Are you struggling with choosing the approach taken to prove mathematical statements? In that case, this article is for you. Here are 4 simple ways to prove or disprove mathematical conjectures. Let’s find out what they are and how they work in this article.
Table of Contents
- Direct proof
- Proof by contradiction
- Proof by mathematical induction
- Proof by counterexample
1. Direct Proof
When you use a direct proof, you extract relevant facts and the information from the conjecture you’ll want to prove and then logically make your way to show that the statement is true. It is suitable for proving statements where, when one statement is true, the other must also be correct. Besides, it’s also useful in proving identities. For instance, if you want to prove that
You’ll have to expand both sides of the identity fully. After that, we get:
In this result, the LHS and the RHS are identical, so the identity above is true.
2. Proof By Contradiction
Sometimes, it’s impossible to prove something directly. In that case, you should prove the statement by contradiction. Firstly, you should assume that your hypothesis is wrong. After that, you should make your way to produce a contradiction to show that the opposite of your conjecture is impossible, thus proving your statement correct.
For example, if you want to prove that the square root of 2 is irrational, you should first assume that it can be expressed as a ratio of two integers, namely √2 = x/y, where x and y are integers. Then, you can infer that 2 = x2/y2, and 2y2 = x2. This is already a contradiction.
To become a perfect square, all exponents in the prime factorization of the integer must be even. For instance, 24 * 32 = 144 is a perfect square because 4 and 2, which are the exponents of the prime powers in its prime factorization, are all even. However, if you multiply a perfect square by 2, the product must not be a perfect square since there will be an odd number in one of the prime exponents. Thus, no integers exist such that 2y2 = x2, which implies that the square root of 2 cannot be expressed as any ratio of two integers, no matter how enormous those numbers are.
3. Proof By Mathematical Induction
Sometimes, instead of proving that an identity or inequality is valid in all cases, you might want to show that it’s true for all integers smaller or greater than a certain number. In that case, you can use mathematical induction to prove your statement.
A theorem proven by mathematical induction is like a set of dominoes that are ready to fall. It provides a method to prove that if it’s true for n, it’s also true for n+1 or n-1, depending on your conjecture. This, in turn, proves that it’s true for n+2 or n-2, and so on. Essentially, the “dominoes” keep falling until infinity, meaning that your assumption is valid for all numbers in a specific range once you can prove that because it is true for an integer n, it’s true for n+1 or n-1 as well.
For instance, this technique has been used to prove that all triangular numbers, which are the sum of the first n integers, are of the form n(n+1)/2. Firstly, you prove that it’s true for n=1 because 1(1+1)/2 = 1. After that, you should show that
Which can be proved using these steps:
Therefore, by mathematical induction, the nth triangular number, or the sum of the first n natural numbers, are of the form n(n+1)/2.
4. Proof By Counterexample
Last but not least, if you want to disprove a conjecture, one way you can do so is to find counterexamples of the statement. It means finding integers that do not comply with the statement to be disproved.
For instance, the conjecture that all numbers of the form 22^n + 1 are primes is not true. At first, it seems like that the assumption is true, given that 3 (21 + 1), 5 (22 + 1), 17 (24 + 1), 257 (28 + 1), and 65537 (216 + 1) are all prime. However, the next integer in the sequence, 4294967297, is a counterexample since it equals 641 * 6700417. Therefore, the conjecture mentioned at the beginning of the paragraph is false.
In this article, we’ve explained the 4 fundamental ways to prove mathematical statements so that you can choose the correct approach to show that a conjecture is true or false. If you want to learn more, please read the articles in the references below.
References and Credits
- (2021, January 17). Direct Proof Fully Explained w/ 11+ Examples! Retrieved December 16, 2021, from https://calcworkshop.com/proofs/direct-proof/
- (n.d.). Mathematics Tools: Proof by Contradiction. Retrieved December 16, 2021, from http://www2.edc.org/makingmath/mathtools/contradiction/contradiction.asp
- Katherine Körner, Vicky Neale. (n.d.). An Introduction to Proof by Contradiction – NRICH. Retrieved December 16, 2021, from https://nrich.maths.org/4717
- Harris Kwong. (2021, July 7). 3.4: Mathematical Induction – An Introduction. Retrieved December 16, 2021, from https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/03%3A_Proof_Techniques/3.04%3A_Mathematical_Induction_-_An_Introduction