These seventeen equations were mentioned in Ian Stewart’s book.These are briefly described as follows.
Pythagoras theorem is a foundational piece of mathematics underlying all of geometry. It’s useful in defining the mathematics of the circle and the so-called circular functions or trigonometric functions such as sine and cosine. It’s crucial to defining the mathematical concept of distance and underlies the theory of vectors. Without Pythagoras we don’t have surveying and without surveying it’s hard to imagine the concept of modern property ownership.
What it says:
knowing two sides of a certain type of triangle tells you the third side.
Why it changed things:
it showed that shapes and numbers are connected, and allowed geometry and measurement to be used to describe things in the world. Pythagoras and his followers took this one step further and believed that mathematics was what the world was made of. This belief still bumps around today in various ways.
As the equation shows, logarithms turn multiplication into addition (and division into subtraction, and exponentiation into multiplication, and roots into division). Once logarithmic tables had been worked out (a very tedious and laborious task that took years), complex calculations became much simpler. This in turn made engineering and science easier and so accelerated the pace of engineering and science throughout the 1700s and 1800s. My very first ‘personal computer’ was a slide rule (anyone under the age of 40 might need to look this up on wikipedia 🙂 ), a ‘calculator’ that was based on the concept of logarithms and for 100 years before the 70’s, the slide rule was the only way engineers did the calculations to design bridges, buildings, dams, airplanes, cars, and so on.
What it says: By transforming numbers we can turn multiplication into addition
Why it changed things: it allowed us to do complex calculations more simply, making problem solving using numbers much easier. It also showed how complicated things could be turned into simpler ones, then back. That idea has had a lot of implications in math and science.
Without calculus there is no modern engineering and science. Period. Everything that has happened over the past 345 years in the applied sciences (engineering, chemistry, economics, medicine, biology, etc. etc) yes EVERYTHING, is in some way dependent on the mathematics of calculus. Of all the mathematical concepts on this list, calculus has had, by far, the greatest influence on how we live. In fact, eleven of the remaining fourteen equations would not be possible without using calculus.
What it says: we can define a mathematical way to say how things change, or what the results of change are.
Why it changed things: It made complicated problems involving describing change and putting together the effects of change. This made it possible to apply math to describe whole new areas of the world, including most of physics.
4.Newton’s Equation of Universal Gravitation
Newton’s law of gravity (along with the mathematics of calculus which he also invented) changed how we look at the universe. Before Newton, no one could figure out exactly how the earth and the other planets fit together with the sun. Various people (Copernicus, Galileo, Kepler, etc.) kept getting closer, but none of them could get it quite right, until Newton and his law of gravity, which explained the motion of planets beautifully and stood the test of time for the next 230 years until Einstein came up with a better theory. Newton’s laws are still good enough to calculate the orbits of satellites and the paths of spaceships – at least until we invent hyperdrive 🙂
What it says: Objects attract one another in a way that depends on how far apart they are and how much mass each has.
Why it changed things: It showed that one idea could explain motion both here on earth and for the planets in the solar system. This not only showed that the entire universe (as known at the time) obeyed the same rules and was therefore one thing (not two separate things as people had assumed before) but also showed that we could understand the rules of the whole universe, it wasn’t random or arbitrary but had deep order in it.
5.The square root of minus one:Iota(i)
Humans have had to invent new number concepts as they became more sophisticated at attempting to count and measure things. First there came the concept of negative numbers, then the concept of zero, then the idea of the square root of 2 which can’t be expressed as the ratio of two whole numbers, leading to the concept of irrational numbers, and next came the concept of numbers such as pi, which can’t be expressed as the solution to polynomial equations (equations that rely on extracting roots to solve the equation), leading to the concept of the transcendental numbers and to the concept of the real numbers. And eventually this chain of logic led to imaginary numbers defined with the square root of minus 1 because there are equations that can’t be solved without this concept. In fact the equations behind quantum mechanics aren’t possible without this concept. Euler used this idea, combined with calculus, to set up the equations that led to the infinite series that are used to calculate sines and cosines, which in turn lead to ways to accurately calculate a value for pi. Today the mathematics used in modern electronics have their basis in the concept of the square root of minus one. And finally, the square root of minus 1 completes the number system. All of mathematics can now rest on a combination of real and imaginary numbers, no further concept of number is necessary.
