Where do numbers come from, anyways?

A short history of imaginary numbers

Mathematicians first came up against imaginary numbers in the mid 16th century and it wasn’t until the mid 19th century that they saw how awesome complex numbers could be. Before we look at how imaginary numbers came to be, let’s look at some other familiar number systems.

Number Systems Solve Problems

The first, most obvious, number system is the integers, or counting numbers. We have integers to answer really useful question that we see all the time in day-to-day life like, how many grapes can I really stuff into my mouth at a time? (About 9)

picture1

The next number system we might think about is the rational numbers, or fractions. These also serve to answer really useful day-to-day questions like that involve division like, if I have 6 roommates but only 1 pint of ice cream, what portion of the tub can I eat?

This assumes that I’m a fair roommate who would never eat more than her share of the communal ice cream- which leads us to our next number system. Negative numbers are used to measure debt; like how much ice cream I might owe my other roommates.

picture2

With these two systems we can count and divide stuff, but we also might have other sorts of problems like how to measure things. Like, for instance, I might need to walk 1 block south and 2 blocks east around a park to get to school, but since I’m inherently lazy (a good quality for all mathematicians), I cut straight through the park, and find that I’ve walked √5 blocks to get to school, which is totally irrational. We have to deal with irrational numbers when we measure distances because it turns out (to the Greeks’ great sorrow) that not all distances can be measured with rational numbers.

So what about imaginary numbers?

Where did they come from, and what are they good for?

We’ve got Real Problems: Imaginary Numbers give Real Results

In the mid 16th century a mathematician named Tartaligia came up with a general solution for finding the roots of 3rd degree polynomial, but he held his method as a closely-guarded secret. Another mathematician named Cardano eventually managed to convinced the reluctant Tartaligia to tell him the method, on the condition that he would never ever tell anyone else. Well, I think they should make a soap opera about 16th century mathematics because in 1545 Cardano completely betrayed Tartaligia by publishing the solution in his book ‘Ars Magna’.

Tartaligia’s method is really important in the history of imaginary numbers because there are some perfectly good 3rd degree polynomials with perfectly good real roots that this method doesn’t make sense for. When you use Tartaligia’s method for these certain polynomials, you get a nonsense step in the middle of the calculation where you have to take the square root of a negative number.

Consider for example the equation:

x3 = 15x + 4.

 This cubic has a real root x = 4, but when we apply Cardano’s formula we get:

x = ∛[ 2 + √(-121) ] + ∛[ 2 – √(-121) ]

The real problem (pun intended) was that even though everyone knew that taking the square root of -121 was totally ridiculous, they also knew that the root x=4 was a totally reasonable real solution. There was this breakdown in what the equation was trying to communicate.

The first mathematician to really break through this mold was Rafael Bombelli, who got around this problem with the crazy proposition that, well let’s just imagine that there’s some number that’s negative when we square it. With this assumption he was able to manipulate Tartaligia’s equation, for instance the example above becomes:

 ∛[ 2 + (√-121) ] + ∛[ 2 – (√-121) ] = (2 +(√-1) ) + (2 –  (√-1))   (**!)

= 4 – 2(√-1) + 2(√-1) – (√-1)2

= 4 (!)

Conveniently, the ‘imaginary’ numbers cancel out, leaving good real roots! Way to take a leap of faith, Bombelli!

** Okay, hold on, what just happened there? Well it turns out (2 +(√-1))3 is :

(2 +(√-1))3 = (2 +(√-1))*(3 + 4i) = (2 + 11i) = 2 + (√-121)

 Same goes for ( 2 – (√-1)). Neat.

About a half-century later in 1637, Descartes coined the term “imaginary” when he wrote about roots of nth degree polynomials in his book ‘La Geometrie’. He wrote that these polynomials might have as many as n solutions, but sometimes they have fewer, as some of the solutions are ‘impossible’, ‘improbable’ and ‘imaginary’. He meant it in a demeaning way- like we should be doing ‘real math’, not ‘day-dream math’.

At this point in history mathematicians swept imaginary numbers under the rug; they cautiously imagined that they might exist but only for long enough to cancel out and yield real solutions. It wasn’t conceivable that they might be useful by themselves.

It’s sort of a complex story

 The complex number system was really first understood as the incredibly powerful mathematically tool that it is in the 19th century when Gauss took an interest in imaginary numbers. He came up with a geometric interpretation for complex numbers (which, to be fair, was also independently discovered by the Norweigan mathematician Wessel and the French bookstore manager and amateur mathematician Argand). Gauss’ interpretation was that the imaginary number line is just like the real number line, so a complex number (a number with a real and an imaginary part) is actually a coordinate in a plane, like in the image below. We just say that real numbers lay on the horizontal axis, while imaginary numbers lie of the vertical axis.

picture3

The really amazing and exciting thing about this description is that it’s totally consistent with operations we might like to do on complex numbers, like addition and multiplication. Consider what happens when we multiply by i, for instance 1*i. We rotate 90 degrees, from the coordinate (1, 0) to the coordinate (0, 1), so we can say that multiplying by i is the same as rotating by 90 degrees. Then consider i*i (which is i2): we rotate another 90 degrees and end up at -1! Neat!

Later in the 19th century complex numbers got a lot of traction because they turned out to be very good at describing waves. At this point in history, physicists were developing ways to describe electricity and magnetism, and complex numbers enabled them to really understand these phenomena.

The neat thing about complex numbers is they show up everywhere in our day-to-day lives. Anything you have that uses electricity only works because some engineer somewhere knew how to build it using their understanding of imaginary numbers. Can you even imagine your life if you couldn’t send your mom photos of other people’s dogs? Any time you snap a photo or make a phone call your phone does a Fast-Fourier-Transform, which is a method based on complex numbers, to compress the data into just tiny amounts of storage.

So do imaginary numbers really exist?

Complex numbers are great representations for lots of natural phenomena, like electricity. Remember how we used our other number systems- like how we used integers to count how many grapes I could fit in my mouth? In some sense, it’s just the grapes that exist, not that the integers. The integers exist mathematically- they’re only there to describe the real world, and this is true for every number system. In this sense, not only do imaginary numbers ‘exist’ mathematically, but they’re first-class citizens because they describe so many awesome things that we use every day.

Sources:

http://www.bbc.co.uk/programmes/b00tt6b2

http://en.wikipedia.org/wiki/Imaginary_number

http://www.storyofmathematics.com/16th_tartaglia.html

http://en.wikipedia.org/wiki/Niccol%C3%B2_Fontana_Tartaglia

http://mathfaculty.fullerton.edu/mathews/c2003/ComplexNumberOrigin.html

http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

http://en.wikipedia.org/wiki/Caspar_Wessel

http://www.ms.uky.edu/~sohum/ma330/files/eqns_4.pdf

http://www.und.edu/instruct/lgeller/complex.html

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2 thoughts on “Where do numbers come from, anyways?

  1. Madhav

    Nice crisp summary. I liked it. Can you give a specific example of how imaginary numbers represent something like, say, electricity. What does an imaginary number represent in electricity?

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    Reply
    1. drevelynlamb

      Thanks for your comment. The Wikipedia page on complex numbers has some information about their use in many practical applications, including electromagnetism: http://en.wikipedia.org/wiki/Complex_number#Electromagnetism_and_electrical_engineering
      The Fourier transform is one way that complex analysis ends up having real implications. Fourier analysis is used to analyze or synthesize basically any signal that has some sort of frequency, including sounds and images.
      http://en.wikipedia.org/wiki/Fourier_transform & http://en.wikipedia.org/wiki/Fourier_transform & http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face

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