Why can’t you invent an imaginary number for division by zero like you can for a square root of a negative?

In: Mathematics

There are two ways to tackle this question: symbolically and physically. Let’s go with the physical first.

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Consider what division represents, using the expression `6 / 2 = 3` as a guide.

>You have six apples. Dividing them into two groups leaves you with three apples per group.

Similarly, you can extend this to division by fractions: `6 / .5 = 12`.

>You have six apples. Dividing them into half a group means that one whole group would have twelve apples.

So far, so good, right?

But: `6 / 0 = ?`.

Let’s divide six apples into zero groups. How many apples per group? …Well, there are zero groups, so…you can’t answer the question.

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Now, symbolically. Let’s do what you suggest, and invent a new number to represent the multiplicative inverse of zero — the number such that z = 0^-1 .

This means that 0z = 1.

But we know that 0z = 0.

By defining a number to be the multiplicative inverse of 0, we end up attempting to assert that 1 = 0, which we know to be false. Therefore, there can be no number for division by zero.

You can. All the people saying it’s impossible are wrong. For example, you could add a value called ∞ to the real numbers, and say that x/0 = ∞ for any nonzero value of x, [turning the number line into more of a number circle](https://en.wikipedia.org/wiki/Projectively_extended_real_line).

However, this sort of thing is not as useful as the idea of imaginary/complex numbers, for two reasons. First, the resulting system isn’t that interesting: it’s basically just the real numbers with a single extra point, and that point has rather boring properties like ∞ + x = ∞ and ∞ – x = ∞ for any real number x. The complex numbers have far more interesting behavior. Second, and more importantly, the complex numbers provide additional insight into the real numbers, especially in calculus and related fields. The mathematician Jacques Hadamard once said that “the shortest and best way between two truths of the real domain often passes through the imaginary one”. Adding the point ∞ to the real numbers, on the other hand, doesn’t tell us much at all.

You can. People have; one example is the projectively extended real line, which is more or less the real numbers and a point at infinity. In this number system, 1/0 = ∞. The real question is “why don’t we use a number system where we can divide by 0 in everyday life?”

Part of it is cultural; “we just don’t.” But there are good reasons for it. Number systems that allow division by 0 inevitably lose some useful properties of the real numbers (side note: the real numbers aren’t any more or less “real” than any other kind of number; mathematicians just suck at naming things). For instance, ∞ introduces all sorts of weirdness. ∞-∞, in the projectively extended real line, is undefined, just like 1/0 is in the reals. So, now you can say 1/x=∞, but you can’t say x-x=0! Generally, trying to allow division by 0 is more trouble than it’s worth for everyday purposes.

Mathematicians like when things work nicely and in almost every case, allowing division by 0 will make things not work nicely.

For example, one of the nice things about numbers is that we can multiply numbers in whatever order we want and still get the same result. For example (2 * 5) * 4 = 2 * (5 * 4) and 2 * 3 = 3 * 2. (Mathematicians call these “associativity” and “commutativity” of multiplication.)

Another nice property is that a(x + y) = ax + ay. For example, 2*3 + 2*4 is the same as 2(3 + 4). (Mathematicians call this “distributivity” of multiplication over addition.)

Let’s see what happens if we allow division by 0. Let’s just make a new thing called X and we will define that 0 * X = X * 0 = 1. So 1 / 0 = X.

Then (3 * 0) * X = 0 * X = 1. Except that 3 * (0 * X) = 3 * 1 = 3. But 3 is not equal to 1!

So this breaks one of the rules we like about multiplication. Maybe this is no biggie, maybe we can just let it slide. Let’s keep exploring.

Let’s check if distributivity still holds. How about X * (0 + 0)?

X * (0 + 0) = X * 0 = 1, but X * 0 + X * 0 = 1 + 1 = 2. Another problem… we lost distributivity.

If you keep exploring, you’ll likely find even more problems with assuming that X = 1/0 exists.

We now have two options here: First, we can allow division by 0, in which case we would have to abandon a bunch of things that work nicely with arithmetic. Or, second, we could just say that X does not exist. The second option is almost always the best option.

You can divide by zero if you are on the extended complex plane, which is the complex number (x i +y, so the combination of real and imaginary numbers ) and ∞( infinity). It is often represented as the [https://en.wikipedia.org/wiki/Riemann_sphere](https://en.wikipedia.org/wiki/Riemann_sphere)

There is only one ∞, not a +∞ or -∞ that you can get to in limes valuation for a real number.

A number on the complex plane is usually represented with z.

z/0 = ∞ and z/∞ =0 that is if z is not 0 or ∞

∞/0 = ∞ and 0/∞ =0

What is still undefined is ∞/∞ and 0/0

But if you use the extended complex plane you need to know what it lacks some stuff you are used to in for example real numbers.

