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I'm in the tenth grade and I recently posed a question to my Algebra teacher on defining 0/0. Here’s another question at the same level: Defining 0/0 And in fact, assuming any value leads to the same contradiction - unless you are willing to allow all numbers to be equal! For that reason, we have to say that it is simply undefined, so we can change that second rule back: The assumption that \(0\div 0 = 1\) leads to a contradiction. So we just proved that all other numbers n are equal to 1! So 0/0 can't be equal to 1.
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Let's say that 0/0 followed that old algebraic rule that anything divided by itself is 1. (She isn’t really saying that 0/0 itself ever has a value, but that something that looks like it may, in the form of a “limit”.) For the moment, let’s stick to arithmetic, as she continues: Here's another bit of weirdness with 0. She then went into a little demonstration with limits in calculus I’ll be getting into that later. \(n\div n = 1\) for any nonzero number n.\(n\div 0\) is undefined for any nonzero number n.\(0\div n = 0\) for any nonzero number n.Note the clarification: each of the three rules has been proved only under certain conditions:
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If you haven't had calculus yet, just let this sit in the back of your head, and refer to it again later. I'm going to give you an example from calculus where the number 0/0 is defined. It's not true that a number divided by 0 is always undefined. You get into the tricky realms when you try to divide by zero itself. You are right that zero divided by any number (except zero itself) is zero. The first issue is to clarify those three rules.ĭoctor Sonya answered: Zero is a tricky and subtle beast - it does not conform to the usual laws of algebra as we know them.