# Why use absolute value?

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 $-\sqrt2$ – Detmondyou Jun 11 '18 at 19:08
• The absolute value of $i$ is 1, not $-1$. – Xander Henderson Jun 11 '18 at 19:09
• 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. – Chase Ryan Taylor Jun 11 '18 at 20:08
• @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 0:44

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.

• 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. – Detmondyou 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
• @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. – Detmondyou Jun 11 '18 at 19:24
• @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

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 $1$.”

• Yes, but my question is what is it for other than to find the distance from 0 to $x$, which should be intuitive. – Detmondyou 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
• 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. – Detmondyou Jun 11 '18 at 19:30
• @MichaelHoppe But subtraction does that fine. – Detmondyou Jun 11 '18 at 19:31