Why proof in Mathematics is important?

Mathematics is a huge topic name for many complicated and various concepts, although in school teachers will teach us only some of it however, some people might find it a hard time for learning. The reason why this group of people still hate Mathematics is not understand the concept of it for a variety of reasons that you can imagine. Just a simple proof might help everyone that didn’t acknowledge Mathematics to understand even more in a complex topic. Remember the rules of Mathematics are not wrong however, it’s not the most effective way to understand and learn Mathematics. Mathematics competition problems can be very hard for some groups of people who remember but do not understand the theorems or rules. In school, a topic test is just a direct use of what we’ve learned. Now, let’s think about using it in real life, this is why most people don’t find Mathematics useful in daily life.

Knowing how to use and how to adapt Mathematics knowledge is one of the important skills in our life. Had you ever used any Mathematics theorems in daily life? Most people will say NO or NEVER, but the way that Mathematics was built is not to use a theorem all the time. It helps humans to think reasonably by the reason that Mathematics is about understanding and in order to understand it you will need to prove that each theorem is true or read proofs from other Mathematicians and try to understand it. Find the simplest proof for you to understand. A basic example of this is… Do you know why Pythagoras' theorem actually works? If you don’t understand why this article will also prove it for you to understand.

Before reading the main part of this article, let’s get to know some important terms in Mathematics about proof. The first one is “axioms” which means mathematically a statement or proposition that is taken to be true without any proof. As it’s a starting point for further proof, reasoning, or arguments. For example, two straight lines can intersect each other at either zero to one point. The second term is “theorem” you might hear this often more than two other terms but do you actually know what it means? The definition of “theorem” is a statement that can be proven to be true. Pythagoras’ theorem is a perfect example of this term. The last one is “conjecture”, it is also a statement but it’s unproven and can be proven to be either True or False. In a false case, the prover will need one counter-example, compared to a true case which needs to be true in every single example.

Besides Mathematics, logical proof can be used in many circumstances. For example, consider these statements “It will be very hot tomorrow” and “Humans can live without breathing”. Is it true or false? In order to conclude the first statement everyone needs proof (which in this case is the weather forecast) to determine whether tomorrow will be hot or not, therefore, this statement is a conjecture. And if we compared it will the second statement the possibility that it will be true is greater as in the second statement we have one counter-example from scientific proof. If you think of it with proof you are now thinking with logic.

As mentioned before proof is required to think logically and one more huge question is “How to prove?”. Some countries don’t teach students to prove what they have learned which makes students more confused. The proof allows students to understand where this comes from such as any even number squared will get a result of an even number. Basic proof for this is just to assume any even number and square it. Assuming that that even number is 2 now what is $2^2$ ? it’s 4 and since 4 is also an even number. Hence, this statement is true with this proof.

Above there’s a proof of an even number squared equal even number, but it often has more than one proof and gets the same result. Different Mathematicians sometimes have completely opposite ways to prove but return with the same output. They’re more than 371 proofs for Pythagoras’ theorem alone. The latest ways to prove Pythagoras’ theorem comes from a couple of students from New Orleans, USA by using the law of sines. With more ways that people use to prove mathematics, it’s become more accessible for new generations to understand and develop it with their own philosophy.

Proof in Mathematics does not just help make people understand it, it also helps Mathematics to convince or justify each statement. In some proofs, Mathematicians can use it to prove another statement. Without proof, we’ll never know what is the hypothenuse in a right-angled triangle because there’ll be no theorem since people don’t believe it. Further away from Mathematics, proof can help in other subjects as well, such as computer science or science.

Proof and logical thinking are pretty much related one to another. Proof comes from logical arguments that state to make people understand and make other people who are interested in that concept of it develop and fix some issues or comes with a better solution and proof.

“traditionally, justifying or becoming convinced about the validity of a conjecture is the main function attributed to proof”, Michael de Villiers once said in his paper. He also said that sometimes students don’t even understand “what is proof?”. Perhaps some Math problems make me think twice about whether this is True or False, as it looks false but the proof says otherwise.

How to prove it?

In our world today, many things have been proven before but understanding how and why to prove is actually better than just reading the proven statement. It makes us to understand further more than just a concept of it and it will also help us to prove our own concept. Trial and Error are basic for proving any statements. No one knows exactly at the first time which one is the correct way to prove.