What it says: we can define a new kind of number that makes it possible to solve problems that had never had solutions.
Why it changed things: It opened up the number system and finally made it complete. We had been adding new kinds of numbers for a while (can’t solve 5x = 7? Create fractions. Can’t solve x^2 = 2? Create irrational numbers. Can’t solve x + 3 = 2? Create negative numbers.) Now the number system is complete and every equation involving basic operations has solutions (they may be hard to find, but they are there in the competed number system). This also makes solving real problems much easier since you can use the complete system to go around problems and get to that answer.
6.Euler’s formula for polyhedra
Euler’s formula for polyhedra is important in the study of topology and graph theory, but right now I’m having a hard time finding practical examples of how this has changed the world around us, except maybe the topology of the internet, which, if I’m not mistaken, relies on graph theory.
What it says: For a 3D shape made of flat pieces (a polyhedron) the number of corners – the number of edges + the number of facts = 2.
Why it matters: this showed that there are properties of shapes that go beyond shape… Every closed shape has this property, but if it has a hole (like a donut) then V – E + F = 0. So having a hole like a donut makes a new kind of shape. There are other kinds of shape that are described by V – E + F being different. So This formula lets you classify shapes in a way that depends on a deep shared property. We call this topology and it turns out to be quite important in understanding how shapes and space itself work.
The normal distribution is a key concept in statistics. You may be more familiar with it under it’s more commonly known name – the bell curve:
A great many naturally occurring phenomena are normally distributed. The height of human adults is normally distributed for example. The variation of objects coming off an assembly line are typically normally distributed which becomes an important fact for quality control. Today, topics as diverse as economic theory, epidemiology, pharmaceutical testing and effectiveness, political polling, and many, many more, are based on statistical theories that rely on normal distributions.
What it says: When random things happen you can’t predict the result by definition. But when a lot of random things happen and add up they make a pattern that we can predict very well.
Why it matters: Discovering that randomness creates predictable patterns let us actually make reliable predictions in spite of not being able to predict any one event. It also gave us ways to see if things were random or if there was something more being hidden by the randomness. Since everything we measure in the real world has some randomness this was hugely important and still is.
8.The Wave Equation
The wave equation explains vibrations, also known as oscillations, and a great many phenomena in science and nature result in waves. The equation itself results from the fact that energy can’t be created or destroyed, but instead oscillates between kinetic energy (energy resulting from motion) and potential energy (energy resulting from a force) that moves through matter in a self-reinforcing way.
Once something starts to vibrate it has energy and this energy distributes throughout the system. This in turn explains how musical instruments work, how sound waves work, why weights hanging from springs bounce up and down. This is why you have squeaks and rattles in your car. It also explains how energy can be coupled into a system to make it vibrate. This is what happened to the Tacoma Narrows Bridge causing it to collapse.
The wave equation has a great many practical applications in engineering (Electrical, mechanical, civil), and physics.
What it says: In certain circumstances a disturbance can spread out making a wave. This equation says when and why that happens.
Why it matters: waves are the principle way that energy is carried from place to place, and this equation lets us understand that process. It tells us how a wave forms and how fast it travels. Recognising this pattern says “waves be here”. When James Clark Maxwell had this equation pop up in his description of electricity and magnetism, for example, he knew right away that this meant that there must be electromagnetic waves. When he calculated that the speed of these ways was the speed light travelled it seemed likely that light was just such a wave.
What Fourier did was show that any arbitrary cyclical (ie: continuously repeating) wave form could be built up by adding together sines and cosines of varying amplitudes and frequencies. Here’s a good illustration showing the basic idea:
This has great application in modern communications. 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. This makes it possible to put 250 TV channels onto a single cable. It makes it possible for one cell tower to communicate with hundreds of different cell phones. And it makes it possible to squeeze ever greater quantities of information into a single signal, such as a wifi signal, making it possible to stream movies in 4K ultra high definition.