One example is that numbers do not have a well-defined order. 4> 3 is well known but is i >1, i<1 or i=1? The answer is complex numbers do not have a single defined order, you can only compare the absolute value that is the distance from 0 and is represented by |z|

The result is that the distance to |i| and |1| both are 1. This is also why there is only one ∞ and not +∞ and -∞

This shows that if you add stuff like the ability to take the square root of -1 the resulting number will loo some other stuff like an absolute order.

This is true for complex numbers not just if you extend the plane and include ∞. But is a relatively simple example that if you gain stuff you loos stuff too. So you can divide by zero in some situations if you know what to do and what other consequences that have.

It has lots of practical applications and for example, https://en.wikipedia.org/wiki/Control_theory where https://en.wikipedia.org/wiki/Zeros_and_poles is a common tool to know how to control systems.

I think the truth is we got lucky with imaginary numbers. I don’t know the history, but it’s not really correct to say that i is the square root of minus 1, in the way that we normally understand square roots. We sort of stumbled onto a whole system of arithmetic, which contains the real numbers within it, and in which there’s a “number” that when “multiplied” by itself gives – 1. Which is neat. But it wasn’t invented exactly.

Someone might stumble on something analogous to that for 1/0, or might have already (possibly in multiple ways). But it wouldn’t really be division as we know it.

Using imaginary numbers for the square root of a negative number gives something useful. It may not be within the Real number spectrum, but it definitely is a value that exists.

Division by zero does not exist.

Division at its base concept is to take a group and split it into a specific number parts.

Take 10 marbles divided by 5. The result is 5 groups of 2.

It’s the opposite of multiplication. One handy tool I learned when younger is that when imagining multiplication with physical objects, the word “of” often means “multiply”.

Notice: 5 groups **of** 2 is a total of 10. 5 x 2 = 10.

By definition, division needs to be a reversible process.

Imagine dividing 10 by 4 now. You’d need to split it into 4 groups of 2.5.

Into 3, and you’d need 3 groups of 3.333…

Into 2, 2 groups of 5.

Divide by 1, and a single group of 10.

Each of these is reversible multiply each and you’ll get back to 10.

Divide by 0, and how many groups do you split 10 into?

It’s just not a thing.

What about the reverse? 0 times X = 10. X could be anything. It’s not a reversible process with a definitive answer. You could make it anything, and if it can be anything, then dividing by zero cannot be defined.

If you want an example for how dividing by zero breaks everything, [watch this math teacher prove that 1 = 2](https://www.youtube.com/watch?v=hI9CaQD7P6I).

See if you can identify the step where the proof breaks. Hint, there’s a point where he ends up dividing by zero. He just masks it by using variables.

You basically can, this is how calculus works. But it has a very very special set of rules, notation, and requires you to really understand what you are doing or get absurd results.

Disclaimer: In math, we can look at all kinds of different structures which have all kinds of different properties. This leads to a lot of variety, but there are some limitations. One such limitation is, that a structure cannot have contradictory properties.

Example:

>This structure has a smallest number.

This statement is true for some structures like the natural numbers {0,1,2,…} and false for others, like the whole numbers {…, -2, -1, 0, 1, 2, …}.

>This structure does not have a smallest number.

This statement is true for some structures like the whole numbers {…, -2, -1, 0, 1, 2, …} and false for others, like the natural numbers {0,1,2,…}.

But it is not possible for a structure to simultaneously possess both properties, as one is the negation of the other. These properties are contradictory.

The answer to your question is, that division by zero is possible given the right structure (these would be pretty obscure, especially for a layman), but contradictory with the properties of our usual number systems.

To see that, let’s look at some of those properties!

Firstly, let’s look at “neutral elements”. When add/multiply these to another number, the outcome doesn’t change. The neutral element of addition is 0, for multiplication it’s 1.

Example:

>2 + 0 = 2 , 2 * 1 = 2 , in general n + 0 = 0 , n * 1 = n

Secondly, let’s look at multiplication by 0. I’m not going to go into too much detail because this post is already going to be very long, but basically multiplying anything by 0 will be 0.

Example:

> 3 * 0 = 0 , in general n * 0 = 0

Thirdly, let’s look at so called “inverse elements”. If we can combine two numbers to make the neutral element, these numbers are inverses of each other.

Example:

>2 + (-2) = 0 , 2 * (1/2) = 1 , in general n + (-n) = 0 , n * (1/n) = 1

You may have noticed that we haven’t talked about subtraction or division up to this point. In fact, we can simply define these two operations in terms of addition and multiplication using the notion of inverse elements.

