One of the more poorly-named concepts in mathematics, “complex” or “imaginary” numbers are neither overly complicated nor any less valid than the “real” numbers. The name “imaginary” comes from René Descartes in the 17th century as a derogatory term, as he regarded such numbers as fictitious or useless. Rafael Bombelli was the first set down the rules for multiplication of complex numbers in 1572. The concept gained wide acceptance among mathematicians following the work of Leonhard Euler in the 18th century and Augustin-Louis Cauchy and Carl Friedrich Gauss in the early 19th century. Today, complex numbers are a powerful tool with many uses across mathematics, engineering, and physics. Yet many people have no idea what they are.
To begin understanding the complex numbers, let us first review the real numbers. “Real” numbers include:
the counting numbers, like one, two, and three;
rational numbers, like ½ and ¾;
irrational numbers, like √2 (the square root of two) and 𝜑 (phi, the golden ratio);
transcendental numbers, like 𝜋 (pi, the circle constant) and 𝑒 (Euler’s number, the exponential constant);
negative numbers, which are duplicates of the positive numbers but in the “opposite direction”, so to speak;
and zero, which is neither positive nor negative.
All these numbers can be drawn on the real number line:
The existence of zero, which represents the concept of “nothing”; of negative numbers, which do not correspond to a physical quantity of something you could hold in your hand; of irrational numbers, which cannot be represented as a ratio of integers; and of transcendental numbers, which cannot be the root of any polynomial with rational coefficients; all gave mathematicians much consternation before they eventually gained widespread acceptance. All proved to be useful concepts, despite their novelty. Such is the case for the complex numbers, as well.
The imaginary unit 𝑖 is a solution to the equation 𝑥² + 1 = 0. Solving for 𝑥 gives 𝑥² = −1, or 𝑥 = √(−1). No real number satisfies this. The square operation multiplies a number by itself, and no real number, when multiplied by itself, gives a negative result. A positive real number multiplied by a positive real number gives a positive result, and a negative real number multiplied by a negative real number also gives a positive result. So 𝑖 cannot sit on the real number line. Where are we to put it then?
The answer is to create a new number line perpendicular to the real number line, creating a two-dimensional space called the complex plane:
A complex number is a number that can have a real part and an imaginary part, written as 𝑧 = 𝑎 + 𝑏𝑖, where 𝑎 and 𝑏 are real numbers and 𝑖 is the imaginary unit. This works like rectangular coordinates in the complex plane, where 𝑎 is the number’s horizontal (real) position and 𝑏 is the number’s vertical (imaginary) position. Both 𝑎 and 𝑏 can be zero, making a purely real or a purely imaginary number.
Every complex number 𝑧 has two additional quantities associated with it: the distance from zero, known as the absolute value or magnitude, represented as |𝑧| or abs(𝑧) and equal to √(𝑎² + 𝑏²); and the angle measured from the real axis, known as the argument or phase, represented as arg(𝑧) and measured over the interval –𝜋 < 𝜃 ≤ 𝜋 in radians or –180 < 𝜃 ≤ 180° in degrees, where positive values are counterclockwise rotations. This is typically shown by drawing a vector between zero and the complex number.
We can think of multiplying by 𝑖 as a 90° counterclockwise rotation about zero: multiplying any real number by 𝑖 rotates it 90° from the real number line onto the imaginary number line. Multiplying by −1 can be thought of a 180° rotation, rotating a number to the opposite side of zero. Multiplying by 𝑖 twice, or 𝑖², is two 90° rotations, equivalent to multiplying by −1 for a 180° rotation, supporting the definition that 𝑖² = −1. Integer powers of 𝑖 cycle through four values: 1, 𝑖, −1, and −𝑖. Dividing by 𝑖 goes in the opposite direction, equivalent to multiplying by −𝑖.
We can perform all the normal algebraic operations on complex numbers exactly as we can on real numbers if we treat 𝑖 as a variable and remember that 𝑖² = −1.
We add two complex numbers 𝑎 + 𝑏𝑖 and 𝑐 + 𝑑𝑖 by separately adding their real components and their imaginary components to get (𝑎 + 𝑐) + (𝑏 + 𝑑)𝑖. For example, 4 + 2𝑖 plus 6 + 5𝑖 equals 10 + 7𝑖. We can subtract two complex numbers in the same way, but we have to remember to distribute the negative sign across both terms to get (𝑎 − 𝑐) + (𝑏 − 𝑑)𝑖. For example, 4 + 2𝑖 minus 6 + 5𝑖 equals −2 − 3𝑖. We can also think of subtraction as adding the negation of the subtracted number. For example, 4 + 2𝑖 minus 6 + 5𝑖 becomes 4 + 2𝑖 plus −6 − 5𝑖. Visualized in the complex plane, addition (and, using negation to convert it to addition, subtraction) has the effect of stacking the vectors of each complex number end-to-end.
