How to Do Math Proofs

Performing mathematical proofs can be one of the hardest things for students to do. Students majoring in mathematics, computer science, or other related fields will likely encounter proofs at some point. Simply following a few guidelines will help erase the doubt from the validity of your proof.

1. Understand that math uses information that you already know, especially axioms or the results of other theorems.
   
2. 
3. <ins style="POSITION: relative; PADDING-BOTTOM: 0px; MARGIN: 0px; PADDING-LEFT: 0px; WIDTH: 468px; PADDING-RIGHT: 0px; DISPLAY: inline-table; HEIGHT: 60px; VISIBILITY: visible; PADDING-TOP: 0px"><iframe style="POSITION: absolute; TOP: 0px; LEFT: 0px" id="aswift_1" height="60" marginheight="0" frameborder="0" width="468" allowtransparency="allowTransparency" name="aswift_1" marginwidth="0" scrolling="no"></iframe></ins>
6. 2
   Write out what is given, as well as what is needed to be proven. It shows that you will start with what is given, use other axioms, theorems, or math that you already know to be true, and arrive at what you want to prove. True understanding means you can repeat, and paraphrase the problem in at least 3 different ways: pure symbols, flowchart, and using words.
   
   
   

3

Ask yourself questions as you move along. "Why is this so?" and "Is there any way this can be false?" are good questions for every statement, or claim. These questions will be asked by your professor in every step, and as soon as he/she can't verify one of those questions, your grade will go down. Back up every statement with a reason! Justify your process.

Make sure your proof is step-by-step. It needs to flow from one statement to the other, with support for each statement, so that there is no reason to doubt the validity of your proof. It should be constructionist, like building a house: orderly, systematic, and with properly paced progress. There is a very graphic proof of Pythagoras theorem which is found by a simple process [1].

 

5

Ask your professor or classmate if you have questions. It's alright to ask questions every now and then. It's the learning process to do so. Remember: There is no such thing as a stupid question.

1. 6
2. Designate the end of your proof. There are several methods for doing this:
   
   
3. 
4. Q.E.D. (quod erat demonstrandum, which is Latin for "which was to be shown"). Technically, this is only appropriate when the last statement of the proof is itself the proposition to be proven.

• A filled-in square (a "bullet") at the end of the proof.
• R.A.A. (reductio ad absurdum, translated as "a bringing back to absurdity") is for indirect proofs, or proofs by contradiction. If the proof is incorrect, however, these symbols are very bad news for your grade.
• If you're not sure if your proof is correct, just write a few sentences saying what your conclusion was and why it is significant. If you use one of the above symbols and you turned out to be wrong, your grade will suffer.

2. 7
   Remember the definitions you were given. Go through your notes and the book to see if the definition is correct.
   
   
   
   
   
3. Take time to ponder about the proof. The goal wasn't the proof, it was the learning. If you only do the proof and then move on then, you are missing out on half of the learning experience. Think about it. Will you be satisfied with this?
   
1. 
4.