What it means: We can represent a function as composed of other functions. This lets us change a function into a different one in a way that shows different information about the function.
Why it matters: Some information about a function is easy to see, but other information is hidden and hard to extact or understand. By changing the equation into a new one in this way that hidden information pops out and hard or complex things become easy. A simple example of this is turning a complex wave into an analysis of its frequencies, which is vital in speech recognition .
10.The Navier-Stokes Equation
The Navier-Stokes equation defines the mathematics of fluid dynamics. This has a great many applications, but perhaps the one application that has had the greatest effect on how we live is in the design of airplane wings. These equations define the flow of a fluid (air) around airplane wings. It’s used to calculate the velocities and resulting pressures of the airflow at all points around the wing which in turn is dependent on the shape of the wing. It’s what helps aeronautical engineers design more efficient wings.
What it says: the equation describes the flow of a fluid using conservation of mass and momentum.
Why it is important: fluid flow is at the heart of systems we care about, from weather to the design of airplane wings to plumbing to fusion in the sun. These equations apply to all of these and more – though we don’t know how to solve them in general. Learning more about them has been important in science and tech and will continue to be so for a long while to come!
Maxwell’s equations result from the physics of electric fields and magnetic fields. By combining the electric and magnetic fields into a set of four equations they define the key mathematics behind radio waves of all types, or what scientists and engineers call electro-magnetic radiation. And when I say radio waves I don’t just mean your car radio. The mathematics applies to all electro-magnetic radiation, including low frequency radio waves, to microwave and radar, to infrared (night vision goggles), to lasers, to visible light, to x-rays and so on. In other words, all waves that can travel through space at light speed. We wouldn’t be able to design wireless transmission devices without Maxwell’s equations. Radio, TV, radar, cellphones, some forms of medical diagnostic imaging, and a whole lot of other things we take for granted would not have been possible without Maxwell’s equations.
What they say: Electric and magnetic fields are interconnected. Changing an electric field creates a magnetic field and vice versa.
Why they matter: These equations linked electricity and magnetism into a single thing, electromagnetism, and opened the way to almost all our modern technology as well as starting physics on its path to field theory and the current view we have of he universe. These equations were therefore huge game-changers. Hidden in their symmetry are also the clues to the connections between space and time that Einstein used in describing relativity theory (there is a reason that the first paper on relativity is called “On the Electrodynamics of Moving Bodies”!
12.The Second Law of Thermodynics
The second law of thermodynamics is key in understanding all heat engines. This was one of the important basic ideas behind the industrial revolution, the steam engine, the internal combustion engine, the diesel engine, the jet engine. None of these things would be possible without a solid understanding of thermodynamics and the second law in particular.
What it says: there is a property of physical systems which is over and above the standard physical properties we are used to, and which can be thought of as a measure of scatter. This does not naturally decrease, but increases as time passes.
Why it matters: the most willfully misunderstood equation on the list, even including E=mc2 because it has links to order and information (more on that in 15.). It tells us that some processes are not reversible (which may even be where time itself originates) and that there is a physical property of systems that is related to their organization and which has important consequences. It also implies that the universe and everything in it eventually runs down, which kind of sucks but is better than change being impossible which is the probable alternative.
Relativity. Atom bombs, black holes, the big bang, the expansion of the universe, the understanding of galaxies, the whole idea of the fabric of space-time and the many awful science fiction movies that these ideas spawned.
What it is: well, the equation given is really a consequence of relativity but OK. What it says is that things we thought were separate (like energy and mass, or space and time) are really the same thing in some important ways. That changes our understanding of how the universe is put together.
