What’s stopping mathematicians from defining a number for 1 ÷ 0, like what they did with √-1?

Question

What’s stopping mathematicians from defining a number for 1 ÷ 0, like what they did with √-1?

In: Mathematics

Answer 1

I can’t see a comment that has explained it this way: but the answer is you can’t do that *consistently*.

It is obviously quite true that you can define anything in any way you like. But if you say defined a fictitious number “infinity” and said 1/0 = infinity, then you immediately get into difficulties as follows:

1/0 = infinity

1 = infinity * 0 (multiply both sides by zero)

1 = 0 (any number times zero is zero)

If 1=0 that is going to unravel most of your mathematics. Of course you can say “new rule: infinity * 0 = 1” but then you will get into more trouble later on.

If you play around with this you will find that either you will change the rules so much that division is not really division in the usual sense and that you 1/0 doesn’t fit into the normal number system, or you will have to give up with too many inconsistencies.

The complex numbers are different. If you add the following assumptions (1) there is a – let’s call it a number – “i” and (2) i squared is -1, but you do not change any of the other rules of multiplication, division, addition or subtraction, you get a consistent system. Everything continues to work as before. You can solve equations in the same way. You do loose ordering (you can’t use > or < usefully any more – complex numbers are better thought of as a 2D plane) but you still get to do a great deal of mathematics.

The reason dividing by zero doesn’t work is that division is an inverse of multiplication. Multiplication by zero squashes everything down to one number (zero). You can’t invert that to get back your original number.