# A group of order $2n$ has an element of order $2$

I'm having problems proving this. The full question is:

"Let $G$ be a group which order is a pair number. Show that $G$ has an element of order $2$".

Can anyone give me a hint?

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If $x\neq x^{-1}$, remove both $x$ and $x^{-1}$ from the set $G$. Repeat as many times as you can. How many elements remain in the end? What can you say about all but one of them? –  Jyrki Lahtonen Mar 27 '12 at 19:16
Ohh, I think I got this. If it is as stated, then every element $g$ of $G$ is such that $|g|>2$. But $|g|=|g^{-1}|$, and since the neutral element $e$ of $G$ is the only one element with order 1, that leaves an odd number of elements left on the group. If we separate in pairs each element and its inverse, there will be only one element left - and this element "$h$" must be its self-inverse, that is, $h^2=e$, which proves the question. –  Marra Mar 27 '12 at 19:16
It follows from Cauchy's theorem but I like @Balin's answer better. –  MJD Mar 27 '12 at 19:17
@GustavoMarra, not only one element left. See my answer. –  lhf Mar 27 '12 at 19:45
Yeah, not only one element left. But we're supposing that every element of G has order greater than 2 (in what I said). That supposition leads to an absurd. –  Marra Mar 27 '12 at 22:43

If there is no element of order 2 then show that $G$ has odd number of elements. (Hint: think of elements and inverses)

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Hint $\$ Inversion $\rm\:x\to x^{-1}\:$ is an involution $\rm\: (x^{-1})^{-1} = x,$ so the cycles (orbits) of this permutation partition $\rm\:G\:$ into orbits of length $2$ or $1$. Since $\rm\:|G|\:$ is even so too is the number of length $1$ orbits, i.e. fixed points $\rm\:a = a^{-1};\:$ these include $\rm\:a = 1,\:$ hence, having even cardinality, must include at least one other fixed-point, necessarily of order $2$ by $\rm a^{-1}=a\:$ $\Rightarrow$ $\rm\:a^2 = 1$ but $\rm\:a\ne 1$.

For an analogous application of orbit decomposition (without parity) see my prior answer today on Wilson's theorem for groups.

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Partition $G = E \cup A \cup B$, where $E=\{e\}$, $A=\{x \in G : x = x^{-1}, x\ne e\}$, $B=\{x \in G : x \ne x^{-1}\}$. Then $E$ has one element and $B$ has an even number of elements because $B$ is invariant under inversion. Since $G$ has an even number of elements, $A$ must have an odd number of elements. In particular, $A$ has at least one element. Every element in $A$ has order 2.

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Two ways to represent ex falso?

According to ProofWiki, the principle of explosion says that:

$$p \implies (\lnot p \implies q)$$

but sometimes it's also stated as

$$(p \land \lnot p) \implies q$$

Can one be converted to the other? How do you prove or show these?

If we know that $(p \implies q) \equiv \lnot p \lor q$ by definition, then substituting the 2nd representation:

$$((p \land \lnot p) \implies q) \equiv \lnot(p \land \lnot p) \lor q \equiv \lnot p \lor p \lor q$$

Is this just true due to law of excluded middle? Like either $p$ or $\lnot p$ is true, i.e. $p \lor \lnot p \equiv \text{T}$ before we even get to whatever $q$ is. Is that why we're able to say it?

Then the first representation is weird too:

$$(p \implies (\lnot p \implies q)) \equiv (\lnot p \lor (\lnot p \implies q)) \equiv (\lnot p \lor (p \lor q)) \equiv \lnot p \lor p \lor q$$

I mean am I even on the right track here? How do you normally get these without "working backwards" and showing that they happen to be equivalent? Is there a more intuitive derivation of both of these?

• Yes, the two are equivalent. – Namaste Apr 5 '18 at 18:50
• @amWhy Yes but I am asking something different than that – user525966 Apr 5 '18 at 18:52
• @user525966 Those are just two ways to write that a false premise implies any conclusion. – dxiv Apr 5 '18 at 18:56
• @Taroccoesbrocco you say the equivalence has nothing to do with excluded middle (and i don’t think it does) but you use the non intuitionistically valid demorgan law in your proof. – spaceisdarkgreen Apr 5 '18 at 19:28
• @user525966 The “reason” behind the classical validity of a proposition can be difficult to untangle cause it all blends together. Intuitionistic logic is an example where excluded middle is not valid but explosion is. So inasmuch as intuitionistic logic is philosophically coherent for accepting one and not the other (there is another called “minimal logic” that rejects both), LEM cannot be the “reason” for explosion. – spaceisdarkgreen Apr 5 '18 at 19:49

The formulas $p \to (\lnot p \to q)$ and $(p \land \lnot p) \to q$ are equivalent because they are particular instances of a more general fact: the formulas $p \to (r \to q)$ and $(p \land r) \to q$ are equivalent. You can prove that checking that the two formulas actually have the same truth table. An alternative proof is the following:

\begin{align} p \to (r \to q) &\iff \lnot p \lor (\lnot r \lor q) &\text{by definition of } \to \\ &\iff (\lnot p \lor \lnot r) \lor q &\text{by associativity of } \lor \\ &\iff \lnot(p \land r) \lor q &\text{by De Morgan law} \\ &\iff (p \land r) \to q &\text{by definition of }\to \end{align}

• @user525966 - Please, see my last edit. – Taroccoesbrocco Apr 5 '18 at 18:57

The two statements you post are logically equivalent.

\begin{align} p \to (\lnot p \to q) &\equiv \lnot p \lor (p \lor q) \tag{definion of \to}\\ \\ &\equiv (\lnot p \lor p) \lor q \tag{associativity of \lor}\\ \\ &\equiv \lnot (p \land \lnot p) \lor q \tag{DeMorgan's}\\ \\ &\equiv (p\land \lnot p)\to q\tag{definition of \to}\end{align}

You were heading in right direction, starting instead from $(p\land \lnot p)\to q$ and moving toward $p \to (\lnot p \to q)$. You just stopped too early. Note each direction can be proven, because we have a string of equivalencies: so to reverse the direction, just reverse the order of the steps.

\begin{align} (p\land \lnot p)\to q &\equiv \lnot (p \land \lnot p)\lor q\tag{definition of \to}\\ \\ &\equiv (\lnot p \lor p) \lor q\tag{DeMorgan's}\\ \\ &\equiv \lnot p\lor (p \lor q)\tag{associativity of \lor}\\ \\ &\equiv p\to (\lnot p \to q)\tag{definition of \to} \end{align}

First of all, we have $$p\to (q\to r) \vdash (p \land q) \to r\tag{importation}$$ and we have $$(p\land q) \to r \vdash p \to (q\to r)\tag{exportation}$$ Your question is a specific case of these, where $q = \lnot p$. See this link for more information.

• How / why are you using $\vdash$ here? – user525966 Apr 5 '18 at 20:07
• The turnstyle, $\vdash$ is the syntactic entailment operator. $\phi \vdash \psi$, symbolises that $\psi$ is derivable from $\phi$ ( or "...can be proven from..." ) using the syntactic transformations of some logic system. @user525966 – Graham Kemp Apr 6 '18 at 1:05

To get pernickety, we need to distinguish the principle about entailment what is usually called Explosion, i.e.

$(A \land \neg A) \vdash B$,

from the following principle that a conditional with a contradictory antecedent is always derivable:

$\vdash (A \land \neg A) \to B$.

You can have logics with one principle and not the other.

• What's the difference between $\vdash$ and $\to$? I know the former is more "is derivable from" but I always thought that was what implies was, really – user525966 Apr 5 '18 at 19:42
• @user525966 yes one is “derivable from” and is a metalogical relation (ie you use it to talk about the formal system) and the other is “implies” and is a logical connective (used within formal statements in the logic). They do oftentimes correspond exactly in the sense that we have $\vdash A\to B$ iff we have $A\vdash B$. One direction directly follows from the inference rule modus ponens if present. The other direction is called a deduction theorem and can be proved to apply to (for instance) one’s favorite deductive system for first order logic. – spaceisdarkgreen Apr 5 '18 at 20:05
• @user525966 not sure why I forgot to add this yesterday but in natural deduction systems where this holds, the two directions are unified as introduction and elimination rules for implication. Whereas, the “deduction theorem” direction being a nontrivial metatheorem, as described above, is typical in Hilbert-style systems. – spaceisdarkgreen Apr 7 '18 at 2:07
• What is metalogic, metatheory, exactly? – user525966 Apr 7 '18 at 2:39
• @user525966 (You have to @ me so I get pinged). In logic, it is useful to distinguish between statements/proofs/theorems in formal systems and statements and results about formal systems (usually proven informally, using "ordinary mathematical reasoning", though we could formalize this too). The latter are called "meta_". "$\vdash A$" means "there is a formal proof of the statement $A$ in the deductive system under consideration"... that's a meta-statement about the system (and if true, $A$ is a (regular old) theorem of the system). See also: en.wikipedia.org/wiki/Metatheorem – spaceisdarkgreen Apr 7 '18 at 3:06


