The old aphorism “my enemy’s enemy is my friend” has a mathematical equivalent: multiplying two negatives makes a positive. In monetary terms, where a negative number is definitely the enemy, it is the same as saying that reducing a debt is equivalent to making a gain.
The simplest case of two negatives making a positive is:
-1 x -1 = 1
Multiplying a number by itself is known as squaring the number, so the square of -1 is 1, the square of -2 is 4, and the square of -3 is 9, etc. However, the square of 1 is also 1, the square of 2 is 4 and so on. The square is the same whether the number is negative or positive.
Going backwards, a procedure called “taking the square root” reverses the result. The square root of 9 is 3, but it is also -3; there are two solutions to any square root calculation.
All this is well and good, and drummed into most of us in school, but what happens if you try to take the square root of a negative number, such as -9? No ordinary number, when multiplied by itself, yields -9, so how can you do the reverse?
This question stymied mathematicians for years. What they finally figured out was the need for a new type of number entirely. By the 18th century, they extended the number system to include the square root of negative numbers.
This is what they did:
i x i = -1
which, rearranged, reads:
i = √-1
The new species of number here is symbolised by i because in the early days it was considered an “imaginary” number rather than a “real” number.
Many mathematicians were suspicious and even derisory about it. The name has stuck, even though today we accept imaginary numbers are just as real as real numbers. You can get more imaginary numbers by multiplying i by real numbers – 2i, 3i, 4i and so on – and there is no problem combining real and imaginary numbers. For example, 5 + 3i is a perfectly good number. Such combinations are called ‘complex’ numbers, though the rules for manipulating them are very simple.
What are imaginary numbers good for? It turns out that by embracing i, the scope and power of mathematical manipulations are enormously broadened, opening the way to a plethora of new shortcuts and tricks.
Exponential growth and decay, when raised to the power of i, superimposes a wave-like oscillation on to the pattern of growth or decay. The world is replete with quantities that oscillate and either grow or decay at the same time. For an example of how this works, think of a swinging pendulum that gradually slows – it oscillates as it “decays”. Using i also greatly simplifies the mathematical description of systems that use complicated oscillating waveforms, such as acoustic and electronic signals.
But imaginary numbers are not just a computational convenience. Mother Nature got there long before mathematicians. We have known since Einstein’s theory of relativity that space and time are not independent but fundamentally tied together by the speed of light into a unified “spacetime”.
Though related, space and time are not the same: it is i that allows us to combine them. To measure the spacetime interval between two cosmic events, for instance, you have to express the time interval in the same units as spatial distance – achieved by multiplying by i. One can, therefore, say that space is “imaginary time” (in the technical sense of imaginary numbers), a term popularised by Stephen Hawking in A Brief History of Time. Hawking discusses a theory of the Big Bang in which the universe started out with four space dimensions, so time was imaginary (in the √ -1 sense) at the outset.
Nature also uses complex numbers in quantum mechanics. If you were the Great Cosmic Designer and tried to come up with laws for atomic processes using only real numbers, the resulting properties of atoms would be very different from what we observe.
There’s another reason to appreciate the number i: its elegance. It won a public beauty contest earlier this year when the BBC asked people to vote for the most elegant mathematical relationship of all time. The winner was declared to be
eiπ+ 1 = 0
where e stands for exponential. This formula was discovered in 1748 by the brilliant Leonhard Euler, known as “the Mozart of mathematics”. By invoking i, Euler was able to combine e with three of the most basic elements of the entire number system: 0, 1 and π. Euler’s formula is a profound relationship that seems to be speaking to us from some sort of mathematical nirvana.
All of which raises the question of how much mathematical beauty still remains hidden from us because of limitations on our existing number system. Is there a future Euler out there who (to borrow another aphorism) will help us to behold it?