From Pythagoras to Euler: 5 math equations that changed the world

Major reforms and inventions often begin with a line of mathematics. For students, equations can look abstract on a classroom board. Yet some of them reorganised science, engineering and economics. Understanding these equations is less about memorising symbols and more about seeing how ideas travel from paper to practice.

Here are five that changed the modern world.

a² + b² = c²
​
Pythagoras’ Theorem states that in a right angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Described in ancient Greek mathematics and recorded in Euclid’s Elements, it became a foundation of geometry.

For students, this theorem is often the first proof they encounter. Its influence extends beyond school exercises.

​log(xy) = log x + log y
​
Logarithms were formalised by John Napier in the seventeenth century to simplify long calculations. The key idea is that multiplication can be turned into addition. This reduced the computational burden in astronomy, navigation and finance before the invention of electronic calculators.

d f / d t = limit as h approaches 0 of [f(t + h) − f(t)] / h
​
Calculus was independently developed by Isaac Newton and Gottfried Wilhelm Leibniz in the seventeenth century. It studies change and motion. The derivative measures how a quantity changes at an instant.

For students, calculus marks a shift from static algebra to dynamic systems.

F = G m1 m2 / r²
​
Isaac Newton’s Law of Universal Gravitation states that every mass attracts every other mass with a force proportional to their masses and inversely proportional to the square of the distance between them. Published in Philosophiæ Naturalis Principia Mathematica in 1687, it unified the motion of falling objects and planetary orbits.

For students, this equation connects mathematics and physical reality.

e^(iπ) + 1 = 0
​
Leonhard Euler’s identity links five fundamental constants: the base of natural logarithms, the imaginary unit, pi, one and zero. Derived from complex analysis in the eighteenth century, it connects exponential functions and trigonometry.