In 16th century Venice, formulae for solving equations were closely guarded intellectual property. Of particular interest to ballistics and fortifications expert Niccolo Tartaglia were quadratic and cubic equations, which model the behaviour of projectiles in flight amongst other things. These may well ring a bell with you from school maths – quadratic equations have an x2 term in them and cubics an x3 term. Tartaglia and other mathematicians noticed that some solutions required the square roots of negative numbers, and herein lies a problem. Negative numbers do not have square roots – there is no number that, when multiplied by itself, gives a negative number. This is because negative numbers, when multiplied together, yield a positive result: -2 × -2 = 4 (not -4).
Tartaglia and his rival, Gerolamo Cardano, observed that, if they allowed negative square roots in their calculations, they could still give valid numerical answers (Real numbers, as mathematicians call them). Tartaglia learned this the hard way when he was beaten by one of Cardano’s students in a month-long equation-solving duel in 1530.
Mathematicians use i to represent the square root of minus one. This is called the imaginary unit – it is not a real number, does not exist in ‘real’ life. We can use it to find the square roots of negative numbers though. If I want to calculate the square roots of -4, I can say that -4 = 4 × -1. This means that the square root of -4 is the square root of 4 multiplied by the square root of -1. In symbols:
The square root of 4 is 2, and the square root of -1 is i, giving us the answer that the square root of -4 is 2i. We should also note that -2 is also a square root of 4 for the reasons stated above. This means that the square roots of -4 are 2i and -2i.
The arithmetic of i itself initially posed an obstacle for mathematicians. I stated above that a negative times a negative gives a positive and we are innately familiar with the idea that a positive times a positive gives a positive. With the imaginary unit, this seems to break down, with two positives multiplying to give a negative:
i × i = i2 = -1
Equally, here two negatives multiply to give a negative:
-i × -i = i2 = -1
This was a problem for some time and made some people feel that using them in formal mathematics was not rigorous. Rafael Bombelli, another Italian renaissance man, wrote a book called, simply, Algebra in 1572 where he tried to explain mathematics to people without degree-level expertise, making him an early educational pioneer. In Algebra, he explains how to perform arithmetic on positive, negative and imaginary numbers, making the case that the imaginary unit (i wasn’t used as the symbol until the 18th century) was neither positive nor negative and hence did not obey the usual rules of arithmetic.
The work of these mathematicians on imaginary numbers allowed the development of what is now called the Fundamental Theorem of Algebra. In basic terms, the number of solutions to an equation is always equal to the highest power of the unknown in the equation. For instance, when I was working out the square roots of -4 above, I was solving the equation x2= -4. The highest (and only) power of the unknown x in the equation is two, and lo and behold we found two answers, 2i and -2i.
With a cubic equation, where the highest power is three, I should get three solutions. Let’s look at x3 + 4x = 0, which is the same form of cubic equation that Tartaglia dealt with. x = 0 is a solution, as 03 – 4 × 0 = 0 – 0 = 0, fulfilling the equation. But what about the other two solutions we expect from a cubic?
Well, there are no more real solutions to the equation, but there are imaginary ones. In fact, 2i and -2i are solutions to this equation too, giving us our three solutions in total.
Listen to Science Focus Podcast episodes about maths:
- What’s the deal with algorithms? – Hannah Fry
- What happens when maths goes horribly, horribly wrong? – Matt Parker
It wasn’t until a few hundred years after Bombelli that the fundamental theorem of algebra was rigorously proven by Parisian bookshop manager Jean-Robert Argand in 1806. Argand was also a pioneer in relating imaginary numbers to geometry via the concept of complex numbers.
Complex numbers are numbers with a real part and an imaginary part. For instance, 4 + 2i is a complex number with a real part equal to 4 and an imaginary part equal to 2i. It turns out that both real numbers and imaginary numbers are also complex numbers. For example, 17 is a complex number with a real part equal to 17 and an imaginary part equalling zero, and iis a complex number with a real part of zero.
Another Frenchman, Abraham de Moivre, was amongst the first to relate complex numbers to geometry with his theorem of 1707 which related complex numbers and trigonometry together. Argand then developed Argand diagrams, which are like a normal graph with an x and y axis, except his axes are the real and the imaginary numbers. These breakthroughs allowed complex algebraic problems to be solved using geometry.
Like so many developments in mathematics, all of this was of purely academic interest until the modern electronic age. Complex numbers turn out to be incredibly useful in analysing anything that comes in waves, such as the electromagnetic radiation we use in radios and wifi, audio signals for music and voice communication and alternating current power supplies. Equally, quantum physics reduces all particles to waveforms, meaning that complex numbers are instrumental in understanding this strange world that has allowed us to enjoy modern computers, fibre-optics, GPS, MRI imaging, to name but a few. Thank goodness that mathematicians, from 500 years ago to the present day, decided that imaginary numbers were worth investigating after all.