In Googlesheets and Excel, =PERCENTILE.EXC(A2:A234,0.25) returns quartile 1 for the scores in cells A2 through A234 (our 233 reaction times). The lower quartile $Q_1$ is the point such that the area up to $Q_1$ is one quarter of the total area. The other name for quartile is basically cuts. Use MathJax to format equations. To work out where the quartiles are, we need to use our equations again: $$\frac{(40)}{4}={10}$$ so the LQ is the 10th number. What is the third quartile when there are no data above the median? It is possible to fit up to six Distribution Functions simultaneously from a total of 19 continuous and discrete distributions. The "Interquartile Range" is from Q1 to Q3: To calculate it just subtract Quartile 1 from Quartile 3, like this: Example: The Interquartile Range is: Q3 − Q1 = 7 − 4 = 3. There are several quartiles of an observation variable. Thanks for contributing an answer to Stack Overflow! 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Do you find the quartiles and medians by dividing up the bars (e.g. Histogram vs. A histogram is used in statistics to plot grouped data. We came across quartiles in Chapter 1 when discussing measures of variation for skewed data. It can be helpful to plot two variables in the same boxplot to understand how one affects the other. Fig. Hope this helps. To find the number of Q1 for grouped data, use n/4. But avoid …. Histogram. $$\frac{3(40)}{4}={30}$$ so the UQ is the 30th number. This just means that you're assuming each value in the histogram is distributed evenly in each class - the proportion of how many items into its class the median is = the proportion of how far the value of the median is into the range of its class. The upper quartile is the value which is three quarters of the way into our data: $$\frac{3}{4}$$ of the (number of values+ 1) or $$\frac{3{(11+1)}}{4}$$ = 9. Biology, 23.07.2019 00:30. Funnily enough, this use to be really easy for me. Not Helpful 6 Helpful 15. b) Find the lower quartile of the scores. 17) The histogram shows information about the times taken by some students to finish a puzzle. It is possible to fit up to six Distribution Functions simultaneously from a total of 19 continuous and discrete distributions. So Q1= 64+64/2=64 and Q3= 77+81/= 79. Primary Study Cards. Question. These were defined as the data values which had: 1 4 of the data values below it: lower quartile 2 4 of the data values below it: median 3 4 of the data values below it: upper quartile The lower quartile was the 2nd number in the list, so the upper quartile must be the 6th number in the list ($$2 \times 3 = 6$$). Clash Royale CLAN TAG #URR8PPP Also on Histogram of both the variables Ozone and Wind, we can see that there is a gap between observations at extreme i.e. Quartiles: lower quartile = 2 upper quartile = 11 highest value = 15 Median = 7 Lowest value = 0 Hence our boxplot looks like this: /**/ Like stem-and-leaf diagrams, these diagrams are best used when we have a relatively small set of data or when the data has already been summarised into its median and quartiles. You are required to calculate all the 3 quartiles.Solution:Use the following data for the calculation of quartile.Calculation of Median or Q2 can be done as follows,Median or Q2 = Sum(2+3+4+5+7+8+10+11+12)/9Median or Q2 will be –Median or Q2 = 7Now since the number of observations is odd which is 9, the median would lie on 5th position which is … a) Complete the frequency table for this information. PC ATX12VO (12V only) standard - Why does everybody say it has higher efficiency? Problem. Reading from the graph, the lower quartile is 38. So the interquartile range is 38 - 15 = 23. The lower quartile $Q_1$is the point such that the area up to $Q_1$is one quarter of the total area. The median or Q1 will be in the first class whose cumulative frequency includes their number. Median, which is middle quartile tells us the center point and upper and lower quartiles tell us the spread. The IQR can be calculated by subtracting the lower quartile from the upper. To do the lower quartile, the same applies, you just need to half the median, so you'd go to 20 on the y-axis. On box plot, how many quartiles can we see? You can use many of the other features of the quantile function which we described in our guide on how to calculate percentile in R.. The result is 811.5. Finding Interquartile Range from Cumulative Frequency Histogram Polygon 1. 400. Are the longest German and Turkish words really single words? Upper quartile (middle value of the upper half) = 36. a) Complete the frequency table for this information. A box plot displays the five-number summary for a univariate data set, giving a quick impression of the distribution. Example: Find the median, lower quartile, upper quartile, interquartile range and range of the following numbers. Reading from the graph, the lower quartile is 38. Practice Questions; Post navigation. 1 st quartile or lower quartile basically separate the lowest 25% of data from the highest 75%. 3 rd quartile or the upper quartile separate the highest 25% of data from the lowest 75%. "Data" is a plural noun; the singular form is "datum." Histograms can be drawn with regular or irregular class intervals and with mean, median, mode, lower and upper quartile values displayed. Below is the comparison of a Histogram vs. a Box Plot. Cluster Duck. A quartile divides the set of observation into 4 equal parts. In order to calculate this value we must first understand what the lower quartile, median and upper quartile are: The lower quartile is the value at the first quarter (once your data has been put in order). Thanks! a) Complete the frequency table for this information. The upper quartile $Q_3$is the point such that the area up to $Q_3$is three quarters of the total area. GCSE Revision Cards. Thanks! Questions in other subjects: Mathematics, 23.07.2019 00:30. 5.3.3. I will use a simple dataset to learn how histogram helps to understand a dataset. LQ is $$\frac{(7+1)}{4}$$ = the second number. What is the highest road in the world that is accessible by conventional vehicles? 34 / 50 Marks The Maths GCSE test scores of 280 students are shown in the histogram below. The lower quartile, or Q1, is the value which lies 1/4th from the lowest value. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Please help! 3 4 5 6 7 8 9 10 0 5 10 15 20 25 30 x c.f What prevents a government from taxing its citizens living abroad? To find these from the graph we draw two lines across from the vertical axis, one at 10 and one at 30. How to connect a flex ribbon cable to a screw terminal block? Example 5: Quartiles, Quintiles, Deciles, Percentiles & Many More. In its broadest sense, Statistics is the science of drawing conclusions about the world from data. Next Bar Charts, Pictograms and Tally Charts Practice Questions. Sampling data - Intermediate & Higher tier - WJEC, Cumulative frequency – Intermediate & Higher tier - WJEC, Home Economics: Food and Nutrition (CCEA). The interquartile range is the difference between the lower quartile and the upper quartile. 3) Median score: Lower quartile: Submit Answer. The Lower quartile or 1st quartile has four values as well as the higher quartile or 3rd quartile has four values. [The questions ask which statement is true, like "the lower quartile value of the number of hits made by the gold team is equal to the lower quartile value of the number of hits made by the blue team]. Get Answer. Further, they could be given as the upper quartile or the lower quartile. Hope this helps. Therefore Interquartile range= 79 … Can you suggest how can I find first and third quartiles and median after building histogram. The. The upper quartile is 15.6346 years . Data come in many forms, most of which are numbers, or can be translated into numbers for analysis. c.) Which interval contains the upper quartile? There are 40 babies in the table, so to find the lower quartile, find $$\frac{1}{4}$$ of 40, which is the 10th value. Question. I want to compute the quartiles of X ∼ Γ(1,β)? Quartile Formula Quartiles are given as values dividing the complete list into quarters. The Corbettmaths video tutorial on median and quartiles from frequency tables and Histograms - linear interpolation Corbettmaths Videos, worksheets, 5-a-day and much more Relevance and Uses of Quartile Formula. 5-a-day Workbooks. The line in the middle shows the median of the distribution. There are 0 observations below the Lower Quartile(Q1) Fence(Whisker) There are 0 observations above the Upper Quartile(Q3) Fence(Whisker) Notes from above all Plots: From the above Boxplots we can see that in Ozone and Wind Variables there are Outliers (the red dots). 4.2.1. However, it would have had a massive effect on a normal range. It will cut the list into four equal parts. 12, 5, 22, 30, 7, 36, 14, 42, 15, 53, 25, 65 . Find the median of the following data 12, 8, 3, 15, 1, 9. What is 4? The upper quartile $Q_3$ is the point such that the area up to $Q_3$ is three quarters of the total area. The box encloses the interquartile range. In the example below, we’re going to use a single line of code to get the quartiles … Sometimes we have to estimate the interquartile range from a cumulative frequency diagram. In this case Quartile 2 is half way between 5 and 6: Q2 = (5+6)/2 = 5.5. Why does my advisor / professor discourage all collaboration? Not Helpful 4 Helpful 10. First quartile (Q 1 / 25th percentile) : also known as the lower quartile q n (0.25), is the median of the lower half of the dataset. 1. Acylinder and its dimensions are shown. From the histogram I have shown frequency on each bar to the last bar being 691. The median $Q_2$ is the point such that the area under the bars each side of $Q_2$ is equal. A histogram is a very specific type of graph. The median $Q_2$is the point such that the area under the bars each side of $Q_2$is equal. What is 7? Accidentally ran chmod +x /* - How bad did I just mess up? What is a range? Histogram vs. Each bin has a bar that represents the count or percentage of observations that fall within that bin.Download the CSV data file to make most of the histograms in this blog post: Histograms.In the fie… The line in the middle shows the median of the distribution. Has a state official ever been impeached twice? That is, half the monarchs started ruling before this age, and half after this age. It only takes a minute to sign up. Quartile 1 is the 25th percentile: it is the score that separates the lowest 25% from the highest 75% of scores. My Tweets. Answer. The histogram is still preferable for a more detailed look at the shape of the distribution. Please be sure to answer the question.Provide details and share your research! Subscribe to this blog. The lower quartile is the point where one-quarter of the values are below it. Read about our approach to external linking. When the temperature reaches 90°, approximately how many glasses of lemonade will be sold? Interquartile range = upper quartile - lower quartile = 47 - 38 = 9 cm. We read the values for the quartiles from the graph and arrive at 38 and 47. We calculated the upper and lower quartiles and the median. Quartile 1 is the 25th percentile: it is the score that separates the lowest 25% from the highest 75% of scores. The interquartile range (IQR) is therefore 18 - 4 = 14. b) Find an estimate for the lower quartile of the times taken to finish the puzzle. The box in the Box Plot extends from the lower quartile to the upper quartile. e. The first quartile (first 25%) is at 4. f. The third quartile (75%) is at 8. g. Middle 50% of the data ranges from 4 to 8. h. The interquartile range is 4. The first quartile, or lower quartile, is the value that cuts off the first 25% of the data when it is sorted in ascending order.The second quartile, or median, is the value that cuts off the first 50%.The third quartile, or upper quartile, is the value that cuts off the first 75%.. b) Find an estimate for the lower quartile of the times taken to finish the puzzle. Search for: Contact us. - Mr Kim 2. A box plot displays the five-number summary for a univariate data set, giving a quick impression of the distribution. 4.2.1. Mathematics, 21.06.2019 17:00, adreyan6221. This means that 25% of our scores are lower than 811.5 milliseconds. The middle term, between the median and first term is known as the first or Lower Quartile and is written as Q 1.Similarly, the value of mid term that lies between the last term and the median is known as the third or upper quartile and is denoted as Q 3.Second Quartile is the median and is written as Q 2. You will notice that the fact there is an outlier in this data (60) which has had no bearing on the calculation of the interquartile range. Consider a data set of following numbers: 10, 2, 4, 7, 8, 5, 11, 3, 12. What kind of wool do you get from sheering a sheep with the easter egg jeb_? Do I have to stop other application processes before receiving an offer? Photos (of the mark schemes I am looking at): 599722 599722599724 I dont have a clue how they worked out the median and Q1 and Q3 for t, even with t However, we can use the probs argument to get basically any quantile metric that we want. And the result is: Quartile 1 (Q1) = 3; Quartile 2 (Q2) = 5.5; Quartile 3 (Q3) = 7; Interquartile Range. What would cause a culture to keep a distinct weapon for centuries? Wiki User Answered . To find the class which the median or Q1 lies in, create a cumulative frequency table for the data. Problem. These graphs take your continuous measurements and place them into ranges of values known as bins. Answers. To find the upper quartile you'd go to 60, as this is 40 and the lower quartile or 20. To do the lower quartile, the same applies, you just need to half the median, so you'd go to 20 on the y-axis. I have X=c(20 ,18, 34, 45, 30, 51, 63, 52, 29, 36, 27, 24) With boxplot, i'm trying to plot the quantile(X,0.25) and quantile(X,0.75) but this is not realy the same lower and upper quartiles in Positive skewed histograms. Our tips from experts and exam survivors will help you through. 4.2.1. Box Plot. What is the interquartile range of the following data? In Googlesheets and Excel, =PERCENTILE.EXC(A2:A234,0.25) returns quartile 1 for the scores in cells A2 through A234 (our 233 reaction times). Median, quartiles, percentiles from a grouped frequency table. Is there a security reason to require email address and password in separate steps? Top Answer. That is, half the monarchs started ruling before this age, and half after this age. write your answer as a fraction in simplest form. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. As discussed above, the quartile formula helps us in dividing the data into four parts very quickly and eventually makes it easy for us to understand the data in these parts. A histogram is drawn like a bar chart, but often has bars of unequal width. To get the five number statistics, you use the fivenum() function. As I told you before, the quantile function returns the quartile of the input vector by default. The histogram is still preferable for a more detailed look at the shape of the distribution. 2 nd quartile or middle quartile also same as median it divides numbers into 2 equal parts. Frequency density O 90 100 110 120 130 140 150 160 170 180 190 200 Score a) Find the median score. Asking for help, clarification, or … Asked by Wiki User. There are 40 babies in the table, so to find the lower quartile, find \ (\frac {1} {4}\) of 40, which is the 10th value. To fine the inter quartile range you'd do what ever you got for you upper quartile, minus what ever you got for your lower quartile. The box in the Box Plot extends from the lower quartile to the upper quartile. Histograms Practice Questions Click here for Questions . Calculation of range and median along with Box-and-whisker plots and Cumulative frequency tables are effective ways to compare distributions and to summarise their characteristics. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Histogram. A quartile divides the set of observation into 4 equal parts. The difference here is that after dividing the data into two groups, instead of considering the data in the lower half, you consider the data in the upper half and then you proceed to find the Median of this subset of data using the methods described in the section on Averages. UQ is $$\frac{3(7+1)}{4}$$ = the sixth number. Box Plot. The histogram shows information about the times taken by some students to finish a puzzle. It is acceptable when dealing with an even total cumulative frequency to use LQ = $$\frac{n}{4}$$ and UQ = $$\frac{3n}{4}$$. The result is 811.5. 1 2 3. 100. In descriptive statistics, a box plot or boxplot is a method for graphically depicting groups of numerical data through their quartiles.Box plots may also have lines extending from the boxes (whiskers) indicating variability outside the upper and lower quartiles, hence the terms box-and-whisker plot and box-and-whisker diagram.Outliers may be plotted as individual points. What does the expression "go to the vet's" mean? Median location = (691+1)/2 = 346. The upper quartile for the data set given below is a0. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. MathJax reference. A box plot displays the five-number summary for a univariate data set, giving a quick impression of the distribution. Calculation. To calculate a quartile in R, set the percentile as parameter of the quantile function. Fig. Fig. The […] Histogram takes only one variable from the dataset and shows the frequency of each occurrence. b) The lower quartile = Q₁ is given by the value at the (n +1)/4th position = (17 + 1)/4 = 4.5th position From the cumulative frequency column, the 4.5th value is in the 90 - 100 score range Similar to the previous question, we find the value of the lower quartile using the weighted average as follows; Our lives are filled with data: the weather, weights, prices, our state of health, exam grades, bank balances, election results, and so on. Histograms can be drawn with regular or irregular class intervals and with mean, median, mode, lower and upper quartile values displayed. 3) Maria could not remember her scores from five math tests. With the following R codes, we can calculate the median… [The questions ask which statement is true, like "the lower quartile value of the number of hits made by the gold team is equal to the lower quartile value of the number of hits made by the blue team]. Asking for help, clarification, or responding to other answers. Why do some microcontrollers have numerous oscillators (and what are their functions)? 200. the lower quartile is the median of the lower half of the data. The third quartile is found in a similar manner to the first quartile. The interquartile range is another measure of spread, except that it has the added advantage of not being affected by large outlying values. To find the lower quartile using the formula, use Q1 = 1/4 (n+ 1), or enter 1 into the Excel function instead of 3. The, the upper quartile is the median of the upper half of the data. To learn more, see our tips on writing great answers. For the standard normal distribution N(0,1) , find the median, quartiles and interquartile range. Which interval contains the lower quartile? Third quartile (Q 3 / 75th percentile) : also known as the upper quartile q n (0.75), is the median of the upper half of the dataset. Which set of numbers correctly identifies the lower extreme, the lower quartile, the median, the upper quartile, and the upper extreme for the data set given? Is it possible to calculate the mean and standard deviation from a median and quartiles? Please refer to the attachment to answer this question. Yes No. Community Answer. Are there any stars that orbit perpendicular to the Milky Way's galactic plane? The Lower quartile is 13.3625 years. It is the area of the bar that tells us the frequency in a histogram, not its height. The histogram shows information about the times taken by some students to finish a puzzle. The middle term, between the median and first term is known as the first or Lower Quartile and is written as Q 1.Similarly, the value of mid term that lies between the last term and the median is known as the third or upper quartile and is denoted as Q 3.Second Quartile is the median and is written as Q 2. To find the actual value of the median or lower quartile, use linear interpolation.


