 Hans Schule, 1939. Photo: simpleinsomnia / Flickr

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.

# 1. Chaos Theory “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. Mathematicians later discovered that those two categories aren’t enough (are they ever satisfied?) Some processes seem predictable, but small changes would bring catastrophic consequences. For example, consider a double pendulum swinging wildly. The locations of the two joints seem easy to calculate, but can vary wildly with the initial acceleration and position.

# 2. Law of Gravity Technically, it’s gravity – not money – that makes the world go round and round. For millennia, astronomers have known that planets revolve in the sky (although many incorrectly thought they revolve around the Earth instead of the Sun), but they couldn’t explain why.

That all changed with Issac Newton’s publication of the Principia in 1687. In the treatise, Newton not only concluded that planets revolve around each other because of gravity, he gave the exact formula that calculates how much force is between two large objects given their masses.

# 3. Fourier Transformation If you have ever pointed a camera at a monitor or a TV, you have probably noticed some wavy lines that you couldn’t see with your eyes. Those are caused by Fourier Transformation. 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.”

You can read more about the Fourier Transformation on MathExchange.

# 4. 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.

# 5. Logarithms 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.

# 6. Maxwell Equations 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.

# 7. Black-Scholes Equation If you ever hear anyone mention “Black-Scholes”, your mind should immediately jump to finance. Developed by Fischer Black, Myron Scholes and Robert Merton in 1970s, the Black-Scholes equation calculates the profit on a financial derivative based on the stock price.

Here’s a crash course on a complex financial instrument called derivative. Let’s say you believe the stock price of Chipotle Mexican Grill (stock symbol CMG) will go from \$450 to \$600 in a year, but you don’t want to buy the shares now – either because you don’t want to take the risk, or you don’t have enough funds right now. For just \$70, you can purchase a contract – called a ‘call option’ – that gives you the right to buy Chipotle in a year at \$450, no matter what happens to the stock.

This contract is called a financial derivative because it’s value is derived from the stock price.

As you can imagine, the contract’s value goes way up if the stock price of CMG increases to \$600, since anyone holding that contract can buy a share at \$450 and immediately sell it on the open market for \$600 – netting a profit of \$150. However, if the price of CMG stays at \$450 in a year, the contract becomes worthless. The Black-Scholes equation allows you to model the value of a financial derivative at various stock prices.

# 8. Navier-Stokes Equation The Navier-Stokes equation represent the flow of incompressible fluid. Incompressible fluids are generally viscous and can’t be drastically reduced in volume under pressure.

The equations can be used to model things such as weather, ocean currents, and flow of hot air. Interestingly, even though scientists have successfully used the equations in real-life applications, no one has ever been able to prove its correctness.

The Navier–Stokes mystery is currently considered one of the seven most important open problems in math. There’s an outstanding \$1 million bounty for the prover of the equations. So far, no one has been able to.

# 9. Normal Distribution Suppose a professor gives out an exam that’s scored out of 100. Most students would score somewhere in the 80s (of course, certain diabolical professors will skew the average down), few may score in the high 90s, and few may score in the low 70s.

This distribution – where the highest frequency happens in the middle, with two skewed tails – is a normal distribution, and it’s one of the most common distributions in nature and widely applicable in population statistics and any predictions of the average.

# 10. Euler’s Polyhedra Formula Polyhedra are to three dimension as polygons are to two dimensions. Each side of a polyhedra is made up of a flat, hole-less surface of a polygon joined by straight lines.

Euler’s polyhedron formula tells us that if you take the number of vertices (V) on a polyhedron, minus the number of edges (E), plus the number of faces (F), you will always end up with two.

Let’s take a cube for example. It has 8 vertices (V = 8), 12 edges along the shape (E = 12), and 6 flat surfaces (F = 6). We see that 8 – 12 + 6 = 2. Voila!

# 11. Pythagorean Theorem We’ve all seen this in grade school because it’s one of the fundamental building blocks of geometry. The Pythagorean Theorem lays out the special relationship between each side of a right triangle: the squares of two shorter sides of a right triangle add up to the square of the third side. There are currently over 130 different proofs for the Pythagorean Theorem, ranging from geometric arrangements to differential calculus.

# 12. Calculus 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.

The ‘discovery’ of calculus is commonly attributed to Isaac Newton, exemplified by the often-repeated story of an apple falling onto his head, leading the young genius to develop theories of planetary motion and inventing calculus as a by product.

Later, it was found that another mathematician – equally as bright as Newton, though perhaps not as famous – named Sir Gottfried Leibniz independently discovered calculus on his own. Neither knew of the other’s discovery, and the glorious title of “inventor of calculus” became a bitter feud between the two men.

Despite a lack of clean founding story, Calculus became a fundamental new way to think about problems. 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.

# 13. Theory of Relativity, Mass–Energy Equivalence Before Einstein came along, classic physicists thought mass and energy were separate things. With the now-famous E = MC2 equation, Einstein showed that mass and energy are interchangeable concepts, and relate to each other through the square of the speed of light.

There are two categories of relativity: special relativity, and general relativity.

In special relativity, Einstein also showed that the speed of light is the absolute fastest speed any object can travel in the universe, precisely at 299,792,458 meters per second.

In general relativity, Einstein postured that mass not only creates a gravitational field around them, they actually bend the space time around it. The more massive the object, the greater the bend. In fact, some objects such as black holes are so massive that time actually slows down around it.