Divide by 0 alternative

Question

Cutting to the chase.

I know you can't divide by zero. And I have read a good few explications for this. And I am happy with this as a fact.

BUT my question is based on this:

X / N = A "should match" A * N = X

At least one side of this has a problem when you get to N = 0.

I see no reason why we picked on divide as being the one to have the problem.

Is it not equally reasonable to have said that anything * 0 is undefined and that anything / 0 = 0?

While this solves no problems what so ever as we still have a system that cannot be undone, I would just like to know out of interest if this would have at lease been a plausible option when making up the rules!

Thanks.

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TL;DR: Is there a consistent formulation of mathematics (i.e. addition, subtraction, multiplication, division) in which we have the theorems $$ x/0 = 0 \quad \text{for all numbers } x\\ x*0 \quad \text{ is undefined}? $$

Edit

To be clear here, My question is. could we have "picked" X / 0 = 0 and X * 0 = 'Undefined' and still continued to enjoy mathematics in the same way we do today (allowing for the difficulty we already have with dividing by 0 except only now with multiplication instead)

Answer 1

Nobody is picking on division, it is simply a matter of definitions.

Think about what how the binary operation $X/N$ is defined: $X/N = X \times \frac{1}{N}$ which, in words, means "$X$ times the multiplicative inverse of $N$".

This usually does not cause any trouble, because every real number has a multiplicative inverse except for $0$. The multiplicative inverse of $0$ is, simply, undefined. So the expression $X/0$ is, simply, undefined.

Answer 2

Think about it this way.

Lets say we have this product:$$123\times x$$You can "intuit" that the value of this product will become smaller and smaller as you reduce the value of $x$. In the limit, as $x$ tends to zero, this product will, therefore, also tend to zero.

Now think about this division:$$\frac{123}{x}$$You can "intuit" that the value of this division will get bigger and bigger as you reduce the value of $x$. In the limit, as $x$ tends to zero, this division will, therefore tend towards an unimaginably large number and is therefore undefined.

Answer 3

Is it not equally reasonable to have said that $\text{anything} \cdot 0$ is undefined and that $\frac{\text{anything}}{0} = 0$?

It is not equally reasonable for several reasons, and I'm sure other users could explain many of them. However, I will give one reason. Assume that we have not defined multiplication or division by $0$ yet, but we want to (where it makes sense). Well, one way is to look at the limit as we approach $0$.

What can we say about $\text{anything} \cdot x$ as $x$ gets closer to zero? We see that the product, in fact, approaches $0$; and this is true whether $x$ is positive or negative.

What can we say about $\frac{\text{anything}}{x}$ as $x$ gets closer to zero? Not only does the expression not get close to any real number, the value of the expression is also highly dependent on whether $x$ is negative or positive.

Therefore we are more justified in defining it the way it is.