# How many graphs are there on 4 nodes with degrees 1, 1, 2, 2?

I know just the basic operations on graphs using Mathematica. But I want to know how to write a code that prints all the possible combinations of a graph with a specified number of edges.
Take for example:
How many graphs are there on 4 nodes with degrees 1, 1, 2, 2?
I know the answer mathematically is 12, but I want Mathematica to print these combinations. \Please help me and explain to me how can I write the code.

• Hint: Since you only have four nodes, could you imagine building a table where each entry is a list of whether or not each node is connected to each other node for every possible graph? Could you check that table has the correct number of entries? And then could you think of a way of discarding entries in that table that do not have the appropriate degrees? Is the number of entries that remain what you want to know? And is that number equal to 12? Does this hint make the programming problem more concrete so you can make some progress on doing this? – Bill Mar 6 at 6:36

am = Tuples[{0, 1}, {4, 4}];

4X4 adjacency matrices with no self-loops:

simpleam = DeleteDuplicates[(1 - IdentityMatrix[4]) # & /@ am];

Adjacency matrices for undirected graphs on 4 nodes:

undirectedam = Select[# == Transpose@# &]@simpleam;

Adjacency matrices for graphs with vertex degree sequence {1,1,2,2}:

vdegree1122am = Select[Sort[Total[#]] == {1, 1, 2, 2} &]@undirectedam;

Counts:

Length /@ {am, simpleam, undirectedam, vdegree1122am}

 {65536, 4096, 64, 12}

Associated graphs:

AdjacencyGraph[#, VertexLabels -> "Name"] & /@ vdegree1122am // Column

We can get the 12 graphs more directly, by noting that two periphery nodes (nodes with vertex degree 1) can be selected in 6 ways:

peripherynodes = Subsets[Range[4], {2}]

 {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}

For each pair in peripherynodes the remaining two nodes can be ordered in 2 ways:

interiornodes = Permutations[Complement[Range[4], #]] & /@ peripherynodes;

We combine peripherynodes with associated interiornodes to get the edgelists of 12 graphs:

edgelists = BlockMap[UndirectedEdge @@ # &, #, 2, 1] & /@ (Join @@
{peripherynodes, interiornodes}])

Graph[#, VertexLabels -> "Name"] & /@ edgelists // Column

• Thank you for the solution. The second method of obtaining the graphs is exactly what I did to obtain them mathematically, I considered (4C2) then (2C1) to obtain the final result, and I wanted to know how to implement this idea in Mathematica. So thanks! – Mubarak Alsaeedi Mar 6 at 20:17

I had an old program lying around that will generate all realizations of a degree sequence. The first vertex will have degree equal to the first element in the input list, the 2nd degree equal to the 2nd element, and so on.

The program requires my IGraph/M package for graphicality testing.

For details on how it works, see http://bolyai.cs.elte.hu/egres/tr/egres-11-11.pdf

The answer here is trivial: there are only two realizations, which are isomorphic. They have an automorphism group size of 2 (they are symmetric to reflection, see also IGBlissAutomorphismCount). Thus not all 4! = 24 permutations of vertices create distinct labelled graphs, only 4! / 2 = 12.

What if the degree sequence is a bit more complicated?

degSeqGraphList[{3, 3, 2, 2, 1, 1}] // DeleteDuplicatesBy[CanonicalGraph]

There are now 5 non-isomorphic realizations:

There have different automorphism group sizes:

In[]:= IGBlissAutomorphismCount /@ %
Out[]= {8, 1, 4, 2, 4}

There are 6! permutations of vertices, but many of these generate the same labelled graph. We can compute the total number of graphs like this:

In[]:= 6!/%
Out[]= {90, 720, 180, 360, 180}

In[]:= Total[%]
Out[]= 1530

The code for generating all labelled realizations of a degree sequence:

Needs["IGraphM"]

removeZeros[list_] := TakeWhile[list, # != 0 &]

iter[ds_, vert_] :=
With[{pos = Position[ds, 0]}, {Delete[ds, pos], Delete[vert, pos]}]

ClearAll[exactGen, list]; (* list is an inert head *)

exactGen[{}, edges_, _] := edges
exactGen[degseq : {d_, rest__}, edges_, vertices_] /; IGGraphicalQ[degseq] :=

Module[{ds = {rest}, temp, newDs, newVertices},
Function[subs,
temp = ds;
temp[[subs]] -= 1;
If[IGGraphicalQ[temp],
{newDs, newVertices} = iter[temp, Rest[vertices]];
exactGen[newDs,
Join[edges, list @@ Thread[First[vertices] <-> Rest[vertices][[subs]]]],
newVertices],
{}
]
] /@ Subsets[Range@Length[ds], {d}]
]

degSeqGraphList[degseq_?IGGraphicalQ] :=

With[{verts = Range@Length[degseq]},
Graph[verts, #, VertexLabels -> Automatic] & /@ List @@@ Cases[
exactGen[degseq, list[], verts],
_list,
Infinity
]
]

• Thanks. I appreciate your solution and your generalization. – Mubarak Alsaeedi Mar 6 at 20:21

A one-liner:

Length@Select[Graph /@ Subsets[EdgeList[CompleteGraph[4]]],
Sort@VertexDegree[#] == {1, 1, 2, 2} &]

EdgeList[CompleteGraph[4]] will give us all the possible edges on four vertices, Subsets will give us all the possible subsets of those edges, and Graph will turn those into graphs. (When a vertex is isolated, it will not be included in the graph, but that's not a problem for us.)

VertexDegree gives the degrees of all vertices of a graph, so Sort@VertexDegree[#] == {1, 1, 2, 2} checks if a graph has the degree sequence you want.

We Select all graphs with that degree sequence, and then find the Length` of that list.

(Like the accepted answer, this would be inefficient for large graphs, in which case Szabolcs's answer is the way to go.)

