28 November 2023

Mathematics is a fascinating discipline that has developed over thousands of years and has enabled the advancement of science and technology in many fields. At the heart of mathematics There are theorems, which are fundamental statements that have been shown to be true through rigorous, logical reasoning. A theorem is a proposition that has been proven from other propositions already accepted as true, such as axioms, hypotheses, and other previously proven theorems.

Theorems are essential to mathematics, since they allow us to establish universal and fundamental truths that apply in a wide variety of situations. Although theorems are common in mathematics, are also used in other disciplines such as physics and engineering to establish fundamental truths in those fields. In this article, we’ll explore the definition of a theorem in mathematics, the different types of theorems, their origin and history, and some famous examples of theorems in use today.

## Definition of theorem in mathematics

In mathematics, a theorem is a proposition or statement that can be logically and rigorously proved from a series of axioms or hypotheses previously established. In other words, a theorem is a mathematical statement that is held to be true and whose proof is possible using a specific set of rules and principles.

In general, theorems are used to establish universal truths and essentials in mathematics. Theorems are used in all areas of mathematics, including geometry, algebra, trigonometry, and physics.

### What is a theorem in geometry?

In geometry, a theorem is a mathematical statement that can be proved using previously established principles and rules. Geometry is one of the areas of mathematics in which theorems are especially important, as they are used to establish fundamental truths about shapes and spatial relationships.

The Pythagorean theorem is a well-known statement in geometry. It states that, in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. This theorem is very valuable in solving problems involving the measurement of the sides and angles of right triangles.

Another relevant theorem in geometry is Thales’s theorem. This postulates that, if you draw a parallel line to one of the sides of a triangle, then the other two sides are divided into segments proportional to the length of the parallel line drawn.

### What is a theorem in trigonometry?

In trigonometry, a theorem is a mathematical statement that can be proved using the relationships between the angles and the lengths of the sides of a triangle. Trigonometry is a branch of mathematics which focuses on the study of triangles and trigonometric functions, such as sine, cosine, and tangent.

The cosine theorem is one of the most significant postulates in trigonometry. It states that the square of one side of a triangle is equal to the sum of the squares of the other two sides, minus twice the product of the lengths of those sides times the cosine of the angle between them. This theorem is very useful for solving problems involving the measurement of angles and side lengths of triangles.

Another important theorem in trigonometry is the tangent theorem, which states that the tangent of the sum or the difference of two angles is equal to the sum or the difference of the tangents of the angles divided by one minus the multiplication of the tangents of the angles.

## What is a theorem used for in mathematics?

The theorems have many applications in different fieldssuch as physics, engineering, computer science, economics and many others. Theorems are tools used to prove universal truths in mathematics and other disciplines.

For example, the pythagoras theorem it is used in measuring distance on a map or in solving problems related to the construction of structures, such as bridges and buildings. The cosine theorem is used in solving problems in physics and in navigation. In general, theorems are used to establish essential truths in mathematics and to provide a solid foundation for solving problems in a wide variety of fields.

## What are the types of theorems in mathematics?

There are several types of theorems in mathematics, some of which are used in several areas of mathematics, while others are specific to a single area. Next, Some types are presented of most common theorems:

• Algebraic theorems: they are used in algebra and focus on solving equations and systems of equations.
• Geometric theorems: They are used in geometry and focus on the spatial relationships between geometric objects, such as points, lines, and planes.
• Mathematical analysis theorems: They are used in calculus and mathematical analysis and focus on the study of functions and their properties.
• The theorems of number theory: They are fundamental postulates in the branch of mathematics that focuses on the study of numbers and their characteristics.
• The theorems of set theory: They are essential statements in the mathematical discipline that focuses on the analysis of sets and their properties.

These are just Some examples of the different types of theorems in mathematics.

## What is the origin and history of the theorem?

The concept of theorem dates back to ancient Greece, where mathematicians began to prove geometric propositions by deductive arguments. The term ‘theorem’ comes from the Greek ‘theorein’, which means ‘contemplate’ or ‘observe’.

The idea of ​​theorems dates back to the ancient Greeks, who focused on the study of universal and essential truths in mathematics. Theorems were one of the main results of this effortas they allowed mathematicians to establish universal truths about shapes and spatial relationships.

The Greek mathematician Euclid is known for having systematized the study of geometry and for his work ‘The Elements’, which became a benchmark for many centuries. In ‘The Elements’, Euclid presented a series of propositions and proofs based on axioms and postulates, including the famous ‘Pythagorean Theorem’.

Throughout history, theorems have been developed and refined by mathematicians around the world, from ancient India to medieval Europe and modern. Often theorems have been discovered as solutions to practical problems or as a consequence of theoretical investigations.

In the 19th century, the German mathematician Carl Friedrich Gauss and others developed number theory, which is a major branch of mathematics that focuses on the study of whole numbers and its properties. This theory has given rise to many important theorems, including the ‘Prime Numbers Theorem’.

From ancient Greece to the present, theorems have been the basis of research and the discovery of new mathematical concepts and tools. Mathematicians have worked for centuries to establish new theorems and to prove existing theorems, which has allowed the advancement of science and technology in a wide variety of fields.

## Examples of theorems in mathematics

Below are shown some examples of mathematical postulates:

• The Pythagorean theorem indicates that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
• The law of sines states that, in an arbitrary triangle, the relationship between the length of a side and the sine of the angle opposite that side is invariable.
• The Cosine Theorem states that, in any triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides, minus twice the product of those lengths by the cosine of the angle opposite the first side .
• Fermat’s theorem holds that there are no three positive integers a, b, and c that satisfy the equation a^n + b^n = c^n for any integer n greater than 2.
• The four color theorem indicates that any map can be colored with only four different colors, in such a way that two contiguous regions never share the same color.

These are just a few examples of theorems in mathematics, but there are many others that are used in a wide variety of fields. Theorems are a fundamental part of mathematics and are used to establish universal and essential truths that apply in a wide variety of situations.

Throughout history, theorems have allowed the advancement of science and technology in many fields and have been fundamental to development of mathematics as a discipline. The different types of theorems in mathematics make it possible to establish truths in specific areas of mathematics and have enabled the resolution of complex problems in a wide variety of fields, from physics and engineering to biology and economics.

The proof of a theorem is a rigorous process that requires logical and precise reasoning, and is often requires a lot of time and effort to make it. However, once a theorem has been proven, its truth becomes an essence in mathematics and is used as the basis for developing new theories and solving complex problems.

In short, theorems are an essential part of mathematics. The analysis of these mathematical postulates it is essential for any individual with an interest in understanding and using mathematics. Both in daily life and in scientific research.