Book Preface Why Equations? Equations are the lifeblood of mathematics, science, and technology. Without them, our world would not exist in its present form. However, equations have a reputation for being scary: Stephen Hawking’s publishers told him that every equation would halve the sales of A Brief History of Time, but then they ignored their own advice and allowed him to include E = mc2 when cutting it out would allegedly have sold another 10 million copies.
I’m on Hawking’s side. Equations are too important to be hidden away. But his publishers had a point too: equations are formal and austere, they look complicated, and even those of us who love equations can be put off if we are bombarded with them. In this book, I have an excuse. Since it’s about equations, I can no more avoid including them than I could write a book about mountaineering without using the word ‘mountain’.
I want to convince you that equations have played a vital part in creating today’s world, from mapmaking to satnav, from music to television, from discovering America to exploring the moons of Jupiter. Fortunately, you don’t need to be a rocket scientist to appreciate the poetry and beauty of a good, significant equation. There are two kinds of equations in mathematics, which on the surface look very similar. One kind presents relations between various mathematical quantities: the task is to prove the equation is true. The other kind provides information about an unknown quantity, and the mathematician’s task is to solve it – to make the unknown known. The distinction is not clear-cut, because sometimes the same equation can be used in both ways, but it’s a useful guideline.
In 2012, Mathematician Ian Stewart came out with an excellent and deeply researched book titled 'In Pursuit of the Unknown: 17 Equations That Changed the World.' Jan 17, 2012 Ian Stewart 17 Equations That Changed the World Topten List From Author - Download as PDF File (.pdf), Text File (.txt) or read online.
You will find both kinds here. Equations in pure mathematics are generally of the first kind: they reveal deep and beautiful patterns and regularities.
They are valid because, given our basic assumptions about the logical structure of mathematics, there is no alternative. Pythagoras’s theorem, which is an equation expressed in the language of geometry, is an example. If you accept Euclid’s basic assumptions about geometry, then Pythagoras’s theorem is true. Equations in applied mathematics and mathematical physics are usually of the second kind.
They encode information about the real world; they express properties of the universe that could in principle have been very different. Newton’s law of gravity is a good example. It tells us how the attractive force between two bodies depends on their masses, and how far apart they are. Solving the resulting equations tells us how the planets orbit the Sun, or how to design a trajectory for a space probe. But Newton’s law isn’t a mathematical theorem; it’s true for physical reasons, it fits observations. The law of gravity might have been different.
17 Equations That Changed The World Ebook
Indeed, it is different: Einstein’s general theory of relativity improves on Newton by fitting some observations better, while not messing up those where we already know Newton’s law does a good job. The course of human history has been redirected, time and time again, by an equation. Equations have hidden powers. They reveal the innermost secrets of nature. This is not the traditional way for historians to organise the rise and fall of civilisations. Kings and queens and wars and natural disasters abound in the history books, but equations are thin on the ground. This is unfair.
In Victorian times, Michael Faraday was demonstrating connections between magnetism and electricity to audiences at the Royal Institution in London. Allegedly, Prime Minister William Gladstone asked whether anything of practical consequence would come from it. It is said (on the basis of very little actual evidence, but why ruin a nice story?) that Faraday replied: ‘Yes, sir. One day you will tax it.’ If he did say that, he was right. James Clerk Maxwell transformed early experimental observations and empirical laws about magnetism and electricity into a system of equations for electromagnetism. Among the many consequences were radio, radar, and television. An equation derives its power from a simple source.
It tells us that two calculations, which appear different, have the same answer. The key symbol is the equals sign, =. The origins of most mathematical symbols are either lost in the mists of antiquity, or are so recent that there is no doubt where they came from. The equals sign is unusual because it dates back more than 450 years, yet we not only know who invented it, we even know why.
The inventor was Robert Recorde, in 1557, in The Whetstone of Witte. He used two parallel lines (he used an obsolete word gemowe, meaning ‘twin’) to avoid tedious repetition of the words ‘is equal to’. He chose that symbol because ‘no two things can be more equal’. Recorde chose well.
His symbol has remained in use for 450 years. The power of equations lies in the philosophically difficult correspondence between mathematics, a collective creation of human minds, and an external physical reality. Equations model deep patterns in the outside world.
By learning to value equations, and to read the stories they tell, we can uncover vital features of the world around us. In principle, there might be other ways to achieve the same result.
Many people prefer words to symbols; language, too, gives us power over our surroundings. But the verdict of science and technology is that words are too imprecise, and too limited, to provide an effective route to the deeper aspects of reality. They are too coloured by human-level assumptions. Words alone can’t provide the essential insights. Equations can. They have been a prime mover in human civilisation for thousands of years. Throughout history, equations have been pulling the strings of society.
