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So, there’s this thing called absolute value, or a modulus function that basically says how far away any real number $n$ is from $0$. For example, $|2|=2$ because $2$ is $2$ units away from $0$. Furthermore, a negative number’s $($such as $-3)$ absolute value is simply its positive counterpart. So, $|-3|=3$.

This got me thinking, what is the usage of such a function other than to turn negatives into positives?

Edit: Xander Henderson commented about the Wikipedia article for absolute value, and honestly, the article literally repeats what I already know. This is the case for every video or website I go to. In this post, I want to know if there is any OTHER use for absolute value other than telling an integer’s distance from $0$.

  • Complex numbers? Based on if you're referring to the actual notation, absolute value bars can mean the determinant in linear algebra, magnitude with complex numbers, etc. – Andrew Li Jun 11 '18 at 19:07
  • No, the absolute value of $i$ is -1, so something like $2i$ would have an absolute value of $-\sqrt2$ –  Jun 11 '18 at 19:08
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    The absolute value of $i$ is 1, not $-1$. – Xander Henderson Jun 11 '18 at 19:09
  • Then $|2i|=2$. This doesn’t really change anything. (Or maybe $\sqrt{2}...$) –  Jun 11 '18 at 19:10
  • You are correct. $|2i| = 2$. – Xander Henderson Jun 11 '18 at 19:24
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    Again I am saddened that an astute question has been downvoted because it covers rudiments. This question meets all the criteria of a good post, and no one has offered advice on how to make it better! You shouldn’t have to be a math pro to ask well received questions here. – gen-ℤ ready to perish Jun 11 '18 at 20:08
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    @ChaseRyanTaylor You have assumed that people have downvoted and/or voted to close this question because it is rudimentary. While I have done neither, I have sympathy for both: the question basically comes down to "what use is the absolute value?" which is extremely broad, and displays a shocking lack of research (the first paragraph of the Wikipedia article would be a good place to start). – Xander Henderson Jun 12 '18 at 00:44
  • I just saw something even better (not to disregard your comment @XanderHenderson): our OP is a seventh grader. Not only is s/he not overly versed in mathematics, but s/he probably doesn’t have much knowledge of how to conduct personal research or pose questions that are directed towards online forums. Let’s all play nice; you never know—we might be, after some years of schooling and academic cultivation, be dealing with a future Nobel Laureate. I’m getting a bit lofty here, though, so I’ll leave things at that. – gen-ℤ ready to perish Jun 12 '18 at 02:55
  • @ChaseRyanTaylor Not to disregard your comment either, but are we supposed to be age-impartial in terms of judging the post's quality? Someone who is young is still able to write a good post, but this is just too broad in my view. Even a little research, as Xander mentioned, would go a long way even if you're just a middle schooler. – Andrew Li Jun 12 '18 at 16:51
  • @AndrewLi Yes, I totally agree. The post is not without objective flaw, and I can’t reasonably defend it anymore than I have. Yes, I offered a bias in the positive direction, but I just wanted to balance the opinions out. I’m not sure where to go from here though $\ddot\smile$ – gen-ℤ ready to perish Jun 12 '18 at 17:13
  • @ChaseRyanTaylor I might be in 7th grade, but not to brag, I know basically all of 9th grade math, so I think I know what I’m talking about. As for the research, I do admit I could’ve included some hint about it, which I will do. –  Jun 13 '18 at 11:21
  • @Detmondyou Don’t bite the hand that’s defending you . – gen-ℤ ready to perish Jun 13 '18 at 17:22

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In $\mathbb{R}$, the absolute value function may seem too simple to be useful. But the 'idea' of an absolute value is generalizable and quite important, because it captures the concept of distance between two points. For example, in $\mathbb{R}$, $\lvert x-y \rvert$ tells us how far $x$ is from $y$. It measures distance. Now move up to $\mathbb{R}^3$. We can grasp the idea of distance between points $x$ and $y$ in $\mathbb{R}^3$, but how do we denote it? We can write $\lvert x-y \rvert$. Now, this again gives the distance between $x$ and $y$, but it is not the same simple function as it was in $\mathbb{R}$; however, it captures the same idea.

