The material and scientific changes that our current world experiences are the result of the real scientific application of mathematical equations discovered through the ages.

Mathematics is an abstract science based on observation and deduction, and from here comes the role of applied sciences in translating mathematical formulas in various fields such as physics, engineering, programming, and all other applications.

Explore how mathematical equations have shaped our modern world, from Pythagoras’ equation to Einstein’s theory of relativity. Learn about the vital role of mathematics in applied sciences, and how mathematical equations have contributed to radical changes in the human world.

**Mathematical Equations That Made a Difference**

Here are some of the most important formulas and equations discovered by scientists and worked upon, which solved cosmic puzzles and made changes in the human world:

**Pythagorean Theorem**

This equation studies the relationship between the lengths of the sides of a right triangle and states that “the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.”

The Pythagorean theorem is used to calculate right triangles and the distance between two points in a plane; it is used by engineers in the design of buildings and structures for calculating the lengths of walls and upright roofs, and in astronomy to calculate the distance between celestial bodies and planets, and in scanning technologies as well, and it has been used to calculate the distance between the source and the receiver.

**Calculus Theory**

Considering that differentiation and integration are opposite concepts; they were linked according to the calculus theory which is a two-part theory:

**The first part** states that definite integration can be reversed by differentiation.

**The second part** states that the definite integration of a function over a certain range can be calculated using one of its indefinite derivatives.

Calculus involves calculating rates of change and the slope of curves, while integration involves calculating the areas enclosed between the curves and the coordinate axes.

Calculus enables us to calculate definite integrals using inverse derivatives, developed by **Isaac Newton and Leibniz**, and these concepts have evolved into various applied sciences in engineering, economics, and physics in mechanics and electricity.

**Euler’s Equation**

This equation is classified as the most beautiful equation ever because it combines five famous mathematical constants: the number 0, the number 1, the natural number e, the approximate value, and the imaginary unit i. Euler’s equation takes the form: eiπ+1=0

Because Euler’s equation relies on the imaginary part, its usefulness in the real world lies in translating one pattern of motion into another, allowing engineers to transform trigonometric problems into simpler algebraic problems.

It also transforms all the positive shapes in quantum mechanics, electronics, signal processing, and even in computer systems.

**Euler-Lagrange Equation**

This second-order differential equation is characterized by its constant solutions, which determine the curve on which a weighted particle falls to a fixed point in a specific period of time regardless of the starting point, discovered in the 1750s, and this equation resembles Fermat’s principle in calculus.

The scientists Euler and Lagrange developed this equation and its application to mechanics, which led to the formulation of a new mechanics known as Lagrangian mechanics, which led to the calculation of variabilities.

**General Theory of Relativity**

This is the theory of gravity formulated by Albert Einstein in 1915, which states that “the noticeable effect of gravity results from the curvature of spacetime by these masses.”

General relativity has evolved to become the foundation of modern astrophysics; the reality of black holes was understood through relativity, and general relativity is part of the standard framework for the Big Bang model in cosmology.

**Special Theory of Relativity**

Einstein published his special theory of relativity, which reconciled Newton’s laws of motion and electrodynamics, presenting a new framework for physics by proposing new concepts of time and space.

The principle of special relativity states that “there are no differences or peculiarities in the natural laws among different inertial systems,” and the special theory of relativity replaced Newton’s theory of time and space, providing solutions to sterile problems regarding electromagnetic waves in general and light in particular.

These laws and equations are not the only ones; there are other equations and theories that have played a fundamental role in the development of the human world, such as the law of universal attraction.

Faraday’s equation (electric charge), quantum mechanics law (Schrodinger equation), Fourier’s theory and logarithms, information theory, and the law of electromagnetic waves.

Each of the mentioned equations has played its role in changing the world by solving one of the mysterious puzzles of the universe to become clear to humanity, and it should be noted that there is a new theory being worked on that will lead to major changes in the world and adjustments to many previous theories, namely the theory of chaos and the science of the unexpected, which we will discuss in a future article, If Allah is willing.

**Author: Mr. Abdelhamid Al-Diban, Mathematics teacher at Masarat Initiative**