By the early nineteenth century complex numbers had been in use for three hundred years, since Cardano and Bombelli, and Euler had worked wonders with them. Wessel, Argand and Gauss had drawn them as points of the plane. But the unease remained: what, after all, is $\sqrt{-1}$? A number whose square is negative seemed a useful fiction, not an object of mathematics.

Hamilton settled the question without drawing anything. In the essay Theory of Conjugate Functions, or Algebraic Couples, presented to the Royal Irish Academy in 1833 and 1835 and published in 1837, he proposed that a complex number is an ordered couple of real numbers, and nothing more.

The rules of the game

For couples $(a, b)$ and $(c, d)$, Hamilton defined

$$(a, b) + (c, d) = (a + c,\; b + d), \qquad (a, b)\cdot(c, d) = (ac - bd,\; ad + bc).$$

Couples of the form $(a, 0)$ behave exactly like real numbers: $(a, 0) + (c, 0) = (a + c, 0)$ and $(a, 0)\cdot(c, 0) = (ac, 0)$. So we may simply write $a$ instead of $(a, 0)$. And the couple $(0, 1)$ has a special property:

$$(0, 1)\cdot(0, 1) = (0 \cdot 0 - 1 \cdot 1,\; 0 \cdot 1 + 1 \cdot 0) = (-1, 0) = -1.$$

Calling $(0, 1)$ by the name $i$, the couple $(a, b)$ is $a + bi$, and $i^2 = -1$. The “root of minus one” stopped being a mystery: it is a couple of real numbers obeying a well-defined rule of multiplication. Everything done with complex numbers can be checked with the arithmetic of real numbers.

Why that rule?

The rule of multiplication looks arbitrary, but it has a precious property that Hamilton called the law of the moduli. The modulus of $(a, b)$ is $\sqrt{a^2 + b^2}$, the distance from the origin. And the modulus of the product is the product of the moduli:

$$(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2.$$

It is an old identity, known to Diophantus and to Brahmagupta, and it is what makes multiplication of complex numbers a geometric operation: multiplying by a complex number of modulus 1 rotates the plane. Multiplying by $i$ rotates it by $90°$.

The next step

If couples describe the plane and its rotations, might there not be triplets $(a, b, c)$, or $a + bi + cj$, that do the same in three-dimensional space? Hamilton wanted a multiplication of triplets obeying the law of the moduli: the size of the product should be the product of the sizes. That question occupied him for more than ten years, and the surprising answer lies in the quaternions.

Exercise: multiplying couples

Compute $(1, 2)\cdot(3, 4)$ and check the law of the moduli.

$(1 \cdot 3 - 2 \cdot 4,\; 1 \cdot 4 + 2 \cdot 3) = (-5, 10)$. The squared moduli are $1 + 4 = 5$ and $9 + 16 = 25$, and that of the product is $25 + 100 = 125 = 5 \cdot 25$. The law of the moduli holds.

Exercise: rotating with i

Show that multiplying a couple $(a, b)$ by $i = (0, 1)$ gives the couple rotated by $90°$ anticlockwise.

$(a, b)\cdot(0, 1) = (a \cdot 0 - b \cdot 1,\; a \cdot 1 + b \cdot 0) = (-b, a)$. The point $(a, b)$ goes to $(-b, a)$, which has the same length and is perpendicular to it, a quarter turn further on. For example, $(1, 0)$ goes to $(0, 1)$, and this goes to $(-1, 0)$.

Sources: William Rowan Hamilton, “Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time”, Transactions of the Royal Irish Academy (1837) · Thomas L. Hankins, Sir William Rowan Hamilton (1980) · Paul J. Nahin, An Imaginary Tale (1998).

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