• Yes, I can see that. But when I tried to solve it mathematically, I used the approach of Kglr in his second solution, where he used the idea of peripherynodes and interiornodes. Is this an efficient approach? – Mubarak Alsaeedi Mar 6 at 20:25
• It's situational: it doesn't give a general algorithm for an arbitrary degree sequence. So it's hard to say how efficient this is, given that for this specific problem, all of these solutions work instantly. – Misha Lavrov Mar 6 at 20:37


How can three vectors be orthogonal to each other?

In a scenario, say that:

Vectors $$\mathbf{U}$$, $$\mathbf{V}$$ and $$\mathbf{W}$$ are all orthogonal such that the dot product between each of these $$(\mathbf{UV}\;\mathbf{VW}\;\mathbf{WU})$$ is equal to zero.

I imagine that for any potential vector space $$\mathbf{R}$$ this would only be possible in two situations.

1) $$\mathbf{U}$$, $$\mathbf{W}$$ and/or $$\mathbf{V}$$ is the zero vector.

2) $$\mathbf{U}=(1, 0, 0)$$, $$\mathbf{V} = (0, 1, 0)$$ and $$\mathbf{W} = (0, 0, 1)$$.

Is there any other situation where three vectors are all orthogonal to each other?

• What about $u=\big(\tfrac{1}{\sqrt 2},\tfrac{1}{\sqrt 2},0\big)$, $v=\big(\tfrac{1}{\sqrt 2},-\tfrac{1}{\sqrt 2},0\big)$ and $w=(0,0,1)$? Sep 29, 2019 at 0:24
• Yes, infinitely many. For example, rotate your basis in 2. Sep 29, 2019 at 0:25
• Pick two non-zero and non-parallel vectors $x$ and $u$ in $\mathbb R^3$. Then set $v=x\times u$ and $w=u\times v$. Now $u,v,w$ are mutually orthogonal. Sep 29, 2019 at 3:53
• @Azif00 More generally, $(a,b,0)$, $(b,-a,0)$, $(0,0,c)$ for any values of $a$, $b$, and $c$. Orthogonal vectors don't have to be unit vectors. The OP also seems to have made that mistake in the second example. Sep 29, 2019 at 10:34
• Hold out your hand. Arrange your fingers so that your index finger, middle finger, and thumb are all mutually perpendicular. Now rotate your hand around. All of these configurations correspond to a distinct set of 3 mutually orthogonal vectors. Sep 29, 2019 at 12:53

Think it like this: for a given vector $$u\neq 0$$ in $$\mathbb{R}^3$$, what is the space of all vectors perpendicular to it? It is a plane $$P$$ containing the origin, whose perpendicular direction is obviously $$u$$.

Now take a $$v\neq 0$$ in that plane. The space of all vectors perpendicular to $$v$$ is a new plane $$P'$$. In order to get a vector $$w$$ perpendicular to both $$u,v$$, we need $$w\in P\cap P'$$. What does $$P\cap P'$$ look like?

By the way, notice that $$v$$ is ANY non-zero vector, not only $$(1,0,0), (0,1,0)$$ or $$(0,0,1)$$

There are infinitely many triple of non zero orthogonal vectors obtained by the three you have indicated by scaling of each one and rotations of the triple all togheter.

To construct any othogonal triple we can proceed as follows:

1. choose a first vector $$v_1=(a,b,c)$$
2. find a second vector orthogonal to $$v_1$$ that is e.g. $$v_2=(-b,a,0)$$
3. determine the third by cross product $$v_3=v_1\times v_2$$
• You can also have $v_3 = v_2 \times v_1 = - v_1 \times v_2$, but of course you can just as easily interchange $v_1$ and $v_2$ to do that, so it's not really an extra degree of freedom. Still, chirality is sometimes useful to talk about. Sep 30, 2019 at 3:33
• @Kevin We ca see that as a simple scaling by a factor $-1$ for $v_3$. There are of course many ways! It was just to give a simple procedure. To be more genral we can assume $v_1=(a,b,c)$, $v_2=(d,e,-ad-ce)$ and $v_3$by $v_1\times v_2$.
– user
Sep 30, 2019 at 5:54

How about $$v_1= (\frac{1}{\sqrt2}, \frac{1}{\sqrt2},0),v_2=(-\frac{1}{\sqrt2},\frac{1}{\sqrt2},0),v_3=(0,0,1)$$ ? Geometrically take any frame and rotate in any direction.

Here is an example of just what you seek:

• I don't know what this GIF looks like to you, but on my setup, it flickers between some sort of geometry animation and a mostly-white frame with a red rectangle in the upper left, before eventually settling down to just the white frame with red rectangle. Sep 29, 2019 at 9:25
• I don't know how you created that gif, but it had a lot of blank frames in it seemingly at random, causing it to appear to flicker. I've taken the liberty of editing the gif to remove the blank frames. Sep 29, 2019 at 15:10
• Gif works fine for me Sep 29, 2019 at 17:11
• @B.Swan Yes, it's fixed now. You can see the edit history to compare with the old version. Sep 29, 2019 at 17:45

There are infinitely many possibilities. $$(1,0,0), (0,1,-1)$$ and $$(0,1,1)$$ is one example.

The word "orthogonal" really just corresponds to the intuitive notion of vectors being perpendicular to each other. Draw out the unit vectors in the $$x$$, $$y$$ and $$z$$ directions respectively--those are one set of three mutually orthogonal (i.e. perpendicular) vectors, just like you observed. But if you rotate those three vectors together in any way that you like in 3D space, without changing the angles between them, then of course they will still remain orthogonal.

Furthermore, if we scale these vectors by changing their lengths independently by any arbitrary nonzero factor, we will still end up with a set of orthogonal vectors, since the only thing that matters is the directions and not the lengths of the vectors. In this manner we end up with a description for an infinite family of orthogonal vectors, which hopefully makes it easy for you to convince yourself intuitively.

In a more general vector space, of course, this sort of pictorial intuition might no longer hold, but the idea of orthogonality can be easily generalised. That's the reason we define orthogonality abstractly and independent of the usual geometric notion of perpendicularity.


# Foundations of Computational Economics #31¶

by Fedor Iskhakov, ANU

## Function approximation in Python¶

https://youtu.be/liNputEfcXQ

Description: How to approximate functions which are only defined on grid of points. Spline and polynomial interpolation.

### Interpolation problem¶

• $f(x)$ is function of interest, hard to compute
• Have data on values of $f(x)$ in $n$ points $(x_1,\dots,x_n)$
$$f(x_1), f(x_2), \dots f(x_n)$$
• Need to find the approximate value of the function $f(x)$ in arbitrary points $x \in [x_1,x_n]$

#### Approaches¶

1. Piece-wise approach (connect the dots)
• Which functional form to use for connections?
1. Use a similar function $s(x)$ to represent $f(x)$ between the data points
• Which simpler function?
• What data should be used?
• How to control the accuracy of the approximation?

#### Distinction between function approximation (interpolation) and curve fitting¶

• Functions approximation and interpolation refers to the situations when data on function values is matched exactly
• The approximation curve passes through the points of the data
• Curve fitting refers to the statistical problem when the data has noise, the task is to find an approximation for the central tendency in the data
• Linear and non-linear regression models, econometrics
• The model is over-identified (there is more data than needed to exactly identify the regression function)

#### Extrapolation¶

Extrapolation is computing the approximated function outside of the original data interval

Should be avoided in general

• Exact only when theoretical properties of the extrapolated function are known
• Can be used with extreme caution and based on the analysis of the model
• Always try to introduce wider bounds for the grid instead

### Spline interpolation¶

Spline = curve composed of independent pieces

Definition A function $s(x)$ on $[a,b]$ is a spline of order $n$ ( = degree $n-1$) iff

• $s$ is $C^{n-2}$ on $[a,b]$ (has continuous derivatives up to order $n-2$),
• given knot points $a=x_0<x_1<\dots<x_m=b$, $s(x)$ is a polynomial of degree $n-1$ on each subinterval $[x_i,x_{i+1}]$, $i=0,\dots,m-1$

#### Cubic splines = spline of order 4¶

• Data set $\{(x_i,f(x_i), i=0,\dots,n\}$
• Functional form $s(x) = a_i + b_i x + c_i x^2 + d_i x^3$ on $[x_{i-1},x_i]$ for $i=1,\dots,n$
• $4n$ unknown coefficients:
• $2n$ equations to make sure each segment passes through its interval points + $2(n-1)$ equations to ensure two continuous derivatives at each interior point
• Additional 2 equation for the $x_0$ and $x_n$
• $s''(x_0)=s''(x_n)=0$ (natural spline)
• $s'(x_0)=\frac{s(x_1)-s(x_0)}{x_1-x_0}$, $s'(x_n)=\frac{s(x_n)-s(x_{n-1})}{x_n-x_{n-1}}$ (secant-Hermite)
In [1]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

np.random.seed(2008) # fix random number sequences
x  = np.sort(np.random.uniform(-5,10,12)) # sorted random numbers on [-5,10]
xr = np.linspace(-5,10,12) # regular grid on [-5,10]

func=lambda x: np.exp(-x/4)*np.sin(x) + 1/(1+x**2) # function to interpolate

In [2]:
def plot1(ifunc,fdata=(x,func(x)),f=func,color='b',label='',extrapolation=False):
'''helper function to make plots'''
xd = np.linspace(-5,10,1000) # for making continuous lines
plt.figure(num=1, figsize=(10,8))
plt.scatter(fdata[0],fdata[1],color='r') # interpolation data
plt.plot(xd,f(xd),color='grey') # true function
if extrapolation:
xdi = xd
else:
# restriction for interpolation only
xdi=xd[np.logical_and(xd>=fdata[0][0],xd<=fdata[0][-1])]
if ifunc:
plt.plot(xdi,ifunc(xdi),color=color,label=label)
if label:
plt.legend()
elif label:
plt.title(label)

In [3]:
plot1(None,label='True function')

In [4]:
from scipy import interpolate # Interpolation routines
fi = interpolate.interp1d(x,func(x)) # returns the interpolation function
plot1(fi,label='interp1d')

In [5]:
help(interpolate.interp1d)

