Love of numbers: How equations changed the world
February 14th, 2012
08:00 AM ET

Love of numbers: How equations changed the world

Ian Stewart is a mathematician at the University of Warwick in the United Kingdom. His new book "In Pursuit of the Unknown" is published by Basic Books in the United States in March. In the United Kingdom it is available from Profile Books with the title "Seventeen Equations That Changed the World."

I was one of those annoying kids who actually liked equations. I collected them in a notebook. I loved the way you could plug a few numbers into an equation and find out how bright the Sun would be if you were standing on Pluto. Or work out how big a rainbow looks from the refractive index of water and the time of day.

I realize I am a rarity in that respect. Stephen Hawking’s publishers allegedly told him that every equation he put into his runaway bestseller "A Brief History of Time" would halve its sales. So, if he’d left out Einstein’s E=mc2, he would have sold another 10 million copies. But his publishers had a point. Although the great equations have had more impact on humanity than all the kings and queens in the history books put together, they can look very off-putting.

That’s why I wrote "In Pursuit of the Unknown: Seventeen Equations That Changed the World." We need to stop being put off, and learn to value our equations. It’s hard to write a book about equations without including any, so I decided to follow the age-old theatrical advice: ‘if you’ve got a wooden leg, wave it.’ Make equations the main characters in a story of the rise — and occasional fall — of humanity.

Let’s not overdo it, though. I’m a mathematician; it’s my job to understand the nuts and bolts of equations. You don’t have to. But I reckon it may well be worth your while to appreciate where equations came from, what they say, and what they’ve done for us. Without actually doing any of the sums. It’s like enjoying music without knowing how to compose it or how to read a score. All gain and no pain.

The book came about by accident and serendipity. A Dutch publisher asked my English publisher if they had a book in the pipeline about equations for non-specialists. They didn’t, but that can always be fixed. The more we thought about the idea, the better we liked it. We moved a couple of my other projected books up a year to make room to write it.

The first task was to choose the equations. I decided that unless they really had changed the world, in a big way, then they didn’t go in. However pretty they were, however fascinating they were to mathematicians... no major impact on humanity meant they got the chop. I managed to cut my original list down to about 20. To avoid unnecessary overlaps, I started combining closely related equations into a single chapter. So Einstein’s equation could be a placeholder for the whole of special and general relativity, and a calculus equation would take care of Newton’s laws of motion as well. That whittled the number down to 17—a nice, mysterious, seductively precise yet totally arbitrary figure.

As the book evolved—no book of mine ever follows the original plan exactly, that’s more of a negotiating platform—I found myself spending more and more time trying to picture what the world had been like before the equation was invented or discovered. Only then could I explain what it had changed. Before James Clerk Maxwell wrote down his equations for electromagnetism, streets were lit by burning gas and vital messages went on horseback. Afterwards we had radio, and television soon followed. In between came electric lighting and the telegraph. I learned a lot of history and managed to avoid a few myths.

Most of the great equations emerged from the needs of science. Relativity and quantum mechanics encompass almost the whole of modern physics. Newton’s law of gravity explains most of what we know about the movements of the stars and planets. General relativity plugs a few gaps where Newton’s law isn’t accurate enough. But some came from elsewhere—the Black-Scholes equation for pricing options originated in the needs of global finance. The equation for information came from code-breaking and communications. The equations of statistics began as aids for gamblers.

A key theme in the book is: What do equations do for us—today, in our homes, in daily life? Maxwell and the wave equation gave us radio, TV, cellphones and wireless computer peripherals. The Fourier transform, introduced to understand the flow of heat, makes digital cameras feasible by compressing image data. Engineers use Newton’s law of gravity to design orbits for spacecraft and to launch satellites, giving us thousands of TV channels and global communications. Without both special and general relativity, satnav would never work. Without quantum mechanics, leading to memory chips and most modern electronics, there would be no laptops, iPads, or Amazon Kindles.

I now find that as I go about my everyday activities, it’s as if little labels keep popping up in my head: Equations Inside. I sense them in the supermarket—a pack of carrots pops up a label with equations from statistics, used to develop new breeds of vegetables suitable for storage and transportation. A passenger jet passes overhead, and up pops the Navier-Stokes equation for fluid flow, implemented through computational fluid dynamics, whose applications also include the design of stents to keep arteries open and prevent heart attacks. And I realize that the title of my book is no exaggeration. Equations really have changed the world, repeatedly altering the course of history. And they’ll do it again.

soundoff (33 Responses)
  1. physics science

    Thanks for some other informative blog. The place else may I am getting that kind of info written in such a perfect method? I have a project that I am just now operating on, and I have been on the glance out for such information.

    April 3, 2012 at 11:24 pm |
  2. GovernmentCheese

    There are only 10 types of people in the world – those that understand binary and those that don't.

    February 15, 2012 at 9:21 am |
    • Arpit

      pink bracelets. BTW, Was this thuhgot of by the same guy who wanted to include more shopping to appeal to women? If it is, it's definitely time to get some pink Doc Martins. Stompy Stompy.

      October 13, 2012 at 1:18 am |
  3. helenecha

    Uh, E=mc^2 + 0

    February 14, 2012 at 10:23 pm |
  4. Peter Aleff

    for another story told by the numbers about the creations of both the world and also of the numbers themselves, look at the ancient Egyptian numeral system, as explained at

    February 14, 2012 at 10:17 pm |
  5. Peter Aleff

    For another set of equations that were once deemed to rule the world, see the creation story of Genesis 1 and its expression as quasi-equations between mathematical constants, as shown at Enjoy those numbers and their neat fit!