> 4 – 2 is the same as 4 + (-2)

> 5 / 3 is the same as 5 * (1/3)

We’re mainly interested in multiplicatve inverse elements, which means we’re going to take a look at rational numbers because we need fractions.

What is a fraction?

Basically, a fraction is written *a / b* where *a* is an integer and *b* is a natural number bigger than zero with the property

> If you multiply this number by *b* it will be equal to *a*.

Now let’s take a look at division by 0. Recall that division is simply multiplication with the inverse. So imagine there was a fraction of the form *1 / 0*. Let’s look at the term

> 0 * (1 / 0) = ?

This is where we run into our contradiction, because we can apply two different rules to get different outcomes.

By the definition of a fraction, *1 / 0* is number becomes 1 when multiplied by 0, so the answer should be 1.

In contrast, anything multiplied by 0 is 0, so the answer should be 0.

So we can see that introducing this fraction *1 / 0* would lead to a contradiction, that’s why we explicitly forbid 0 to be the denominatior of a fraction in the definition.

You can, and people have.

Creating an imaginary number that breaks the rules of squaring a negative produces a useful concept that can be used to explore things like imaginary roots of polynomials, which helps with things like typing solutions to differential equations. So far, there has been absolutely no useful application of a concept that allows you to divide by zero, and it also comes with the added complication of creating many more problematic hiccups in your math, such as allowing 1=2.

I challenge you to define a new symbol, &, where &/0=1, and produce a useful result with it.

As a math major, the closest I’ve come to dealing with division by zero is using limits. This means you divide by numbers approaching zero and see how the result responds. You must do this both from the negative (very small negative numbers) and positive very small positive numbers. There are basically 4 cases:

1. Both sides approach positive infinity. Therefore the limit of x/0 = infinity

2. Both sides approach negative infinity => negative infinity

3. Both sides approach 1 (since it’s essentially 0/0) => 1

4. The sides approach different values (infinity on one side, negative infinity on the other side, etc). In this case, it’s undefined

To put it simply, if we do, all starts of funky things can start happening, like proving that 1 = 2.

Since 1 isn’t equal to 2 we can’t have anything for division by zero

My favorite explanation for this is the t shirt dilemma. Say you have $100 and t shirts cost $5. Then you can buy 100/5=20 t shirts. But what if t shirts are free? You could buy 1, or 10, or 50, or literally any number of shirts. So we say that 100/0 is “undefined” because there are multiple possible correct answers. It’s not that we can’t invent something to represent the value of 100/0, it’s that there is no single correct such value.

As others have said, you can also see this algebraically using an equation. But that’s not appropriate for most five year olds.

To put it in non-math terms. Take a pizza and slice it up into 8 pieces. You divided it by 8 and every piece is 1/8th of the whole. But if you divided it into zero pieces, there are no pieces at all, the pizza can’t exist if there are zero pieces. The result can’t be represented by anything because it no longer exists.

First off, I’d like to say that we didn’t invent imaginary numbers, they were with us the whole time. “i” was just called imaginary by an extremely prominent mathematician of the time named Rene Descartes as a derogatory way of telling another person that they were asking stupid questions and inventing stupid answers and the term stuck. (Do not judge Descartes to harshly here, he made a mistake on this, but has done a lot of good like cartesian coordinates which are named for him.)

Now to get into things a bit deeper on what is actually going on with division by zero.

There are three types of numbers, prime composite and identities. 0 is the additive identity for the real numbers. This is why it behaves quite oddly in the first place. It isn’t truly meant to be part of multiplication, and as such does not have a multiplicative inverse to reach 1, the multiplicative identity. Basically

0 × r =/= 1 for any real number r.

This also has the effect of any number times 0 equalling 0. And since division is multiplication by the inverse we get

0 × r = 0 for any real number r

=> 0 ÷ (1/r) = 0/r = 0 for any real number r.

So if I divide some number by 0 and then run it through the basic process to turn that into a multiplication problem rather than a straight division problem I get

1 ÷ 0

= 1 ÷(0/r)

= 1× (r/0)

= r/0

But r can be literally any number at any time. And the problem also just recycled back on itself by dividing r by zero, which will just happen over and over again because zero does not have a multiplicative inverse to stop the process.

If you’re still interested in learning more about what’s going on here, this is an idea in group theory. Work at it and keep asking questions like you are, and you too many end up in a theoretical math program asking some really neat questions and getting some truly fascinating answers.

You can. In fact there are multiple, conflicting ways to do it. It is sometimes convenient to extend the real line by compactification in such way that infinity = -infinity, like an infinite ‘circle’. Sometimes it is convenient to keep these separate. So we don’t take any of them as *default*, and in the reals themselves we leave 1/0 as undefined.

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