# Why is divide by zero not the square root of infinity?

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Searching through past posts I didn’t see anything specific to this. As I understand it, we can approximate what occurs when dividing by zero if you graph the function [y=1/x](https://i.imgur.com/MebU9l3.png) (ripped from google)

As X approaches zero, it becomes both infinity or negative infinity, which results in it being undefined.

Couldn’t positive or negative infinity be defined as the square root of infinity? or the square root of infinity squared? Obviously not all infinities are equivalent.

Thanks

Sincerely, a person who failed math

In: 0

No. Infinity is a concept not a number. And you can’t divide anything into groups of nothing. Since infinity isn’t actually a number talking about it’s square root is nonsensical.

You can define “plus infinity” and “minus infinity” as special values and then postulate that dividing x by zero gives that infinity value with the sign determined by the sign of x (provided that x itself is non-zero and non-infinity). The result is called the “extended real number line”, you can look it up on wiki. This leads to certain complications down the road, however. For one thing, the values of expressions like “infinity minus infinity” and “zero times infinity” will have to be left undefined anyway, and that limits the useful things you can do with that version of the real number line.

Mainly because that doesn’t really make any sense.

Consider mainly what division is, division is taking a group of objects and splitting it into that many groups how many objects are in each group.

Take 50, divide it into 2 groups, and each group contains 25 objects.

How do you take a group of objects and divide them into no groups?

The thing is that in math, you can define whatever you want.

So, you can define a function called squaring however you want. You can make it be a map from the set {all real numbers and +/- infinity} to itself (here infinity and -infinity are purely symbols) which sends -infinity and infinity to infinity. Under this map, the elements that gets sent to infinity is infinity and -infinity. In this sense, the squareroot of infinity is infinity and -infinity. You can also define a function called division, which takes in two elements of the set {all real numbers and +/- infinity}, and spits out another element in this set, however you want. You can make it so that 1 divided by 0 is the symbol infinity.

With these definitions, the square of (1 divided by 0) is infinity.

Such definitions aren’t always nice though. For example, with these definitions, some usual properties of arithmetic will need to be violated.

It’s very important in math to say exactly what we mean in order to avoid confusion about things like this question!

Say what you’re saying, but be more precise about it. As x approaches zero from the left, y approaches (not becomes) negative infinity. As x approaches zero from the right, y approaches (not becomes) positive infinity. There is nothing wrong with this happening. It does mean that the limit as x approaches 0 is undefined, which again is perfectly acceptable as a result. Nothing has to happen in math just because it feels like it should. Instead, what happens, happens.

Note that you are using the word “approaches” very loosely, but it has a very technical definition. When we say something approaches infinity or negative infinity we in no way treat infinity as a number, or define it at all even. It’s typical to gloss over these technicalities in a first year calculus class.

Creating a new definition is very tricky. If you want to define positive or negative infinity as the square root of infinity then you need to be clear about what this even means. You can define anything to be anything, but this can lead to meaningless results.

If you try to define a thingy called “inf” such that sqrt(inf) = inf = -inf then you can see how it plays with numbers. You will probably find that it creates paradoxical situations, which means that the definition does not play well with the rules that numbers follow. You may need to get rid of other important rules in order for “inf” to be treated as a number.