you meant something like: $G$ contains a collection of simple cycles such that each vertex belongs to exactly one of these cycles. Is that right? – Fimpellizieri Sep 22 at 21:04
• Yes, that is correct. – S. Dolan Sep 22 at 21:06
• I think your answer would benefit from better writing. I think you are trying to handle cases here rather than actually going 'without loss of generality' and it makes parsing the answer confusing. $G$ can very trivially have a simple cycle of length $5$, but I guess you're dismissing it as the easiest case for which what we need to prove is basically handed to us on a platter. – Fimpellizieri Sep 22 at 21:18
• If G has a simple cycle of length 5 then, by definition, G contains a simple cycle which include every point precisely once. – S. Dolan Sep 22 at 21:24
• Yes, that is my point. It's not 'Without loss of generality, we can suppose $G$ has no simple cycle of length $5$' and it most definitely isn't 'By definition, $G$ has no simple cycle of length $5$'. It's just that when $G$ has a simple cycle of length $5$, there is nothing to prove, so this case is done. Your answer is tackling cases, but is structured and written as something else, which makes it harder to parse. – Fimpellizieri Sep 22 at 21:28

This is my take on the graph theoretic approach by S. Dolan. Credits go to him.

Let $$M$$ be the adjacency matrix of a graph $$G$$, so that $$G$$ is a graph on $$5$$ vertices with no loops and such that each vertex has degree $$2r$$.

Claim: It is enough to prove that $$G$$ contains a collection of simple cycles such that each vertex belongs to exactly one of these cycles.

Indeed, if $$H$$ is the $$G$$-subgraph that corresponds to this collection, then each vertex of $$H$$ has degree exactly $$2$$. We can thus take the adjacency matrix of $$H$$ as one of the matrices in our integral linear combination that adds up to $$M$$.

We are hence left with the same problem, except now the rows of $$M$$ add up to $$2(r-1)$$; equivalently, except the vertices of $$G$$ have degree $$2(r-1)$$. Induction then takes care of finding the other matrices in our integral linear combination.

Let $$v_1,v_2,v_3,v_4$$ and $$v_5$$ be the vertices of $$G$$. We will prove the claim by studying different cases.

## Case $$(1)$$: $$G$$ has a vertex which is joined to only one other vertex.

Assume without loss of generality that $$v_1$$ is joined only to $$v_2$$. Since all vertices have equal degrees, $$v_2$$ must also be joined only to $$v_1$$.

This implies that if some $$u$$ in $$\{v_3, v_4,v_5\}$$ were also joined to only one other vertex $$w$$, we would have $$w \in \{v_3, v_4,v_5\}\setminus\{u\}$$. Equality of degrees would then require that the remaining vertex be joined to itself, in contradiction to $$G$$ having no loops. It follows that each of $$\{v_3, v_4,v_5\}$$ is connected to the other two.

The collection of simple cycles $$\{(v_1,v_2), (v_3,v_4,v_5)\}$$ satisfies the claim and handles this case.

## Case $$(2)$$: Each vertex of $$G$$ is joined to at least two other vertices.

By Veblen's Theorem, $$G$$ can be written as the union of disjoint simple cycles. We break down into subcases.

$$\qquad (2.1)$$: $$G$$ has a simple cycle of length $$5$$.

In this case, the claim is obviously satisfied and there is nothing to prove.

$$\qquad (2.2)$$: $$G$$ has a simple cycle of length $$4$$.

Let $$(v_1,v_2,v_3,v_4)$$ be the simple cycle.

$$\qquad\qquad(2.2.1)$$: $$v_5$$ is joined to two vertices that are adjacent in the cycle.

If that were the case, then we could enlarge the simple cycle to have length $$5$$. This reduces the problem to case $$(2.1)$$, which we have already handled.

$$\qquad\qquad(2.2.2)$$: No two vertices adjacent in the cycle are both joined to $$v_5$$.

Without loss of generality, suppose $$v_5$$ were joined to vertices $$v_1$$ and $$v_3$$. Notice that $$v_5$$ cannot be joined to any other vertex.

If $$v_2$$ and $$v_4$$ weree joined, then we could take the simple cycle $$(v_1,v_2,v_4,v_3,v_5)$$, handled by case $$(2.1)$$.

If $$v_2$$ and $$v_4$$ were not joined, then each of $$\{v_2, v_4,v_5\}$$ would be joined only to $$v_1$$ and $$v_3$$. This would contradict each vertex having equal degree, and finishes this case.

$$\qquad (2.3)$$: $$G$$ has a simple cycle of length $$3$$.