Why it matters: Well, it changes our understanding of how the universe is put together! It also shows that our ideas of how all that works were based on some very wrong ideas (like the existence of an objective time in which things happen). Everyone gets hung up on the light speed limit or the idea that this has something to do with bombs, but really it just shows that the universe can only make sense (in a certain way) if spacetime has a simple and elegant geometry underlying it. This geometry was already hidden in the way that Maxwell’s equations work (see 11.) and just making our understanding consistent gives everything else as a consequence.
Schrödinger’s equation was a key contribution to the understanding of quantum mechanics. It defines what we call the ‘wave equation’ that is famously used in the philosophical discussion of Schrödinger’s cat, which exists in both a dead state and an alive state inside a closed box, and we don’t know which (dead or alive) until we observe by opening the box, thus ‘collapsing’ the wave function and determining the fate of the unfortunate cat.
However, the cat has a lot more to do with philosophy than physics. Physicists don’t think much about the cat. Instead, they just know the equation works and describes the behavior of matter at the scale of atoms. Today, all of our semiconductors (transistors, integrated circuits, Intel CPU chips, etc. etc.) rely on the science of quantum mechanics, and (the cat not withstanding) we wouldn’t have been able to understand quantum mechanics without Schrödinger’s equation. So thank Mr. Schrödinger for the computer you’re using to read this right now.
What it is: Forget the cat. This equation describes how the waves that QM theory says are reality change and evolve in time. So it tells you how, if you know the state of the system at one time, you can determine what the state of the system will become in the future.
Why it matters: This is the essence of QM, and the equivalent of total F=ma for classical physics (and how is that not on the list?) This equation, and its generalizations, turns the basic idea of QM into something that can be used to describe real systems and see how they change and develop. Without that the theory wouldn’t really say anything!
Information theory defines the basics behind how we code and transmit information. It defines the minimum number of 1’s and 0’s needed to encode a specific piece of information, and how we differentiate between signal and noise. It’s a key concept behind modern communications.
What it is: The Shannon Entropy. This says that systems containing information have a property that looks a lot like the entropy from equation 12.
Why it matters: it tells us how to do with information what entropy does for physical systems (and that gave us steam engines and everything thereafter). It hints at connections between information and physical systems and implies that information has physical aspects, which is a bit surprising. It is also often overgeneralized and misunderstood.
16 Chaos Theory
Chaos theory is used to gain greater mathematical insight into weather prediction, and into unstable systems such as turbulence in fluid flows, instability in finance and economic systems, and so on.
What it is: I don’t know this particular equation, but it looks like a basic iterative process, which is the sort of thing central to chaos theory. What it says is that if you have a perfectly well described system, that doesn’t mean you can figure out what will happen. In the long term tiny differences in initial conditions add up to huge differences in output.
Why it matters: well, it means you’ll never predict the weather too much in advance. And that there are limits on how well we can describe interacting systems. But it also shows that there are patterns within the unpredictable behaviour, so that we can say interesting meaningful things about such systems in spite of not being able to make full pedictions.
Black-Scholes is used to price more exotic financial instruments that are based on more basic underlying instruments. It’s based on assumptions from statistics and economic theory. It’s used to price stock options that are based on the underlying stock. Financial instruments like this are called derivatives because they derive their value from the underlying financial instrument that they’re based on. Financial derivatives can be very sophisticated and complex, much more so than the relatively basic stock option, and the modern financial system relies on derivatives – maybe too much so. It can be argued that the great recession of 2008 was made much worse because of the great many sophisticated derivative products (credit default swaps for example) that were spread throughout the financial world, and it turned out that despite Black-Scholes, these products weren’t accurately priced after all (for a lot of reasons), and once bankers and investors realized that they could no longer depend on these prices being accurate panic ensued resulting in a lot of innocent people being hurt financially.
What it is: a mathematical model for investment, which relates the trading of a risky item (like stocks) vs a risk-free item (like cash).
Why it matters: it shows how to adjust your use of the risky asset to make the risk less and still get the advantage of returns it offers. This is now very widely used and has reshaped the stock market etc. Some think this is a Good Thing and some think it is a very Bad Thing. But it is definitely a Thing.
Contents-God created the integers by Stephen Hawking.