Help on class interp1d in module scipy.interpolate.interpolate:

class interp1d(scipy.interpolate.polyint._Interpolator1D)
|  interp1d(x, y, kind='linear', axis=-1, copy=True, bounds_error=None, fill_value=nan, assume_sorted=False)
|
|  Interpolate a 1-D function.
|
|  x and y are arrays of values used to approximate some function f:
|  y = f(x).  This class returns a function whose call method uses
|  interpolation to find the value of new points.
|
|  Note that calling interp1d with NaNs present in input values results in
|  undefined behaviour.
|
|  Parameters
|  ----------
|  x : (N,) array_like
|      A 1-D array of real values.
|  y : (...,N,...) array_like
|      A N-D array of real values. The length of y along the interpolation
|      axis must be equal to the length of x.
|  kind : str or int, optional
|      Specifies the kind of interpolation as a string
|      ('linear', 'nearest', 'zero', 'slinear', 'quadratic', 'cubic',
|      'previous', 'next', where 'zero', 'slinear', 'quadratic' and 'cubic'
|      refer to a spline interpolation of zeroth, first, second or third
|      order; 'previous' and 'next' simply return the previous or next value
|      of the point) or as an integer specifying the order of the spline
|      interpolator to use.
|      Default is 'linear'.
|  axis : int, optional
|      Specifies the axis of y along which to interpolate.
|      Interpolation defaults to the last axis of y.
|  copy : bool, optional
|      If True, the class makes internal copies of x and y.
|      If False, references to x and y are used. The default is to copy.
|  bounds_error : bool, optional
|      If True, a ValueError is raised any time interpolation is attempted on
|      a value outside of the range of x (where extrapolation is
|      necessary). If False, out of bounds values are assigned fill_value.
|      By default, an error is raised unless fill_value="extrapolate".
|  fill_value : array-like or (array-like, array_like) or "extrapolate", optional
|      - if a ndarray (or float), this value will be used to fill in for
|        requested points outside of the data range. If not provided, then
|        the default is NaN. The array-like must broadcast properly to the
|        dimensions of the non-interpolation axes.
|      - If a two-element tuple, then the first element is used as a
|        fill value for x_new < x[0] and the second element is used for
|        x_new > x[-1]. Anything that is not a 2-element tuple (e.g.,
|        list or ndarray, regardless of shape) is taken to be a single
|        array-like argument meant to be used for both bounds as
|        below, above = fill_value, fill_value.
|
|      - If "extrapolate", then points outside the data range will be
|        extrapolated.
|
|  assume_sorted : bool, optional
|      If False, values of x can be in any order and they are sorted first.
|      If True, x has to be an array of monotonically increasing values.
|
|  Attributes
|  ----------
|  fill_value
|
|  Methods
|  -------
|  __call__
|
|  --------
|  splrep, splev
|      Spline interpolation/smoothing based on FITPACK.
|  UnivariateSpline : An object-oriented wrapper of the FITPACK routines.
|  interp2d : 2-D interpolation
|
|  Examples
|  --------
|  >>> import matplotlib.pyplot as plt
|  >>> from scipy import interpolate
|  >>> x = np.arange(0, 10)
|  >>> y = np.exp(-x/3.0)
|  >>> f = interpolate.interp1d(x, y)
|
|  >>> xnew = np.arange(0, 9, 0.1)
|  >>> ynew = f(xnew)   # use interpolation function returned by interp1d
|  >>> plt.plot(x, y, 'o', xnew, ynew, '-')
|  >>> plt.show()
|
|  Method resolution order:
|      interp1d
|      scipy.interpolate.polyint._Interpolator1D
|      builtins.object
|
|  Methods defined here:
|
|  __init__(self, x, y, kind='linear', axis=-1, copy=True, bounds_error=None, fill_value=nan, assume_sorted=False)
|      Initialize a 1D linear interpolation class.
|
|  ----------------------------------------------------------------------
|  Data descriptors defined here:
|
|  __dict__
|      dictionary for instance variables (if defined)
|
|  __weakref__
|      list of weak references to the object (if defined)
|
|  fill_value
|      The fill value.
|
|  ----------------------------------------------------------------------
|  Methods inherited from scipy.interpolate.polyint._Interpolator1D:
|
|  __call__(self, x)
|      Evaluate the interpolant
|
|      Parameters
|      ----------
|      x : array_like
|          Points to evaluate the interpolant at.
|
|      Returns
|      -------
|      y : array_like
|          Interpolated values. Shape is determined by replacing
|          the interpolation axis in the original array with the shape of x.
|
|  ----------------------------------------------------------------------
|  Data descriptors inherited from scipy.interpolate.polyint._Interpolator1D:
|
|  dtype

In [6]:
fi = interpolate.interp1d(x,func(x),kind='linear')
plot1(fi,label='Linear')

In [7]:
for knd, clr in ('previous','m'),('next','b'),('nearest','g'):
fi = interpolate.interp1d(x,func(x),kind=knd)
plot1(fi,label=knd,color=clr)
plt.show()

In [8]:
for knd, clr in ('slinear','m'),('quadratic','b'),('cubic','g'):
fi = interpolate.interp1d(x,func(x),kind=knd)
plot1(fi,color=clr,label=knd)

In [9]:
# Approximation errors
# x = np.sort(np.random.uniform(-5,10,11))  # generate new data
fi = interpolate.interp1d(x,func(x),kind=knd,bounds_error=False)
xd = np.linspace(-5,10,1000)
erd=np.abs(func(xd)-fi(xd))
plt.plot(xd,erd,color=clr)
print('Max error with  %s splines is %1.5e'%(knd,np.nanmax(erd)))

Max error with  slinear splines is 1.05142e+00
Max error with  quadratic splines is 3.89974e-01
Max error with  cubic splines is 4.35822e-01

In [10]:
# Approximation errors for regular grid
fi = interpolate.interp1d(xr,func(xr),kind=knd,bounds_error=False)
xd = np.linspace(-5,10,1000)
erd=np.abs(func(xd)-fi(xd))
plt.plot(xd,erd,color=clr)
print('Max error with  %s splines is %1.5e'%(knd,np.nanmax(erd)))

Max error with  slinear splines is 4.63043e-01
Max error with  quadratic splines is 3.48546e-01
Max error with  cubic splines is 1.89578e-01

#### Accuracy of the interpolation¶

How to reduce approximation errors?

• Number of nodes (more is better)
• Location of nodes (regular is better)
• Interpolation type (match function of interest)

In economic models we usually can control all of these

### Polynomial approximation/interpolation¶

Back to the beginning to explore the idea of replacing original $f(x)$ with simpler $g(x)$

• Data set $\{(x_i,f(x_i)\}, i=0,\dots,n$
• Functional form is polynomial of degree $n$ such that $g(x_i)=f(x_i)$
• If $x_i$ are distinct, coefficients of the polynomial are uniquely identified

Does polynomial $g(x)$ converge to $f(x)$ when there are more points?

In [11]:
from numpy.polynomial import polynomial
degree = len(x)-1 # passing through all dots
p = polynomial.polyfit(x,func(x),degree)
fi = lambda x: polynomial.polyval(x,p)
plot1(fi,label='Polynomial of degree %d'%degree,extrapolation=True)

In [12]:
# now with regular grid
degree = len(x)-1 # passing through all dots
p = polynomial.polyfit(xr,func(xr),degree)
fi = lambda x: polynomial.polyval(x,p)
plot1(fi,fdata=(xr,func(xr)),label='Polynomial of degree %d'%degree,extrapolation=True)

In [13]:
# how number of points affect the approximation (with degree=n-1)
for n, clr in (5,'m'),(10,'b'),(15,'g'),(25,'r'):
x2 = np.linspace(-5,10,n)
p = polynomial.polyfit(x2,func(x2),n-1)
fi = lambda x: polynomial.polyval(x,p)
plot1(fi,fdata=(x2,func(x2)),label='%d points'%n,color=clr,extrapolation=True)
plt.show()

In [14]:
# how locations of points affect the approximation (with degree=n-1)
np.random.seed(2025)
n=8
for clr in 'b','g','c':
x2 = np.linspace(-4,9,n) + np.random.uniform(-1,1,n) # perturb points a little
p = polynomial.polyfit(x2,func(x2),n-1)
fi = lambda x: polynomial.polyval(x,p)
plot1(fi,fdata=(x2,func(x2)),label='%d points'%n,color=clr,extrapolation=True)
plt.show()

In [15]:
# how degree of the polynomial affects the approximation
for degree, clr in (7,'b'),(9,'g'),(11,'m'):
p=polynomial.polyfit(xr,func(xr),degree)
fi=lambda x: polynomial.polyval(x,p)
plot1(fi,fdata=(xr,func(xr)),label='Polynomial of degree %d'%degree,color=clr,extrapolation=True)

#### Least squares approximation¶

We could also go back to function approximation and fit polynomials of lower degree

• Data set $\{(x_i,f(x_i)\}, i=0,\dots,n$
• Any functional form $g(x)$ from class $G$ that best approximates $f(x)$
$$g = \arg\min_{g \in G} \lVert f-g \rVert ^2$$

### Orthogonal polynomial approximation/interpolation¶

• Polynomials over domain $D$
• Weighting function $w(x)>0$

Inner product

$$\langle f,g \rangle = \int_D f(x)g(x)w(x)dx$$

$\{\phi_i\}$ is a family of orthogonal polynomials w.r.t. $w(x)$ iff

$$\langle \phi_i,\phi_j \rangle = 0, i\ne j$$

#### Best polynomial approximation in L2-norm¶

Let $\mathcal{P}_n$ denote the space of all polynomials of degree $n$ over $D$

$$\lVert f - p \rVert_2 = \inf_{q \in \mathcal{P}_n} \lVert f - q \rVert_2 = \inf_{q \in \mathcal{P}_n} \left[ \int_D ( f(x)-g(x) )^2 dx \right]^{\tfrac{1}{2}}$$

if and only if

$$\langle f-p,q \rangle = 0, \text{ for all } q \in \mathcal{P}_n$$

Orthogonal projection is the best approximating polynomial in L2-norm

#### Uniform (infinity, sup-) norm¶

$$\lVert f(x) - g(x) \rVert_{\infty} = \sup_{x \in D} | f(x) - g(x) | = \lim_{n \rightarrow \infty} \left[ \int_D ( f(x)-g(x) )^n dx \right]^{\tfrac{1}{n}}$$

Measures the absolute difference over the whole domain $D$

#### Chebyshev (minmax) approximation¶

What is the best polynomial approximation in the uniform (infinity, sup) norm?

$$\lVert f - p \rVert_{\infty} = \inf_{q \in \mathcal{P}_n} \lVert f - q \rVert_{\infty} = \inf_{q \in \mathcal{P}_n} \sup_{x \in D} | f(x) - g(x) |$$

Chebyshev proved existence and uniqueness of the best approximating polynomial in uniform norm.