Having said that, even in $\mathbb{R}$, I would argue that the absolute value function simplifies notation a lot. For example, when we talk about a sequence $a_n$ converging to a limit $a$, we usually say something like $\forall \epsilon >0, \exists N$ such that if $n>N$, $\lvert a_n - a \rvert < \epsilon$. If we didn't use the absolute value function, we would have written $a_n - a < \epsilon$ if $a_n > a$ and $a - a_n < \epsilon$, otherwise. Obviously, this is more cumbersome.

John
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  • Doesn’t substraction tell us the difference between $x$ and $y$? And if the result is negative, we can just remove the negative sign manually. We don’t need a function for that. –  Jun 11 '18 at 19:13
  • @Detmondyou $x-y$ gives you a difference in one direction, $y-x$ the difference in the other direction; neither of these is generally the distance between points on a number line. Distance is always positive. – David K Jun 11 '18 at 19:18
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    @Detmondyou Why do we need the $\exp$ function, when we can just put the $x$ into $\sum\limits_{n \in \mathbb{N}} \frac{x^n}{n!}$? – Botond Jun 11 '18 at 19:23
  • @David K What do you mean by distance between points? If I do $5-2$, the answer is $3$. $5$ is $3$ units away from $2$. $2-5=-3$. Remove the negative sign, and we get $2$ is $3$ units away from $5$, which it is. –  Jun 11 '18 at 19:24
  • It's also much easier to write $0<|x|<\epsilon$ than $-\epsilon<x<0\lor 0<x<\epsilon$. Especially if you replace $\epsilon$ and $x$ by more complicated expressions. – celtschk Jun 11 '18 at 19:25
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    @Detmondyou “Remove the negative sign”? Do you see what you did right there? You took an absolute value of a number when its sign became inconvenient for you, because you really wanted distance rather than a difference. If the difference were what you wanted, you would have kept the negative sign. – David K Jun 11 '18 at 19:31
  • @David K It seems to me, however, that if what you say is true, then distance and difference are practically synonymous. –  Jun 11 '18 at 19:35
  • @Detmondyou What? David K just said that a sign separates distance and difference... they are not synonymous. The absolute value of difference is distance (because negative distance doesn't make sense). – Andrew Li Jun 11 '18 at 19:42
  • Oh, ok. That makes more sense. –  Jun 11 '18 at 19:44
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You’ve mentioned the usage in your first sentence: to measure the distance from zero, that is to measure the length of a number.

A prominent example: $x^2=1\iff|x|=1$. That reads: “The square of a number equals $1$ iff its distance from zero is $1$.”

Michael Hoppe
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  • Yes, but my question is what is it for other than to find the distance from 0 to $x$, which should be intuitive. –  Jun 11 '18 at 19:26
  • @Detmondyou: What is addition useful for other than for describing the sum of numbers? – celtschk Jun 11 '18 at 19:28
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    No, that’s the only purpose the absolute value was made for and nothing else. And once you can measure distances to zero you may calculate the distance of any two numbers by taking the absolute value of their difference. Isn’t it quite important to measure distances? – Michael Hoppe Jun 11 '18 at 19:29
  • @celtschk Nothing else; it’s made to be simple. That’s why it’s taught in kindergarten. Last I checked, you don’t hear about absolute value in high school, so I figured it was a little more complicated. –  Jun 11 '18 at 19:30
  • @MichaelHoppe But subtraction does that fine. –  Jun 11 '18 at 19:31
  • @Detmondyou: Well, I learned about the absolute value as soon as I learned about negative numbers. – celtschk Jun 11 '18 at 19:31
  • Did you teach yourself or were you taught? OR did you realize that what you were doing was absolute value. –  Jun 11 '18 at 19:31
  • @Detmondyou: I were taught. – celtschk Jun 11 '18 at 19:32
  • Hmm.. Then something’s different up here in Canada, ‘cause we’ve just learnt about reflex angles. ;-; –  Jun 11 '18 at 19:34
  • @Detmondyou: Well, different countries (and sometimes even different schools in the same country) have different curricula. Which shows that the year you learn something is not necessarily related to about how complicated it is. For example, I recently heard that in the US, fractions are taught before negative numbers. But fractions are far more complicated than negative numbers! – celtschk Jun 11 '18 at 19:46