Math is all around us.
Renowned UC Berkeley mathematician Edward Frenkel once said, “mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars.”
Its mysteries weren’t discovered by one person, but by hundreds of talented mathematicians who labored for lifetimes, collaborating on centuries of knowledge that got us to our understanding today.
Here are some of the most important math equations that changed the world.
The Pythagorean Theorem:
One of the fundamental principle in Euclidean Geometry, the Pythagorean theorem, also known as Pythagoras’s theorem deals with the lengths of the sides of a right triangle. The theorem states that: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse. Currently, there are over 130 different proofs for the Pythagorean Theorem, ranging from geometric arrangements to differential calculus.
Isaac Newton’s Law Of Universal Gravitation:
Issac Newton’s publication of the Principia in July 1687 changed the way how we look at the universe, as no one before that knew how the earth and the other planets fit together with the sun. He explained why the planets move the way they do, and how gravity works on earth and the universe. Newton not only concluded that planets revolve around each other because of gravity, but he also gave the exact formula that calculates how much force is between two large objects given their masses. Newton’s Law Of Gravity was the defacto reference equation for more than 200 years until Einstein’s Theory of General Relativity replaced it. However, Newton’s laws are still good to calculate the orbits of satellites and the paths of spaceships.
Albert Einstein’s Theory Of Relativity:
Einstein’s theory of relativity usually covers two interrelated theories: special relativity and general relativity. This theory was proposed in 1905 and depicts the relationship between space and time. Special relativity brought in ideas like the speed of light being a universal speed limit and the passage of time being different for people moving at different speeds. General relativity explains the law of gravitation and its relation to other forces of nature. Einstein’s theories of special and general relativity changed the course of physics and helped the world understand the past, present and future of earth.
“A butterfly flaps its wings, and it starts to rain,” the narrator of one episode of How I Met Your Mother begins begins, “It’s a scary thought but it’s also kind of wonderful.” He continues:
“All these little parts of the machine constantly working, making sure that you end up exactly where you’re supposed to be, exactly when you’re supposed to be there. The right place at the right time.”
It’s portrayed differently in each one, but you see a popularized version of the Chaos Theory everywhere in movies and TV shows – a butterfly flaps its wing, and the course of history is altered forever. The theory isn’t as crazy as you would think.
Traditionally, scientists believed that all natural processes were either deterministic or nondeterministic – meaning we can either predict their behavior, or not at all. Throwing a ball is deterministic, because if you throw a ball at exactly the same angle and speed, you can predict how far it goes. Uranium decay is nondeterministic, because it’s impossible to predict which exact atom will decay at a given time.
The Square Root of -1:
It’s easy to grasp the concept of square roots. The square root of 4 is 2, because 2 * 2 = 4. But what about the square root of -4? There’s no real number, multiplied itself, that will yield to -4.
To represent the strange behavior of numbers, mathematicians came up with imaginary numbers that serve as place holders in solving equations.
For example, in quadratic equations, you’ll often find imaginary roots among your answers. Other applications include rotating a graph on a polar grid.
Math Is Fun explains logarithm succinctly: “How many of one number do we multiply to get another number?”
For example, if we want to find the number of 2s we need to multiply to get to 32, then we define that problem as “log of 32 with base 2.” The answer is 5.
This has useful applications in data storage. For example, all digital data are store in bits of 0 or 1. To figure out how many bits you would need to represent 32 possibilities, you would calculate “the log of 32 with base 2”. The answer indicates that you would need 5 bits to represent 32 possibilities, since 5^2 = 32.
The Second Law Of Thermodynamics:
Rudolf Clausius’second law of thermodynamics states that the total entropy can never decrease over time for an isolated system, that is, a system in which neither energy nor matter can enter nor leave. The total entropy can remain constant in ideal cases where the system is in a steady state (equilibrium), or is undergoing a reversible process. In all other real cases, the total entropy always increases and the process is irreversible. It also states that whenever energy changes or moves, it becomes less useful as it keeps losing energy on the way. It has led to the discovery of inventions like electricity, internal combustion engines, and cryogenics.
The Fourier Transform defines the mathematics that allows us to put many different signals onto one wire, or one radio signal, and to then extract each individual signal at the other end. It is essential to understanding more complex wave structures, like human speech. Basically, it helps in breaking down the complicated signals into simple waves. According to explanation by Boston University alum, Fourier theory “states that any signal, in our case visual images, can be expressed as a sum of a series of sinusoids.”\
As we know it today, there are four fundamental forces in the world: gravitational force, electromagnetic force, weak force, and strong force.
Maxwell’s equations are a set of four equations that describe electromagnetic force. They’re as important to electromagnetism as Newton’s equation is to gravitational force.
For thousands of years, it was naturally assumed that algebra and geometry were the only fundamental fields in math. It wasn’t until 17th century that calculus joined them as one of the pillars of mathematics.
For example, if a car is driving at 20 miles per hour, and steadily increasing it’s speed to 80 miles per hour over 30 seconds, how far did the car travel in that time span? The problem isn’t easily solved by traditional math, because there’s no clear indications of the car’s speed at various points. Calculus takes into account the rate of change of a process, modeling what happens after a infinitesimally small step occurs.
Today, calculus is a fundamental subject taught in schools around the world.