    February 14, 2012 at 10:12 pm |
  6. enkephalin07

    Equations and numbers don't necessarily have anything to do with them. In most mathematics, numbers are just a shorthand attached to letter constants and variables, only dealt with when there's an opportunity to simplify the equation. For higher mathematics, what you need is a love of operators and the language of logic they can be applied to.

    February 14, 2012 at 9:30 pm |
  7. M in Oz

    Is this meant to be a Valentines Day article about a professor who loves Number 1 and Number 2?

    February 14, 2012 at 8:30 pm |
    • It's binary to me

      0 and 1.

      February 14, 2012 at 9:41 pm |
  8. BOBBY

    The rectification of numeric rationalizations is esoteric when compared to the algorithmic genuflection of digital complexity. For one to genuflect on reactive subcutaneous calculative theory one must first understand the digital interface in terms of retro formulaic reciprocity.

    February 14, 2012 at 6:07 pm |
    • TYF

      The idea of a chair is more real than a chair." (Plato?!) The purpose (reasoning, methodical madness) for mathematics is more real than the equations declariing the process for reasoning: equations are simply the labeling or naming of what was (could be or would be) the cause-effect of physical phenomenon after figuring how it could/would/should be, right? BOBBY...

      February 14, 2012 at 11:14 pm |
  9. BOBBY

    Every one knows that the study of rational second order ODEs in the complex plane led to the discovery of new transcendental special functions which are now known as Painlevé transcendents and thats where the real fun is.

    February 14, 2012 at 5:58 pm |
    • dr187

      Finite dimensional systems are for women.

      February 14, 2012 at 6:53 pm |
    • If you say so


      February 14, 2012 at 9:42 pm |
  10. BOBBY

    The 'E" and "mc" are letters....just sayin.

    February 14, 2012 at 5:48 pm |
  11. dr187

    What about Ax=lambda*x? Eigenvalues/Eigenvectors play a pivotal role in Google's page rank algorithm. Hopefully this relationship is included, Ian.

    February 14, 2012 at 5:19 pm |
    • BOBBY

      clearly not relevant.

      February 14, 2012 at 6:00 pm |
    • ajk68

      You know we are in a scientifically illiterate culture when most people have never even heard of eigenvalues/eigenvectors. It's a lot more relevant to people's daily lives than E=mc^2.

      February 14, 2012 at 6:57 pm |
      • Dan, Tx

        very few people take that level of math or physics don't forget the Hamiltonian operator and the equation for electron orbit of hydrogen

        February 14, 2012 at 9:31 pm |
  12. ianstewartyousucknow

    there IS a book out there about equations that doesn't hold back the meat of the subject: Journey Through Genius. it is much better than any of these ian stewart books that is watered down. i wish ian stewart could write a book for mathematicians, not the dilettante who buys the book but will never read it.

    February 14, 2012 at 4:59 pm |
  13. ianstewartyousucknow

    ian stewart. you write too many crappy books. take a break and spend some time writing a really good book that goes deep into mathematics and not some pop-sci book that took you two weeks to write. if you don't, you are a mathematical sell-out.

    February 14, 2012 at 4:56 pm |
    • YoureADoofus

      I think the idea was to broaden the appeal of mathematics – not write another book for people who are already etrenched ("A Dutch publisher asked my English publisher if they had a book in the pipeline about equations for non-specialists.") – why does this make him a sell-out??

      February 14, 2012 at 5:39 pm |
    • BOBBY

      Ya....hey get out of your complexdiffyq world and understand that people like me might want to "dabble" with this stuff and not spend a lifetime enslaved to the most baoring $h!t in the galaxy to nail it.

      February 14, 2012 at 5:50 pm |
  14. Yes1fan

    If Fractal equations were left out of the book, then that is a huge omission.
    Their relative newness, compared to the already huge practical impact they are making in modeling and graphics, mean Fractals rank among the most important.

    February 14, 2012 at 3:00 pm |
    • ajk68

      Fractals aren't that important when compared to Maxwell Equations, Heat Equation, distance equation, etc.

      February 14, 2012 at 7:01 pm |
    • Jean

      Fractals, as fascinating as they are, are just a subset of complexity.

      February 14, 2012 at 10:15 pm |
  15. Marine57

    I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be.
    Lord Kelvin (1824-1907)

    February 14, 2012 at 1:48 pm |
    • TYF

      Would you be interested with employment at SDG&E? To do mathematics, please: for instance, how to figure out what is one day's worth of electrical use from (isolated) one month of 32 days... SDG&E (with the miracle of mathematics still being a mystery) has decided every one day = 3.2 days worth. I've explained how that is a "gross mathematical error"
      but SDG&E is persisting it is intentional malicious fraud. Why should I care, besides having to pay the bill...
      (Despite digression: .01 x 32 = .32 or one day; .1 x 32 = 3.2 or not 1 day). If you get it, could you apply?!

      February 14, 2012 at 11:23 pm |
  16. Unacceptable


    That's the real equation, if you are going to use E=mc^2 please make sure to say that momentum is not included in the equation.

    February 14, 2012 at 12:52 pm |
    • YoureADoofus

      Please bite me

      February 14, 2012 at 4:52 pm |
  17. Ian. UK

    Isn't it E = m c 2 and a bit? Thanks to CERN and those pesky little neutrinos.

    February 14, 2012 at 11:15 am |
    • LoLcAtZ

      e=mc^2 isn't even the full equation here. An Einstein didn't even come up with this. He only developed the Lorentz transforms and simplified the equation into these 3 variables. Currently, the understanding of time and space dilation and length contraction works in every known scenario except for neutrinos. So, either CERN is wrong (which has yet to be proven or disproven) or there is a correction factor in these equations for special types of particles that can be added in. Either way, this has no effect on normal day life, it only means that there are special cases which we can possibly make use of.

      February 14, 2012 at 11:25 am |


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