What is a Proof?Reference > Mathematics > Introduction to Proofs
Many students arrive in a geometry class after several years of elementary and middle school math, and a year of algebra, and feel as though they've hit a brick wall. It feels like everything they've learned up until this point in their mathematics career just got thrown out a window, and the teacher is up there speaking a whole new language.
"It was bad enough when they started introducing the alphabet into math, but what is all this 'proof' stuff?"
That's a good question. And students who can get a good handle on the concept of a proof will have a very good head start in Geometry, because the concept of proving things is fundamental to most geometry curricula. The problem is, that when you take Geometry, you're being introduced to two very disparate topics simultaneously: proofs, and geometry. So if you have a tough time understanding geometry, proofs become impossible, and if you have a tough time understanding proofs, then geometry becomes impossible.
How much better to use a course of study you already know (Algebra) and use those concepts to introduce proofs, without having to deal with any other new concepts at the same time!
So let's talk about what a proof is, and then we'll explore proofs in the context of Algebra.
One definition of "proof" is the following: "a sequence of steps, statements, or demonstrations that leads to a valid conclusion."
Another way of saying it is: it's no longer okay to just state that something is true, we want you to explain, in detail, why it is true.
Look at it this way: when you were taking an algebra class, your teacher probably told you over and over again, "Show your work!" A proof is when you teacher doesn't just tell you "Show your work," but also tells you, "Explain your work!" If you understand it in this way, proofs are not as terrifying as you might have once thought.
Let's take an example. In your algebra class, if you were given a problem like this: "Find x if 3(x + 6) = 27," you would solve it as follows:
3(x + 6) = 27 3x + 18 = 27 3x + 18 - 18 = 27 - 18 3x = 9 3x / 3 = 9 / 3 x = 3
But suppose I said, "Prove that if 3(x + 6) = 27, then x = 3." Now you can't just show me your work; you have to explain your work. That's not hard, right? You know how you did it, so it shouldn't be too difficult to explain. Explaining is simplified by the fact that we have algebraic properties and rules that we can use as explanations.
For example, you know that you're allowed to add the same thing to both sides of a true equation, and the equation will still be true. The name of that property is "Additive property of equality." So if you're ever in the position where you add something to both sides of the equation, "Additive property of equality" is all you have to put for a reason.
So what we're going to do is, we're going to write down exactly the same work we just did, but we're going to do it in a two-column format, in which the first column shows what we did, and the second column shows our reason for why we did it.
So let's begin.
3(x + 6) = 27
How do we know that 3(x + 6) = 27? Easy! We were given that information in the problem! Notice the word "given" in that sentence; that's an important word in many proofs. The word "Given" means exactly what it sounds like; we were given the information in the statement of the problem. So now our proof looks like this:
STATEMENT REASON 1. 3(x + 6) = 27 Given
Our next step was: 3x + 18 = 27. How do we know that's true? Easy! We used the distributive property!
STATEMENT REASON 1. 3(x + 6) = 27 Given 2. 3x + 18 = 27 Distributive property
What did we do next? We subtracted 18 from both sides. Why can I do that? It's the subtraction property of equality, which says I can subtract the same quantity from both sides of a true equation, and the result will still be true. Note that I don't have to explain why I subtracted 18; I just have to explain why I'm allowed to do it.
STATEMENT REASON 1. 3(x + 6) = 27 Given 2. 3x + 18 = 27 Distributive property 3. 3x + 18 - 18 = 27 - 18 Subtraction property of equality
What about my next step? How did I get from where I was to 3x = 9? Well, I guess you could just say I did "arithmetic." (Or, in this case, you could say I "combined like terms")
STATEMENT REASON 1. 3(x + 6) = 27 Given 2. 3x + 18 = 27 Distributive property 3. 3x + 18 - 18 = 27 - 18 Subtraction property of equality 4. 3x = 9 Combine Like Terms
Now I divide both sides by 3. Why am I allowed to do that? Because the Division property of equality says I can divide both sides of a true equation by the same non-zero number, and I'll still have a true equation. Since 3 is non-zero, I can do it!
STATEMENT REASON 1. 3(x + 6) = 27 Given 2. 3x + 18 = 27 Distributive property 3. 3x + 18 - 18 = 27 - 18 Subtraction property of equality 4. 3x = 9 Combine Like Terms 5. 3x / 3 = 9 / 3 Division property of equality
And, of course, we just do a little bit of arithmetic as our last step!
STATEMENT REASON 1. 3(x + 6) = 27 Given 2. 3x + 18 = 27 Distributive property 3. 3x + 18 - 18 = 27 - 18 Subtraction property of equality 4. 3x = 9 Combine Like Terms 5. 3x / 3 = 9 / 3 Division property of equality 6. x = 3 Arithmetic
And with that, you've done your first proof! The proof was nothing more than showing all the steps, and explaining the reason you could do each step. Sometimes, we like to put a funny symbol (three dots in a triangle) at the end of the proof, before the last line. It's a way of saying "Therefore..." It's nice to put that in there, because it lets people know, "Here it is - the grand finale of the proof!" With that in mind, the proof would look like this:
STATEMENT REASON 1. 3(x + 6) = 27 Given 2. 3x + 18 = 27 Distributive property 3. 3x + 18 - 18 = 27 - 18 Subtraction property of equality 4. 3x = 9 Combine Like Terms 5. 3x / 3 = 9 / 3 Division property of equality 6. ∴ x = 3 Arithmetic
As a quick summary, here are some of the most common algebraic rules that you might use:
- Additive property of equality
- Subtraction property of equality
- Multiplicative property of equality
- Division property of equality
- Combining like terms
- Distributive property
- Arithmetic (not really a rule, but it's a way of explaining what you're doing if all you've done is arithmetic operations on numbers!)