Does Zero Equal One?
How to prove that 0 equals 1.
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The “0 = 1” paradox states that the numbers 0 and 1 are equivalent. We can easily prove this. Once done, we’ll show you how you can use this to prove that any two numbers are equivalent!
DISCLAIMER: The math in this article is invalid, as we can’t abuse parentheses in a divergent series. It’s a fun experiment, but it doesn’t count as a valid argument.
We’ll start out with a series which we’ll prove to be equal to both 0 and 1. After that’s done, we can use basic mathematics to prove that 0 equals 1.
First of all, let’s define as follows:
Proving That S = 0
We can group the terms in into parentheses.
Because , we can simplify this.
Because we can add 0s forever and still get zero, we can simplify this even more.
Ta da! We proved that .
Proving That S = 1
We can group the terms in into parentheses as well. Let’s try this.
Because , we can simplify this.
Because we can subtract 0s forever and still get zero, we can simplify this even more.
Just like that, we proved that .
Wait, S is 0 and 1?
Yup! Now we can use substitution to prove that .
Proving That Any Two Numbers Are Equal
By multiplying both sides of the equation we can show that multiple numbers are equal to each other. For example, to prove that , we can use the following:
And just like that, we’ve proved that !