Math Tips 4:


Revenge of the Math Tips!

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Prove it!

“But why?” was the question my math teachers hated to hear most. But it’s a perfectly good question, and any decent math teacher should be able to answer it. If you don’t believe a mathematical fact, the good news is you probably don’t have to accept it on faith. Besides the definitions and axioms, we don’t have to accept anything as true in mathematics unless there’s a proof for it. The proof will usually explain exactly why something is true.
A proof is a step-by-step chain of logical reasoning which takes you from the premises (the given information that we are accepting as true because it is an axiom or it has already been proved) to the conclusion (the statement of the theorem, the thing we are trying to prove).

For example, suppose I don’t understand why it’s true that in order to multiply two exponents with the same base, I can just add the exponents. (In other words, why it’s true that for any numbers a, x, and y, \ a^x \cdot a^y = a^{x+y} .) I can ask for a proof, and it might go something like this:

\begin{align*}a^x \cdot a^y &=& \underbrace{a \cdots a}_\text{x factors of a} \times \underbrace{a \cdots a}_\text{y factors of a} \\
&=& \underbrace{a \cdots a}_\text{x + y factors of a} \\
&=& a^{x+y}\end{align*} \\

If you can’t find the proof in your textbook and you don’t have the luxury of having a tutor go through the proof with you, the good news is you can find many proofs of theorems online.

If you’re confused or have a question about something in math, chances are the explanation is in the proof.

To Be Continued…?

Oh, no! Could this be the end of the daring Math Tips? Tune in next week to find out, true believer! Same math-time, same math-channel! And now, a word from our sponsors:

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