#### Chebyshev polynomials¶

• $[a,b] = [-1,1]$ and $w(x)=(1-x^2)^{(-1/2)}$
• $T_n(x)=\cos\big(n\cos^{-1}(x)\big)$
• Recursive formulas:
$$\begin{eqnarray} T_0(x)=1,\\ T_1(x)=x,\\ T_{n+1}(x)=2x T_n(x) - T_{n-1}(x) \end{eqnarray}$$

#### Accuracy of Chebyshev approximation¶

Suppose $f: [-1,1]\rightarrow R$ is $C^k$ function for some $k\ge 1$, and let $I_n$ be the degree $n$ polynomial interpolation of $f$ with nodes at zeros of $T_{n+1}(x)$. Then

$$\lVert f - I_n \rVert_{\infty} \le \left( \frac{2}{\pi} \log(n+1) +1 \right) \frac{(n-k)!}{n!}\left(\frac{\pi}{2}\right)^k \lVert f^{(k)}\rVert_{\infty}$$

📖 Judd (1988) Numerical Methods in Economics

• achieves best polynomial approximation in uniform norm
• works for smooth functions
• easy to compute
• but does not approximate $f'(x)$ well

#### General interval¶

• Not hard to adapt the polynomials for the general interval $[a,b]$ through linear change of variable
$$y = 2\frac{x-a}{b-a}-1$$
• Orthogonality holds with weights function with the same change of variable

#### Chebyshev approximation algorithm¶

1. Given $f(x)$ and $[a,b]$
2. Compute Chebyshev interpolation nodes on $[-1,1]$
3. Adjust nodes to $[a,b]$ by change of variable, $x_i$
4. Evaluate $f$ at the nodes, $f(x_i)$
5. Compute Chebyshev coefficients $a_i = g\big(f(x_i)\big)$
6. Arrive at approximation
$$f(x) = \sum_{i=0}^n a_i T_i(x)$$
In [16]:
import numpy.polynomial.chebyshev as cheb
for degree, clr in (7,'b'),(9,'g'),(11,'m'):
fi=cheb.Chebyshev.interpolate(func,degree,[-5,10])
plot1(fi,fdata=(None,None),color=clr,label='Chebyshev with n=%d'%degree,extrapolation=True)

### Multidimensional interpolation¶

• there are multidimensional generalization to all the methods
• curse of dimensionality in the number of interpolation points when number of dimensions increase
• sparse Smolyak grids and adaptive sparse grids
• irregular grids require computationally expensive triangulation in the general case
• good application for machine learning!

Generally much harder!