Learning how to prove it is much harder than memorizing the concept for the particular reason that proof requires tons of creativity. At the same time, if you know how to prove Mathematics will become more fun and easier for you because you understand the concept of it.

As mentioned previously in the first paragraph, about proving that $a^2+b^2=c^2$ or Pythagoras’ theorem is a true conjecture. Can you prove it without any hint? There’re tons of ways that people can use to prove Pythagoras’ theorem however, today this article will provide to you with one basic example.

This is a way to prove Pythagoras’ Theorem that is the most simple and well-known proof for this theorem. This method of proof is also used in many schools as it is the most straightforward proof. By using a square!

Image from Math is Fun

We can place a square that has a side of $c$ and another four congruent right-angled triangles around it by placing a hypotenuse side with a side of a square. Since the four triangles are congruent to each other we can label the other two sides as $a$ and $b$ .

Let’s find the area. A total area can be found by using this formula $(a+b)(a+b)$ . Move on to a smaller shape, a small square’s area is $c^2$ , each triangle’s area is $\frac{ab}{2}$ and four of them combined we will get the area of $\frac{4ab}{2}=2ab$ . Adding them up together to find a total area $c^2+2ab$

The area of a large square is equal to a small square plus four identical triangles. This can be written as $(a+b)(a+b)=c^2+2ab$ . Rearrange this form by expanding $(a+b)(a+b)$ and subtract $2ab$ from both sides and it will finally change into a form of $a^2+b^2=c^2$ . This is a brief proof of Pythagoras' theorem. This proof comes from China over 2000 years ago.

Move on from geometry, let’s try to prove algebra! Prove that any real number square is positive. This statement is conjecture that can be proven to be true or false. Now let’s find out whether this is true or false. Remember, a FALSE statement is required only one counter-example, and a TRUE statement is required to be true in every single case.

“Any real number”, from this statement you can imagine any real number and try to square it. Let’s try 5, 5 squared is 25, or -4 squared is 16. But what about 0 squared? Since zero is neither positive nor negative, zero squared is zero. Therefore, this statement this FASLE as we can provide this one counter-example.

As we proved false conjecture, now let’s try to prove some true ones. Consider this statement: “If $n$ is a whole number, then $n \ge 0$ ”. A key for proving this statement is know the definition of “whole number” as we can determine from this statement whether is either true or false. “Whole number” can be defined as an integer starting from zero and going on to one, two, … such as 3494, 129, 42. Now that we can define the word “whole number” we can certainly look at the statement and say that this statement is true by the fact that if $n=0$ , moreover we substitute n in the statement we will get 00 which is true or if we substitute n=89347 we will also find that this statement is TRUE because it’s true in every single case that we’ve tried. If you don’t believe it, try it yourself! Substitute any “whole number” into this inequation.

For any true conjecture, if it has been proved to be true in any possible proofs that humans have tried, it becomes a theorem. But in order for a conjecture to transform itself into a theorem, it required tons of proofs by many mathematicians. Just to be an example Pythagoras’ theorem has more than 370 proofs.

These days, proving plays an important role in the Mathematics world as well as other subjects that you learn. It’s important for teachers all around the globe to express this significant philosophy to all learners.

Just after we grow up, the ability to prove claims on existing pieces of information will become more essential than others. This posed clear support by looking at many universities’ midterm and final examinations, with often required students or learners to think logically rather than using four or five multiple-choice question styles.

Book recommendation

Image from Amazon

There’s a book written specifically around proving in Mathematics called “How to prove it” by Daniel J. Velleman, a famous mathematics professor from Amherst College based in the United States. It points out multiple ideas for mathematical proof moreover it explains proof strategies with some obvious examples.

Conclusion

In conclusion, proving still has an important role in our world since it describes moreover convinces people to understand formulas and theorems in Mathematics. This is a precise summation consisting of several papers and books as presented beneath in the references section. Proof also enables mathematicians into building on existing knowledge and push it to go further. Overall, the importance of proof in mathematics cannot be overstated, and it remains an essential tool for advancing our understanding of the world around us.

How to prove it?

Book recommendation

Conclusion

References