Let $$(v_1,v_2,v_3)$$ be the simple cycle.

$$\qquad\qquad(2.3.1)$$: One of $$\{v_4, v_5\}$$ is joined to two vertices in the cycle.

If that were the case, then we could enlarge the simple cycle to have length $$4$$. This reduces the problem to case $$(2.2)$$, which we have already handled.

$$\qquad\qquad(2.3.1)$$: No two vertices in the cycle are both joined to one of $$\{v_4, v_5\}$$.

Without loss of generality, suppose $$v_4$$ were joined to vertices $$v_1$$ and $$v_5$$. Notice that $$v_4$$ cannot be joined to any other vertex.

If $$v_5$$ were joined to one of $$\{v_2,v_3\}$$, then we could enlarge the simple cycle to have length $$5$$, handled by case $$(2.1)$$. Indeed, if $$v_5$$ were joined to $$v_2$$ we would have the cycle $$(v_1,v_3, v_2,v_5,v_4)$$ and if $$v_5$$ were joined to $$v_3$$ we would have the cycle $$(v_1,v_2,v_3,v_5,v_4)$$.

Suppose instead that $$v_5$$ were joined only to $$v_1$$ and $$v_4$$.
If there were more than edge joining $$v_4$$ and $$v_5$$, the collection $$\{(v_1,v_2,v_3),(v_4,v_5)\}$$ satisfies the claim.
If there were more than edge joining $$v_2$$ and $$v_3$$, the collection $$\{(v_1,v_4,v_5),(v_2,v_3)\}$$ satisfies the claim.
If $$v_4$$ were joined to $$v_5$$ by a single edge and $$v_2$$ were also joined to $$v_3$$ by a single edge, then $$v_1$$ would have degree $$4(2r - 1) > 2r$$, which contradicts our hypotheses and concludes this case.

$$\qquad (2.4)$$: $$G$$ has no simple cycle of length $$3$$ or more.

We show that this case is impossible.

$$G$$ does not have loops, so in this case it would contain only cycles of length $$2$$. $$G$$ would then be bipartite: its vertices could be divided into two disjoint sets $$U$$ and $$V$$ such that every edge of $$G$$ joins a vertex in $$U$$ to a vertex in $$V$$. But $$G$$ has $$5r$$ edges and one of $$\{U, V\}$$ has at most two vertices. This contradicts each vertex having degree $$2r$$ and finishes the last case.


# Number of ways an answers sheet can be fill with at least two consecutive answers?

A student is taking a $$12$$-question multiple choice test. Each question has $$5$$ choices. How many ways can the student fill out the answer sheet so there is at least one place where two consecutive answers are the same?

This is what I did:

Total possible ways the student can fill the answer sheet: $$5^{12}$$ (because he has $$5$$ options in the $$12$$ questions).

Total possible ways the student can fill the answer sheet with no consecutive answers: $$5 \cdot 4^{11}$$ (In the first question he can answer any of the $$5$$ options, however, in the remaining questions he can only answer $$4$$ options to avoid choosing consecutive answers.)

Total possible ways the student can fill the answer sheet with at least two consecutive answers: $$5^{12}- 5 \cdot 4^{11} = 223,169,105$$.

I doubt about the total possible ways the student can fill the answer sheet with no consecutive answers... is my logic right?

• Write out a detailed explanation of the $5\times4^{11}$. This should either convince you that you are right, or show you where you are wrong. Do this for the first part of your answer too. – David May 7 '19 at 5:17
• Sorry, I had a typo on my first statement. – Nico May 7 '19 at 5:33
• Yes, you are correct. – N. F. Taussig May 7 '19 at 9:05

You are correct, $$5\times 4^{11}$$ is the number of ways to fill out the test with no two consecutive equal answers, so $$5^{12}-5\times 4^{11}$$ is the number of ways to fill out the test so that there are two consecutive equal answers.

I am trying to do this question and am getting a different answer. Instead of making a new thread, I hope I am allowed to post my response and see where people think I'm going wrong here. If not please let me know and I will delete.

Denote the answers of the multi-choice test to be A_1,A_2,..., A_12.

Now, for us to have consecutive answers be the same it must be that A_1 = A_2 or A_2 = A_3 or .... or A_11 = A_12. Denote this pair of questions with the same answer to be A_s.

Now, consider a new set which contains 11 elements: A_s, and our other 10 remaining answers to questions. Certainly, we can position A_s anywhere, as remember that we have set A_s to be the two consecutive answers already. Thus, we have no conditions on the remaining choices. We can re-arrange this set 11! different ways (as it is simply the re-arrangements of a set with 11 elements where order matters), and each element can have 5 different options.