## Decimal To Binary

In order to convert a decimal number to its binary equivalent, we will repeatedly divide the decimal number by 2, the base of the binary system. One for "yes", and zero for "no". Binary definition, consisting of, indicating, or involving two. Add the following byte-long (8 bit) two’s complement numbers together, and then convert all binary quantities into decimal form to verify the accuracy of the addition:. This tutorial explains how to convert a decimal IP address in binary IP address and a binary IP address in a decimal IP address step by step with examples. A value can also be in binary, hexadecimal or octal if the appropriate prefix is included. The number that i'm trying to convert is a source address in a decimal form. How to Convert Decimal to Binary and vise versa is crucial to master, such skills will give you the confidence when you deal with Network and Storage devices. It allows the user to specify the bit of the binary number that will be converted to. Example of decimal to binary conversion: The decimal positive integer 330 can be deconstructed: 330 10 = 3*100 + 3*10 + 0*1 = 1*10 2 + 2*10 1 + 5*1 0. Before a floating-point binary number can be stored correctly, its mantissa must be normalized. Converting between Decimal (Base 10) and Binary (Base 2) Base 10 vs. In this tutorial i am going to show you how to convert “Decimal to Binary”, “Decimal to Octal” and “Decimal to Hexadecimal” Check the Java Program below. For example, the decimal. In the octal to binary conversion method, we will convert each digit from the given octal number into three bit binary number starting from the right most octal digit(LSD) to the left most digit(MSD) and at the end combine each three bit binary number to form the binary equivalent of given octal number. Program to convert decimal to binary more than 18 bits. Example :- 7 1). This way people won't think it is the decimal number "101" (one hundred and one). Below is the code for a 3-digit BCD-Counter (binary Coded decimal) I implemented while studying verilog. C program to covert 'Decimal to Binary' and 'Binary to Decimal' - Binary Decimal Conversion. Video created by Hebrew University of Jerusalem for the course "Build a Modern Computer from First Principles: From Nand to Tetris (Project-Centered Course)". This is how we would successfully convert a binary number into decimal, (which you can read about here) but we actually want to do the reverse, and easily convert decimal to binary. Binary to Decimal to BCD Conversion Example. Explanation of how number bases, including binary and decimal, are used for reasoning about digital data. Take this quiz to test yourself on conversion between decimal and binary representations. This way you can convert up to 19 decimal characters (max. To convert fraction to binary, start with the fraction in question and multiply it by 2 keeping notice of the resulting integer and fractional part. Decimal -> Binary Decimal to binary conversion is even more exciting. You can convert to and from binary and the base-10 system typically used by humans. The binary number 1101011001111110 converts to 54910 in Decimal and D67E in hex. Converting decimal fraction to binary. In this example I use 8 bit binary numbers, but the principle is the same for both 8 bit binary numbers (chars) and 32 bit binary numbers (ints). Binary, and other systems of representing numbers is similar to decimal, only the use a different amount of symbols or digits to represent a number. It is much easier to work with large numbers using hexadecimal values than decimal. binary values The following table shows the decimal values of binary numbers. Program to Convert Binary to Decimal number:Number System [crayon-5d4e448ac93bb860279627/] Output : [crayon-5d4e448ac93c5762542184/] Note : This program is for beginners (not for Experts) that’s why we haven’t provided any Validations while accepting the input. Type 110 in binary by clicking the Buttons "1", "1" and "0" in the binary system calculator. 243F6 in a hexadecimal system (base 16), or to 11. It can convert very large and very small numbers — up to hundreds of digits. Hexadecimal is used to convert byte/modern computer numbers into defined binary digits. We use cookies for various purposes including analytics. It provides output as 170 Decimal value. For beginners hexadecimal is base 16 number, while decimal is base 10, Octal is base 8 and binary is base 2 numbers in Number systems. Method1: Use (()) brace expatiation. This video tutorial explains how to convert decimal to binary numbers. It focuses on the following learning objectives: 5b. As it is written, it would be hard to work out whether it is a binary or decimal (denary) value. Each hexadecimal digit represents four binary digits (bits) (also called a "nibble"), and the primary use of hexadecimal notation is as a human-friendly representation of binary coded values in computing and digital. Convert any decimal number between 0 and 255 into binary. Decimal to Binary, Hex & Octal Converter to perform decimal to binary, decimal to hex & decimal to octal conversion online by using simple successive division methods along with step by step calculation & solved example problems. The online BCD to Binary Converter is used to convert BCD (Binary-coded decimal) to binary (Base-2) number. AD3, Alternating Directions Dual Decomposition. It involves dividing the number to be converted, say N, by 2 (since binary is in base 2), and making note of. Then the calculate button is pressed and the textbox should display the corresponding decimal number. TCS Programming Question Solution - 1. Two good examples where we use hexadecimal values are MAC addresses and IPv6 addresses. So the binary (so far) is _ _ _ 1 1. Decimal To Binary Conversion A decimal number system is a base 10 number system. Likewise, we use 0 and 1 to write numbers in the binary number system. Calculator will display the result in total number of hours, minutes and seconds in decimal. This guide shows you how to convert from decimal to binary and binary to decimal. Enter the primary number (in binary; make sure it is valid) first then enter the secondary number (also in binary) for the calculation and click on Calculate. Binary to decimal - Four ways to convert formats in Matlab In this article we're going to present four variations of a binary to decimal conversion in Matlab; we're going to convert binary numbers (numbers with only symbols '0' and '1') to decimal numbers (numbers with 10 different symbols, from '0' to '9'). Binary to hexa decimal decoder / converter. 0 Flash Drive Cute Memory Stick Fruit Vegetable Series Thumb Drive Data Storage Pendrive Cartoon Jump Drive Gift: USB Flash Drives - Amazon. In this system, numbers are represented in a decimal form, however each decimal digit is encoded using a four bit binary number. the radix of binary number system). Specifically, dotted decimal notation is used in various IT contexts, including in Internet Protocol addresses. For example-. Roman to decimal converter. Say we wish to convert an unsigned decimal integer, $$N\text{,}$$ to binary. Binary form of 15 is 1111 Binary form of 10 is 1010 Binary form of 18 is 10010 Binary form of 27 is 11011 In the above program, the DecimalToBinary function has binary value of the decimal number n and is stored in the array binaryNumber[]. Although computers are based on the binary number system, we don’t have to use binary numbers when using one. Especially for IPv6 addresses it's useful to understand how you can calculate from hexadecimal to binary and decimal or the other way around. We use decimal number system for our day to day calculations. C Program to Convert Decimal to Binary. C++ program to convert decimal number into binary. However there are alternate solutions, in case you were just curious to know :) Method #2 [code. Decimal to Binary Table. Decimal -> Binary Decimal to binary conversion is even more exciting. Binary string to decimal conversion. It will convert a decimal number to its nearest single-precision and double-precision IEEE 754 binary floating-point number, using round-half-to-even rounding (the default IEEE rounding mode). The equivalent decimal representation of a binary number is the sum of the powers of 2 which each digit represents. Binary number is a base 2 number because it is either 0 or 1. Binary Fractions Summary. However, if you study the Microsoft Windows calculator application that ships along with the OS, the "programmer" type of calculator takes an input whether the entered number is binary or decimal from the user, then internally in its code it would check whether the entered number was a valid binary or decimal number. Example values of hexadecimal numbers converted into binary, octal and decimal. These 8-bits can represent any decimal number between 0 and 255. The decimal number is equal to the sum of binary digits (d n) times their power of 2 (2 n):. There are no ads, popups or nonsense, just an awesome decimal number to BCD number converter. IP addressing is a core functionality of networking today. Take a look at decimal number 6. And I encourage you to pause the video, and try to work through it out on your own. Press button, get result. Decimal : For denoting integer and non-integer numbers, decimal number system uses 10 different digits, 0,1,2,3,4,5,6,7,8 and 9 i. Number conversion between hexadecimal and decimal. How to Convert from Decimal to Binary. C Program To Convert Decimal To Octal Number. If you work with computers, you may find yourself needing a basic understanding of binary. Entering Repeating Decimals. Binary numbers can contain digits 0 and 1 but no decimal or scientific notation. Here we will learn, How to convert decimal to binary in java with easy examples. Let me show it on example. In binary form, for example, the decimal quantity 1895 appears as 11101100111. If you’d like to dig deeper into Binary, this course helps explain the details, and show how Binary is the foundation of all digital computing. dec = Fix(dec) / 2: After obtaining the first residue the variable "dec" is going to get itself value between 2. Binary Coded Decimal which is also called as BCD is another process for converting decimal numbers into their binary equivalents. Binary numbers have lot of use in the digital world, in fact, binary is the language of computers where 0 and 1 represent true/false, on/off and becomes key for logic formation. In the binary system, each digit represents an increasing power of 2, with the rightmost digit representing 2 0, the next representing 2 1, then 2 2, and so on. Continue multiplying by 2 until you get a resulting fractional part equal to zero. Treat the division as an integer division. binary values The following table shows the decimal values of binary numbers. Then just add them up. It then displays the decimal values and the hexadecimal value of the elements in the array returned by the GetBits method. 0x578FCF (hex) 5738447 (decimal) 025707717 (octal) 10101111000111111001111 (binary) are all different representations of the same number. You can use national decimal items anywhere that other category numeric data items can be used. 25), 2-3 (0. By using user defined logic. About the Decimal/Binary Converter. The binary or base 2 numbering system is the keystone of computer systems and digital electronics. Input and output in binary, decimal, hexadecimal or ASCII. Get the integer quotient for the next iteration. The following function takes in a binary string and returns a bigint with the decimal value:. Decimal to Binary Using Excel: https://www. To use this decimal to binary converter tool, you should type a decimal value like 308 into the left field below, and then hit the Convert button. Write a C program to input any decimal number from user and convert it to binary number system using bitwise operator. A one has the value 1 and a zero has the value 0. This shweet conversion tool will take any text string and convert it into binary code - you know? those little 1's and 0's that make our world go around. 2) Convert the decimal number 143 into binary. Hello guys, Can anyone give me the code required for 'decimal to binary conversion using verilog for 4 bit' please im doing a project. 14159 in a decimal system (base 10), or to 3. Convert number systems units. Department of Labor’s Employment and Training Administration. Watch our tutorial on How To Do Decimals To Binary Numbers from one of Videojug's professional experts. (Binary to decimal). ; In this method of conversion the fractional part of the given decimal number is multiplied by 2 (i. Binary number: In digital electronics and mathematics, a binary number is a number expressed in the base-2 numeral system or binary numeral system. The Quotient method is the new conversion method invented and derived by using Raghavendra's Analysis. There are other methods also available which can also be used in decimal to binary conversion. Convert Binary to Decimal Numbers. Let's see if we can get some experience converting from a decimal representation to a binary representation. Binary form base 8 Octal form base 10 Decimal form (common used) base 16 Hexadecimal form (hexa) Examples. This is a conversion table with decimal numbers next to their binary and hex equivalents. It means that a binary number can be formed by using 0 and 1 (2 base elements) only. Binary to Decimal conversion How to convert decimal to binary Conversion steps: Divide the number by 2. Useful, free online tool that converts plain text to decimal string. Fast Decimal to Binary Conversion. The flip side of the coin is translating a decimal number into binary format. (The old flash version is here. What that means is, everytime you count numbers under base 10, you are counting from zero to nine, then starting over by adding another number in front to make 10 and so on. It casts a string following it to a binary string. Several numeric data types are discussed, including the common "packed" and "comp-3" fields. The computer chips that run PCs work with a binary system. Binary arithmetic is pretty easy once you know what's going on. A circuit is either on or off. 33) convert to binary Ask for details ; Follow Report by Arjun306 7 minutes ago Log in to add a comment. The representation of such a number in decimal and binary and any other such system must be infinite in length and non-periodic. Converting Decimal Fractions to Binary. 625) a_split = (int(a//1),a%1) Convert the fractional part in its binary representation. Program to convert decimal to binary more than 18 bits. A number only becomes hexadecimal, decimal or binary when you format it in a certain way for printing. The right-most. s, To binary is easy: each digit after the decimal point (to the left) represents powers of 2. In the example a 1, 0, and then 1 is added to the listbox. Learn how to convert 26 to binary and create a Java project to convert from base-10 to base-2. decimal synonyms, decimal pronunciation, decimal translation, English dictionary definition of decimal. It takes three binary digits to represent an octal digit. > convert_to_binary(52) 110100 In this program, we convert decimal number entered by the user into binary using a recursive function. 3424 (base 10) = 110101100000 (base 2). This way people won't think it is the decimal number "101" (one hundred and one). I understand the arithmetics of it, but I was wondering how I can add the remainders onto each other using strings. For example, say that N is 23. Here you can make an arithmetic calculation you are interested in values through our online calculators. An understanding of binary numbers,the binary system, and how to convert between binary and decimal is essential for anyone involved in computers, coding, and networking. Below is the code for a 3-digit BCD-Counter (binary Coded decimal) I implemented while studying verilog. For example, decimal 1234. Created by computer nerds from team Browserling. Binary to Decimal Conversion To convert binary number into decimal number we need to multiply each bit of binary by increasing powers of 2 starting from 0. We will use the bitwise operator "AND" to perform the desired task. Just a quick recap, binary is a base-2 number system that uses 0 and 1 as the only digits, and hexadecimal is a base-16 number system that uses sixteen values from 0 to 9 and a to f as digits. , a number with fractional part. In a base 10 (or decimal) number, each column in the number stands for a power of 10. Decimal to binary conversion: two methods to do it with Matlab In this article we're going to present two methods of a decimal to binary conversion in Matlab; that is, we're going to convert decimal numbers (numbers with 10 different symbols, from '0' to '9', or in base 10) into binary numbers (numbers with only symbols '0' and '1', or in base 2). Converting Decimal Fractions to Binary. Binary numbers can contain digits 0 and 1 but no decimal or scientific notation. Each hexadecimal digit represents four binary digits (bits) (also called a "nibble"), and the primary use of hexadecimal notation is as a human-friendly representation of binary coded values in computing and digital. DECIMAL TO BINARY / HEXADECIMAL TO BINARY synonyms, DECIMAL TO BINARY / HEXADECIMAL TO BINARY pronunciation, DECIMAL TO BINARY / HEXADECIMAL TO BINARY translation, English dictionary definition of DECIMAL TO BINARY / HEXADECIMAL TO BINARY. How to Show that a Number is Binary. Repeat step 2 and 3 until result is 0. This way people won't think it is the decimal number "101" (one hundred and one). Java program to convert decimal to binary. The negative of this number is the value of the original binary. The decimal number is equal to the sum of binary digits (d n) times their power of 2 (2 n):. Converting a binary floating point number to decimal. So for example, 10111 is actually equal to 2^4+2^2+2^1+2^0 (2^3 is not included because there is a zero in that spot). It involves dividing the number to be converted, say N, by 2 (since binary is in base 2), and making note of. Welcome to the website Decimal-to-Binary. Example values of hexadecimal numbers converted into binary, octal and decimal. 2) Do conversion by writing your own logic without using any predefined methods. I'm having trouble writing the program, specifically finding a function that converts the items in the listbox to a decimal number. We will see two Python programs, first program does the conversion using a user defined function and in the second program we are using a in-built function bin() for the decimal to binary conversion. Binary has a base of 2. I've been trying to convert the number -441 to binary, but I don't really know how I can accomplish this. This wikiHow will show you how to do this. Returns a string representation of an integer or of a binary type converted to hexadecimal. To show that a number is a binary number, follow it with a little 2 like this: 101 2. How to Convert Decimal to Binary and vise versa is crucial to master, such skills will give you the confidence when you deal with Network and Storage devices. Decimal to Base 2 Algorithm. Input and output in binary, decimal, hexadecimal or ASCII. Converting between binary and decimal numbers is fairly simple, as long as you remember that each digit in the binary number represents a power of two. Byte conversion chart for binary and decimal conversion In Data storage and when describing memory size, a Kilobyte is 2^10, or 1024 bytes. Also Read: Convert Binary to Decimal in Java. Java Convert Decimal to Binary example and examples of string to int, int to string, string to date, date to string, string to long, long to string, string to char, char to string, int to long, long to int etc. This guide shows you how to convert from decimal to binary and binary to decimal. There are many number conversion interview questions which is asked in java interviews like java program to convert decimal to hexadecimal, java program to convert decimal to. Python Program to Check Whether a String is Palindrome or Not; Python Program to Multiply Two Matrices; Python Program to Convert Decimal to Binary Using Recursion; Python Program to Find Factorial of Number Using Recursion; Python Program to Convert Decimal to Binary, Octal and Hexadecimalc; Python Program To Display Powers of 2 Using. A binary number which consists of only 1 and 0, while a decimal number is a number which consists of numbers between 0-9. How computers see IP addresses. 1 mm to decimal = 1 decimal. Quite some time ago I had a particular need to convert Decimal to Binary and vice versa. Repeat this process till quotient becomes zero. Keep calling conversion function with n/2 till n > 1, later perform n % 1 to get MSB of converted binary number. hello i have a string with 11 bits in it in binary and would like to make this into a decimal how can i do it the stuff in the string looks like this 10111100111 and when this is converted to a decimal should be 3023. 30 mm to decimal = 30 decimal. Converts decimal -100 to binary (1110011100) Convert a decimal number to hexadecimal. Java program, asking user for a positive integer and converting it to binary representation. ) Instructions Just type in any box, and the conversion is done "live". Steps for using the binary decoder tool: First select the type of data you want the binary code to be converted. sage code for decimal to binary expansion. In hexadecimal, each digit can be 16 values, not 10. Decimal Code Binary; Decimal Coded Wire. Two’s complement notation really shows its value in binary addition, where positive and negative quantities may be handled with equal ease. Calculator for crc32, md5, sha1, ripemd128 and gost hash algorithms. Binary is the numeral system used to express data stored in computers. Practice these questions for upcoming IBPS PO Mains Examination. Then the calculate button is pressed and the textbox should display the corresponding decimal number. Converting binary to decimal (base-2 to base-10) or decimal to binary numbers (base10 to base-2) can be done in a number of different ways as shown above. Scanner; public class Decimal_Binary { Scanner scan; int num; void getVal() { System. About the Decimal/Binary Converter. Hi, I was asked to program a recursion (in C), which converts a binary to decimal. I've been trying to convert the number -441 to binary, but I don't really know how I can accomplish this. Hence, negative binary numbers are used to represent On or Off in electronic circuits. The maximum value of a 8 bit binary number is 255 in decimal. Unit Descriptions; 1 Gigabyte (binary): A gigabyte (binary) contains 1024 3 bytes, this is the same as a gibibyte. Example values of hexadecimal numbers converted into binary, octal and decimal. You can write it like this: Yes, I've made it up. Watch our tutorial on How To Do Decimals To Binary Numbers from one of Videojug's professional experts. Then the calculate button is pressed and the textbox should display the corresponding decimal number. 3020 3100 3370 3610 Question 4: Convert 11101111 from binary to decimal. Instead of using a base of 10 or 2 respectively, it uses a base of 16. Of course, a 32-bit binary cannot hold the full range of values that you can get from 12 decimal digits. Now press " x " button from the common operators box. program to convert decimal number to binary DESCRIPTION-: This program receive input as decimal number and then calculate its binary equivalent and then display the binary equivalent of that decimal number. - [Voiceover] Let's now see if we can convert a larger decimal representation to binary. To indicate our answer is in binary (binary numbers are base 2), we must include a 2 at the bottom right corner of the sequence. Binary number system is the basis of all computer and digital systems. As written, IniVal is declared, but only temp is declared and typed as string. Hexadecimal number system: It is base 16 number system which uses the digits from 0 to 9 and A, B, C, D, E. Binary, octal and hexadecimal number systems are closely related and we may require to convert decimal into these systems. If it's odd, write a 1. Please feel free to edit/adapt this before posting. Convert time hh:mm:ss to decimal hours, decimal minutes and total seconds. Instead, we enter decimal numbers the computer converts into binary before manipulating them. An Algorithm for Converting a Decimal Number to a Binary Number by Fox Valley Technical College is licensed under a Creative Commons Attribution 4. CBinS16 – converts a decimal signed integer to 16 bit binary string. The numbering system everyone is most familiar with is base 10, also known as decimal or denary. This is a discussion of COBOL Computational fields. First you need to define the properties of the 16 bit binary encoding that you want to use. Calculating binary numbers can be confusing, until you figure out the system. Java program to convert decimal to binary. Decimal to hexadecimal conversion method: Following steps describe how to convert decimal to hexadecimal Step 1: Divide the original decimal number by 16 Step 2: Divide the quotient by 16 Step 3: Repeat the step 2 until we get quotient equal to zero. Hi i am writing a program to convert a user entered integer number into its binary representation, the aim is to repeatedly ask the user to enter a value then output the number in binary and when 9999 is entered end program, i've managed to make it work with positive integers so far only it displays. Compiled on Platform: Windows XP Pro SP2. 001 is decimal. Decimals are nothing more than glorified fractions. If you’d like to dig deeper into Binary, this course helps explain the details, and show how Binary is the foundation of all digital computing. Can you describe a method of converting a binary number to decimal? 2. June 24, 2019. Following fig-1 mentions block diagram of decimal to binary labview vi. 0 to 255) Convert any binary number from 0 to 111111111 into octal; Convert any octal number from 0 to 777 into binary; Convert any binary number from 0 to 1111 1111 1111 1111 into hex. Using Command Line Arguments Program to Convert a Decimal to Binary Number. Hexadecimal is used to convert byte/modern computer numbers into defined binary digits. The program for conversion of decimal to binary depends on the problem specification. Create a program that would convert a decimal number to binary, octal or hexadecimal counterpart. Write your instructions in English. Converting Decimal to Binary numbers can be done in a number of different ways as shown above. 121 163 199 212 Question 2: Convert 101101 from binary to decimal. Let's see if we can convert the number 13 in decimal to binary. Decimal to Binary, Hex & Octal Converter to perform decimal to binary, decimal to hex & decimal to octal conversion online by using simple successive division methods along with step by step calculation & solved example problems. In this system, numbers are represented in a decimal form, however each decimal digit is encoded using a four bit binary number. All the best and keep practicing!. Before a floating-point binary number can be stored correctly, its mantissa must be normalized. We set it equal to the expression in Equation (2. A small point, but IniVal is NOT typed as a string variable in your code. Given a decimal number as input, we need to write a program to convert the given decimal number into equivalent binary number. Example for Binary to Decimal Conversion. In this post, we will see programs to convert decimal number to an equivalent binary number. Decimal ---> Base 10. Repeat the steps until the quotient is equal to 0. Step 2: Divide 10 by 2. decimal synonyms, decimal pronunciation, decimal translation, English dictionary definition of decimal. 500 * 10 9 bytes = 500,000,000,000 = 500 Gigabytes But the operating system determines the size of the drive using the computer's binary powers of two definition of the "Giga" prefix:. Like I said a hexadecimal number doesn't actually name a type: a type is something like long, or int. Hexadecimal is base 16. I first converted 441 to binary which is: 110111001 Then I'm supposed to take the complim. If everything is done right, the result should be 34. Java decimal to binary conversion is the most important core java interview question. 7/2 = Quotient = 3(grater than 1), Remainder = 1. Conversion Hexa to Decimal. Source code DecimalToBinaryConverter. The hexadecimal number system (hex) functions virtually identically to the decimal and binary systems. In computing and electronic systems, binary-coded decimal (BCD) is a digital encoding method for decimal numbers in which each digit is represented by its own binary sequence. You rely on decimal numbers when doing calculations for your business, such as when figuring sales tax or creating a monthly budget. How to Convert Decimal to Binary and vise versa is crucial to master, such skills will give you the confidence when you deal with Network and Storage devices. Convert 13 10 to binary:. If you understand binary because you learned it for understanding how IP addresses work, you usually stop at 2 7 because one octet of an IP address is 8 bits long. Say we wish to convert an unsigned decimal integer, $$N\text{,}$$ to binary. The binary system is the internal language of electronic computers. Set s=0 for a positive number and s=1 for a negative number. How to convert binary to decimal: The binary number system, also known as the base 2 number system; is used by all modern generation computers internally. Online IEEE 754 floating point converter and analysis. BCD-To-Decimal Decoder Binary-To-Octal Decoder The MC14028B decoder is constructed so that an 8421 BCD code on the four inputs provides a decimal (one−of−ten) decoded output, while a 3−bit binary input provides a decoded octal (one−of−eight) code output with D forced to a logic “0”. This free binary calculator can add, subtract, multiply, and divide binary values, as well as convert between binary and decimal values. It took a little digging but eventually came across this little gem and thought it was worth posting for others to use. For more information about the binary representation of Decimal values and an example, see the Decimal(Int32[]) constructor and the GetBits method. How to convert from decimal number system to binary number system using bitwise operator in C programming. Decimal to binary converter Abstract. Decimal : For denoting integer and non-integer numbers, decimal number system uses 10 different digits, 0,1,2,3,4,5,6,7,8 and 9 i. The binary number is constructed from right to left. Convert from binary to decimal algorithm: For this we multiply each digit separately from right side by 1, 2, 4, 8, 16 … respectively then add them. Decimal conversions can be done by either successive division method or successive multiplication method. Then enter or paste your binary code in the first text box and click Decode button. (The old flash version is here. And I encourage you to pause the video, and try to work through it out on your own. The following utility converts the IP (TCP/IP) address to other browser URL addressable forms. Hexadecimal --> Base 16. For more information about the binary representation of Decimal values and an example, see the Decimal(Int32[]) constructor and the GetBits method. This page of labview source code covers decimal to binary labview vi which converts decimal vector to binary vector. 0,1,2,3,4,5,6,7,8,9 are decimal number and all other numbers are based on these 10 numbers. Here you can make an arithmetic calculation you are interested in values through our online calculators. Binary --> Base 2. The syntax is similar to that of the ToInt32 method. , a number with fractional part. That is the most confusing thing that I want to say about the subject. Jun 9, It was a fascinating rabbit hole that exposed me to things like “decimal machines” and “2 out of 5 code”. Short for binary-coded decimal, BCD is also known as packet decimal and is numbers 0 through 9 converted to four-digit binary. So Java Code Online is going to discuss a case when a number entered as binary is to be converted to a decimal number. Dotted decimal notation is a system of presenting numbers that is a little different from the common conventions in arithmetic as it is taught in schools. Here you can find the answer to questions like: Convert decimal number 168 in binary or Decimal to binary conversion. The Land Before Binary.


