Consider the equation
Solving this equation is no sweat, right? Let’s do it. First, we subtract from both sides:
Now we can factor an out of the left side:
Now, if the product of two things is zero, one of them must be zero. So either , or , that is, .
Easy, right? Sure. But what I’d like to talk about over the course of the next few posts, inspired by this curiosity, is a funny alternate number system called the 10-adic, or decadic numbers. In the decadic numbers, the equation has the usual solutions 0 and 1, but it also has two other funny solutions, which don’t correspond to any familiar real numbers.
The first solution, which I will write about in an upcoming post, we’ll call . That is, is some funny sort of number with the property that . (A fancy math word for this is idempotent.) Notice that this means is also a solution, since .
Notice also what happens when we multiply these two solutions: . This is quite strange: two numbers, neither of which is zero, which multiply to give zero! Such numbers are called zero divisors. This is the step in our solution above which fails for the decadic numbers: once we factored to obtain , we reasoned that if two things multiply to give zero, then one of them must be zero. But I’ve just claimed that this is not true for the decadic numbers.
Often mathematicians stay away from systems with zero divisors, since they ruin some nice properties. For example, not every nonzero number has a multiplicative inverse. (Recall that the multiplicative inverse of a number is another number such that ; for example, the multiplicative inverse of is .) For homework, prove that the we discussed above does not have a multiplicative inverse! (Hint: try supposing that it does, and derive a bogus equation…) But we’re not trying to accomplish anything in particular, just to have fun and learn something, so bring on the zero divisors!
OK, so what exactly are these “decadic numbers”? What is ? And what else can be said about the curiosity in my previous post? All this and more, coming soon to a Math Less Traveled near you…