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Apr 10, 2015 · Dividing both sides by a 2 − a b gives 2 = 1. Mathematics is all about proving that certain statements, such as Pythagoras' theorem, are true everywhere and for eternity. This is why maths is based on deductive reasoning. A mathematical proof is an argument that deduces the statement that is meant to be proven from other statements that you ...
Jul 30, 2022 · Proofs are the whole point of mathematics. They are how we verify and explain that we know things instead of merely guess at them. When I personally teach discrete mathematics, the first-day opening that I use to address this issue is this: Consider a function defined on natural numbers n: f(n) = n2 − n + 11.
May 27, 2019 · The reason we do proofs is to teach ourselves how to think logically. It's very easy to see that since these two angles "look" equal that they must be equal. But it takes practice to be able to explain how you actually arrived at that conclusion. Take, for instance, a prosecuting attorney.
Nov 30, 2021 · Here's Euclid's elegant proof. Maths in a minute: The square root of 2 is irrational — Here's one of the most elegant proofs in all of maths, and a great example of how proof by contradiction works. Maths in a minute: Shake to solve — Looking at the same thing in two different ways can help you prove that two things are equal. Here's an ...
Then P(n) is true for all natural numbers n. For example, we can prove by induction that all positive integers of the form 2n − 1 are odd. Let P(n) represent " 2n − 1 is odd": (i) For n = 1, 2n − 1 = 2 (1) − 1 = 1, and 1 is odd, since it leaves a remainder of 1 when divided by 2. Thus P(1) is true.
I hope that explains why you’re being tormented so with proofs. Written proofs are a record of your understanding, and a way to communicate mathematical ideas with others. “Doing” mathematics is all about finding proofs. And real life has a lot to do with “doing” mathematics, even if it doesn’t look that way very often. 3
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It is proof that is our device for establishing the absolute and irrevocable truth of statements in our subject. This is the reason that we can depend on mathematics that was done by Euclid 2300 years ago as readily as we believe in the mathematics that is done today. No other discipline can make such an assertion.
