Therefore, among three subsequent integers, there is always a number that is divisible by 3. Therefore, you see that in each case, one of the numbers n, n + 1 or n + 2 is divisible by 3. Is also divisible by 3, since every third integer is divisible by 3. If you get 2 as the remainder, it means that n − 2 is divisible by 3, because n − 2 + 2 = n. If you get 1 as the remainder, it means that n − 1 is divisible by 3, because n − 1 + 1 = n. If you have 0 as the remainder, it means that the division went okay, so n is divisible by 3. Thus, you have the following possibilities: 1. When you try to divide n by 3, there are three possibilities: You can get 0, 1 or 2 as the remainder (remainder = what is left when a division doesn’t fully go into the divisor). The subsequent numbers are then: n, n + 1, n + 2 Then, the number that is one larger is n + 1 and the number that is one larger than that again is n + 2. You know that subsequent integers can be expressed like this: You call an arbitrary integer n. But, if you accept the assumptions, the proof explains why you also have to accept that the conclusion is true. The proof doesn’t need to convince you that the assumptions are true-you are allowed to think that Socrates is an undercover zebra or that humans can become immortal if they don’t eat carbohydrates. The proof has two assumptions: “Every human is mortal” and “Socrates is a human”. Here is a simple proof:Įvery proof builds on one or more assumptions. What you are trying to prove is a conclusion. Basic mathematics, pre-algebra, geometry, statistics, and algebra are what this website will teach you. But first, a little bit about proofs in general.Ī proof is an explanation for why something is true. The ones you can lean more about on this website are: Direct proofs, proofs by contrapositive, proofs by induction, and one proof of the Pythagorean theorem. Another symbol that is used to show that a proof is complete is the “Halmos symbol”. This is Latin and stands for “quod erat demonstrandum”, which means “which was to be demonstrated”. Instead, proofs often end with the abbreviation Q.E.D. That is unnecessary in a proof since the answer is the whole text. In a regular mathematical problem, you often draw two lines beneath your last expression to show that you have reached a final answer. A proof is a string of implications and equivalences, where the entire text is the answer. In math, being able to prove what you are doing is of great importance.
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