Thus, we have 11!x(5 x 11) = 21,954,424,000 different options.

My answer seems way too big

• You have not accounted for the possibility that there is more than one pair of consecutive answers that are the same. To do so, you could apply the Inclusion-Exclusion Principle, but with $11$ possible pairs of consecutive answers that could be the same, that could prove exceedingly difficult. – N. F. Taussig May 20 '19 at 0:02


MAT334-2018F > MAT334--Lectures & Home Assignments

About the definition of Argument (in book)

(1/2) > >>

Ende Jin:
I found that the definition of "arg" and "Arg" in the book is different from that introduced in the lecture (exactly opposite) (on page 7).
I remember in the lecture, the "arg" is the one always lies in $(-\pi, \pi]$
Which one should I use?

Victor Ivrii:

--- Quote ---Which one should I use?
--- End quote ---
This is a good and tricky question because the answer is nuanced:
Solving problems, use definition as in the Textbook, unless the problem under consideration requires modification: for example, if we are restricted to the right half-plane  $\{z\colon \Re z >0\}$ then it is reasonable to consider $\arg z\in (-\pi/2,\pi/2)$, but if we are restricted to the upper half-plane  $\{z\colon \Im z >0\}$ then it is reasonable to consider $\arg z\in (0,\pi)$ and so on.

Ende Jin:
I am still confused. Let me rephrase the question again.
In the textbook, the definition of "arg" and "Arg" are:
$arg(z) = \theta \Leftrightarrow \frac{z}{|z|} = cos\theta + isin\theta$
which means $arg(z) \in \mathbb{R}$
while
$Arg(z) = \theta \Leftrightarrow \frac{z}{|z|} = cos\theta + isin\theta \land \theta \in [-\pi, \pi)$
which means $Arg(z) \in [-\pi, \pi)$

While in the lecture, as you have introduced, it is the opposite and the range changes to $(-\pi, \pi]$ instead of $[-\pi, \pi)$ (unless I remember incorrectly):
Arg is defined to be
$Arg(z) = \theta \Leftrightarrow \frac{z}{|z|} = (cos\theta + isin\theta)$
which means $arg(z) \in \mathbb{R}$
while arg is
$arg(z) = \theta \Leftrightarrow \frac{z}{|z|} = cos\theta + isin\theta \land \theta \in (-\pi, \pi]$

I am confused because if I am using the definition by the book,
when $z \in \{z : Re (z) > 0\}$
then $arg(z) \in (-\frac{\pi}{2} + 2\pi n,\frac{\pi}{2} + 2\pi n), n \in \mathbb{Z}$

Victor Ivrii:

BTW, you need to write \sin t and \cos t and so on to have them displayed properly (upright and with a space after): $\sin t$, $\cos t$ and so on

Ende Jin:
Thus in a test/quiz/exam, I should follow the convention of the textbook, right?


Differential Equations

Consider the differential equation: dy/dt=y/t^2
a) Show that the constant function y1(t)=0 is a solution.
b)Show that there are infinitely many other functions that satisfy the differential equation, that agree with this solution when t<=0, but that are nonzero when t>0 [Hint: you need to define these functions using language like " y(t)=...when t<=0 and y(t)=...when t>0 and "]
c) Why doesn't this example contradict the Uniqueness Theorem?

I'm trying to do part b and after I separated and integrated I got
ln|y|=(-1/t)+C
I'm not sure if I can get C with the solution they gave in part a)y1(t)=0.
Anyways, I get y(t)=Ce^-(1/t). I don't know where to go from there.

1. 👍
2. 👎
3. 👁
1. dy/y = dt/t^2
ln y = -1/t + c
y = e^(-1/t+c) = e^c e^(-1/t)
=C e^(-1/t) agree

C can be anything so
what if C = 0 ?
then y(t) = 0 for all t

if C = 0 for t</= 0 then C can be anything at all for t>0

1. 👍
2. 👎
2. If y(t) depends on what C is, then how this equation doesn't contradict the uniqueness theorem if it has many solutions?

1. 👍
2. 👎
3. Because the general solution contains an an arbitrary constant C. The value of C depends on your boundary conditions, for example if y =5 at t = 2
then
5 = C e^-(1/2)
5 = C (.606)
C = 8.24

1. 👍
2. 👎
4. scroll down through this:
http://hyperphysics.phy-astr.gsu.edu/hbase/diff.html

1. 👍
2. 👎

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