Proof techniques

In mathematics, proofs are the cornerstone that distinguishes the subject from pure observations or experiences. A proof is a systematic and logical argumentation that shows why a certain statement is true under the given assumptions. Whereas in natural sciences knowledge is often built on experiments and measurements, mathematics is based solely on proofs. Without proofs, mathematics would only be a collection of conjectures.

 

Proof techniques are methods that help us carry out such argumentations. They give structure to reasoning and serve as tools that can be applied in many different contexts. The purpose is not only to reach a result, but to show why it holds and to ensure that others can verify and reuse the reasoning.

 

An important idea often used in proofs is to write numbers in a specific form. All integers are either even or odd, and we can take advantage of this.

 

• An even number can always be written as \( \large 2a \), where \( \large a \) is an integer.
• An odd number can always be written as \( \large 2a+1 \).

 

This notation makes it easy to show properties of numbers, because one can work directly with the symbols instead of concrete examples. A simple example is the claim that the sum of two even numbers is always even. If we write the numbers as \( \large 2a \) and \( \large 2b \), we obtain the sum:

 

$$ \large 2a + 2b = 2(a+b) $$

 

This is again a number that can be written as 2 times an integer, and therefore even. Another example is that the square of an odd number is always odd. Writing the number as \( \large 2n+1 \), we get:

 

$$ \large (2n+1)^2 = 4n^2 + 4n + 1 = 2(2n^2+2n) + 1 $$

 

This is of the form \( \large 2k+1 \), that is, odd. These examples show how a proof explains and documents the relationship so that there is no doubt.

 

 

There is no single universal technique that fits all situations. The choice of method depends on the nature of the problem: sometimes the conclusion can be derived directly from the assumptions, at other times it makes more sense to work backwards or to assume the opposite to show a contradiction. In number theory, algebra, and analysis, induction is also an indispensable tool, because it makes it possible to prove statements that hold for all natural numbers.

 

Proof techniques also have an educational significance. They force us to be precise in our reasoning, reveal hidden assumptions, and sharpen our understanding of the concepts we are working with. Being able to prove a statement is not only about solving a problem, but about contributing to mathematics as a coherent science.

 

In the following articles, some of the most commonly used techniques are presented: direct proof, proof by contradiction, contraposition, and induction. These methods cover most of the situations encountered in elementary mathematics, and they form a foundation that can later be built upon with more advanced techniques. Each method is presented with its own examples and applications, so one can see when and how they are used in practice.