eli5: Why haven’t mathematicians invented a symbol for x/0 like they have pi and imaginary numbers?
In: 277
Let’s try it. Let’s define a symbol (let’s call it, I dunno, Q) and say that x / 0 = Q for all real numbers x.
Now, when we write a / b = c, we mean a = bc. For example, 12 / 4 = 3 because 3*4 = 12. So since, say, 3 / 0 = Q, we would need 3 = 0*Q. And since 5 / 0 = Q, we would need 5 = 0*Q. So 0*Q equals both 3 and 5, and in fact every other real number. We can, therefore, prove that all real numbers are equal.
Needless to say, this is not particularly useful math.
It turns out that adding *i* does not have this sort of consequence. Nothing about complex numbers disrupts, in any way, the arithmetic of real numbers. The same goes for real-but-not-rational numbers like pi: the reals extend the operations on the rationals without disrupting how rational numbers behave in and of themselves.
——
EDIT since some people are objecting and saying you should just define 1/0 = Q and then, say, 3/0 = 3Q. Turns out that doesn’t work either. Consider the expression 2 * 0 * Q. Multiplication is associative, so we can either write:
(2 * 0) * Q = 0 * Q = 1
Or
2 * (0 * Q) = 2 * 1 = 2
So 1 and 2 are both equal to 2 * 0 * Q and therefore to one another. We can make 1 equal any real number x by doing 1 = 0 * Q = (x * 0) * Q (since 0 = x * 0 for all x) = x * (0 * Q) = x * 1 = x.
More generally, the properties of 0 as a number prevent you from defining division by 0 – at least not if you want to be able to *multiply* anything by 0. The fundamental problem is the expression 0 * Q, which immediately generates this kind of contradiction. While there are constructions that effectively define 1 / 0 = Q (usually using the symbol for infinity for this new value), those constructions don’t allow expressions like 0 * Q or Q/Q, so they’re at least as complicated as just leaving division by 0 undefined.
Latest Answers