Tucked away behind the scenes, to be sure – but the influence was there, whether it was noticed or not. This is the story of the ascent of humanity, told through 17 equations. Download Ebook Read Now File Type Upload Date Epub January 30, 2018 Do you like this book? Please share with your friends, let's read it!!:).
When legendary theoretical physicist was setting out to release, one of the most influential science books in modern history, his publishers admonished him that every equation included would halve the book’s sales. Undeterred, he dared include E = mc², even though cutting it out would have allegedly sold another 10 million copies. The anecdote captures the extent of our culture’s distaste for, if not fear of, equations. And yet, argues mathematician Ian Stewart in, equations have held remarkable power in facilitating humanity’s progress and, as such, call for rudimentary understanding as a form of our most basic literacy.
Stewart writes: The power of equations lies in the philosophically difficult correspondence between mathematics, a collective creation of human minds, and an external physical reality. Equations model deep patterns in the outside world. By learning to value equations, and to read the stories they tell, we can uncover vital features of the world around us This is the story of the ascent of humanity, told in 17 equations. From how the, which linked geometry and algebra, laid the groundwork of the best current theories of space, time, and gravity to how the applies to modeling climate change, Stewart delivers a scientist’s gift in a storyteller’s package to reveal how these seemingly esoteric equations are really the foundation for nearly everything we know and use today. Greek stamp showing Pythagoras’s theorem But the case for why we should even care about equations — and mathematics, and science in general — goes back much further. In 1959, physicist and novelist C. Snow lamented — as — the tragic divergence of the sciences and humanities in his iconic lecture,: A good many times I have been presented at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists.
Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative.
Yet I was asking something which is about the scientific equivalent of: ‘Have you ever read a work of Shakespeare’s?’ Snow later added: I now believe that if I had asked an even simpler question — such as, What do you mean by mass, or acceleration, which is the scientific equivalent of saying, ‘Can you read?’ — not more than one in ten of the highly educated would have felt that I was speaking the same language. So the great edifice of modern physics goes up, and the majority of the cleverest people in the western world have about as much insight into it as their Neolithic ancestors would have had. It is no coincidence, then, that some of the most revolutionary of the breakthroughs Stewart outlines came from thinkers actively interested in both the sciences and the humanities. Take, for instance, who is best remembered for his timeless soundbite, Cogito ergo sum — I think, therefore I am. But Descartes’ interests, Stewart points out, extended beyond philosophy and into science and mathematics. In 1639, he observed a curious numerical pattern in regular solids — what was true of a cube was also true of a dodecahedron or an icosahedron, for all of which subtracting from the number of faces the number of edges and then adding the number of vertices equaled 2. (Try it: A cube has 6 faces, 12 edges, and 8 vertices, so 6 – 12 + 8 = 2.) But Descartes, perhaps enchanted by philosophy’s grander questions, saw the equation as a minor curiosity and never published it.
Only centuries later mathematicians recognized it as monumentally important. It eventually resulted in, which helps explain everything from how enzymes act on cellular DNA to why the motion of the celestial bodies can be chaotic. So how did equations begin, anyway? Stewart explains: An equation derives its power from a simple source. It tells us that two calculations, which appear different, have the same answer. The key symbol is the equals sign, =.
The origins of most mathematical symbols are either lost in the mists of antiquity, or are so recent that there is no doubt where they came from. The equals sign is unusual because it dates back more than 450 years, yet we not only know who invented it, we even know why. The inventor was Robert Recorde, in 1557, in The Whetstone of Witte. He used two parallel lines (he used an obsolete word gemowe, meaning ‘twin’) to avoid tedious repetition of the words ‘is equal to’. He chose that symbol because ‘no two things can be more equal’. Recorde chose well. His symbol has remained in use for 450 years.
The original coinage appeared as follows: To avoide the deiouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe lines of one lengthe: =, bicause noe.2. Thynges, can be moare equalle. Far from being a mere math primer or trivia aid, is an essential piece of modern literacy, wrapped in an articulate argument for why this kind of knowledge should be precisely that. Stewart concludes by turning his gaze towards the future, offering a kind of counter-vision to algo-utopians like Stephen Wolfram and making, instead, a case for the reliable humanity of the equation: It is still entirely credible that we might soon find new laws of nature based on discrete, digital structures and systems. The future may consist of algorithms, not equations. But until that day dawns, if ever, our greatest insights into nature’s laws take the form of equations, and we should learn to understand them and appreciate them.
Equations have a track record. They really have changed the world — and they will change it again.