# Maximum Prefix Mex Sum - Editorial

Setter(s) Divyansh Verma
Tester(s) Vichitr Gandas, Mayank Pugalia
Difficulty Easy
Topics Observation, Greedy, Maths, Combinatorics

## Solution Idea

The prefix Mex sum is maximum when we put all unique numbers in increasing order at the start. Once a number is missed from 0,1,2..., this chain breaks and then Mex will remain same for rest of the prefix arrays. Example:
A=[0,1,2,5,7]
Here Mex for first three prefix arrays is 1,2,3 respectively. After that we see that 3 is not present. Hence for rest of subarrays Mex will remain 3 only.
This way, the sum is maximized when we arrange initial unique numbers in order 0,1,2... like this. And if this order breaks or we are left with duplicate items, we can permute them in any way. Here in the beginning, each unique number x can be picked with count[x] ways where count[x] is number of times it occurs in the array. And let’s say number of remaining elements after break is y, they can be arranged in y! ways. Hence total number of permutations with max prefix Mex sum would be count[0]*count[1]*...*count[k]*y! where k is first number where this sequence 0,1,2,... breaks.

Count can be maintained with map. We can reduce the time complexity here by using a vector instead because the sequence if continued won’t exceed n and frequency of rest of the items is not needed, because they will be part of left over items which can be permuted.

### Complexity Analysis

Time Complexity: \mathcal{O}(n \log{n}) for pre calculation and to find the answer which can be reduced to \mathcal{O}(n) using a vector instead of map.
Space Complexity: \mathcal{O}(n) for maintaining the frequency.

## Codes

Setter's Code
#include<bits/stdc++.h>
using namespace std;
#define int long long int

const int MAXN = 1e5 + 10;
const int MOD = 1e9 + 7;

int fact[MAXN];

void preCompute(){
fact[0] = 1;
for(int i=1;i<MAXN;i++) fact[i] = (fact[i-1]*i)%MOD;
}

class FindMexMax {
public:
int getOutput(vector<int> a){

map<int,int> which;
for(int v:a) which[v]++;

int cnt = a.size(),ans = 1;

for(int i=0;i<a.size();i++){
if(which[i] == 0) break;
ans = (ans*which[i])%MOD;
which[i]--;
cnt--;
}

ans = ans*fact[cnt]%MOD;
return ans;
}
};

signed main(){
preCompute();
int t;
cin>>t;
while(t--){
int n; cin>>n;
vector<int> a(n);
for(int &v:a) cin>>v;
FindMexMax fMM;
cout<<fMM.getOutput(a)<<endl;
}
}

Tester's Code
/***************************************************

@author: vichitr
Compiled On: 13th Mar 2021

*****************************************************/
#include<bits/stdc++.h>
#define endl "\n"
#define int long long
using namespace std;

const int MOD = 1e9 + 7;
const int N = 1e5 + 7;
int fact[N];

void init(){
fact[0] = 1;
for(int i=1;i<N;i++){
fact[i] = (fact[i-1] * i) % MOD;
}
}

void solve(){
int n; cin>>n;
assert(n <= 1e5);
int a[n];
map<int, int> M;
for(int i=0;i<n;i++){
cin>>a[i];
assert(a[i] >= 0 and a[i] <= 1e9);
M[a[i]]++;
}
int ans = 1;
int tot = 0;
for(int i = 0; i < n; i++){
if(M[i] == 0){
break;
}
ans *= M[i];
ans %= MOD;
tot++;
}
ans *= fact[n - tot];
ans %= MOD;
cout<<ans<<'\n';
}

signed main(){
int t = 1;
cin>>t;
assert(t <= 5);
init();
while(t--){
solve();
}
return 0;
}

If you have used any other approach, share your approach in comments!