We can multiply two complex numbers 𝑎 + 𝑏𝑖 and 𝑐 + 𝑑𝑖 with standard binomial multiplication techniques like FOIL (first, outer, inner, last) to get 𝑎𝑐 + 𝑎𝑑𝑖 + 𝑏𝑐𝑖 + 𝑏𝑑𝑖². Remembering that 𝑖² = −1, we can rearrange that to get (𝑎𝑐 − 𝑏𝑑) + (𝑎𝑑 + 𝑏𝑐)𝑖. For example, 3 + 1𝑖 times 2 + 2𝑖 equals 4 + 8𝑖. Visualized in the complex plane, multiplication has the effect of adding the numbers’ arguments (rotations) and multiplying their magnitudes (absolute values).
Division is a little trickier. To divide by a complex number, we first multiply the dividend 𝑎 + 𝑏𝑖 and the divisor 𝑐 + 𝑑𝑖 by the complex conjugate of the divisor. The conjugate of 𝑐 + 𝑑𝑖 is 𝑐 – 𝑑𝑖, with the same real part but with the imaginary part multiplied by negative one. Multiplying the divisor by its conjugate cancels out the imaginary term, making it a purely real number. We then divide the new complex dividend by the new real divisor, giving us ((𝑎𝑐 + 𝑏𝑑) + (𝑏𝑐 – 𝑎𝑑)𝑖) / (𝑐² + 𝑑²). For example, to calculate 3 + 1𝑖 divided by 2 + 2𝑖, we multiply both numbers by 2 – 2𝑖, giving us 8 – 4𝑖 divided by 8, which equals 1 – ½𝑖. Visualized in the complex plane, division has the effect of subtracting the numbers’ arguments and dividing their magnitudes.
What about the square root of 𝑖? Do we need to invent even more new concepts for this just like we invented 𝑖 to be the square root of negative one?
Nope! We already have everything we need. The square root operation asks what number, multiplied by itself, gives the input number. Because multiplying two complex numbers has the effect of adding their arguments (rotations), and because 𝑖 can be thought of as a 90° rotation, all we need to do to find the square root of 𝑖 is find an angle that, added to itself, gives 90°. Thus, the square root of 𝑖 is a 45° rotation, which (with a little trigonometry) is (√2 / 2) + (√2 / 2)𝑖. The negative square root −(√2 / 2) − (√2 / 2)𝑖 can be thought of as a −135° rotation which, when applied twice, puts us at −270°, equivalent to 90°. The same thought process works for higher order roots, and for complex numbers other than 𝑖. In general, there are 𝑛 solutions for the 𝑛th root of a complex number. The magnitudes of a complex number 𝑧’s 𝑛th roots will all be the 𝑛th root of the magnitude of 𝑧, while the arguments will be (2𝜋𝑘 + arg(𝑧)) / 𝑛 for integer values of 𝑘 in the interval 0 ≤ 𝑘 < 𝑛, making the roots equally spaced in a circle around the origin.
So far, we have been discussing complex numbers in the rectangular form 𝑧 = 𝑎 + 𝑏𝑖, listing their horizontal and vertical positions in the complex plane. Sometimes it is easier to use a polar form, listing a number’s magnitude (distance from zero) and argument (angle from the positive real axis). This form is written as 𝑧 = 𝑟 𝑒ᶿⁱ, where 𝑟 is the number’s magnitude and 𝜃 is the number’s argument. The polar form uses the infinite series exponential function 𝑒ˣ = exp(𝑥) = 1 + 𝑥 + 𝑥² / 2! + 𝑥³ / 3! + … + 𝑥ⁿ / 𝑛! where 𝑛 goes to infinity (the exclamation marks denote the factorial operation). The formula 𝑟𝑒ᶿⁱ = 𝑟 (cos(𝜃) + sin(𝜃) 𝑖) gives the relationship between the polar and rectangular forms. While the rectangular form 𝑧 = 𝑎 + 𝑏𝑖 is unique for every complex number, the polar form 𝑧 = 𝑟 𝑒ᶿⁱ gives infinitely many ways of writing any given complex number by adding or subtracting integer multiples of 2𝜋 radians from the argument. The value with an argument in the interval –𝜋 < 𝜃 ≤ 𝜋 in radians or –180 < 𝜃 ≤ 180° in degrees is called the principal value.
This polar form 𝑒𝜋𝑖 = −1 + 0𝑖 gives rise to Euler’s Identity 𝑒𝜋𝑖 + 1 = 0, a single equality that unites five fundamental mathematical constants: the additive identity 0, the multiplicative identity 1, the imaginary constant 𝑖, the circle constant 𝜋, and the exponential constant 𝑒.
Multiplication and division of complex numbers are simpler in polar form than in rectangular form, following standard rules for working with exponents. To multiply two numbers 𝑎 𝑒ᵝⁱ and 𝑏 𝑒ᵞⁱ, we multiply the coefficients (magnitudes) and add the exponents (arguments) to get 𝑎 𝑏 𝑒⁽ᵝ⁺ᵞ⁾ⁱ. To divide them, we divide the coefficients and subtract the exponents to get (𝑎 / 𝑏) 𝑒⁽ᵝ⁻ᵞ⁾ⁱ.
Because multiplication and division of complex numbers are analogous to rotation and scaling, complex numbers are closely related to trigonometry. As a result, complex numbers can be used to derive trigonometric identities. For example, the angle-addition identities: since multiplying two complex numbers gives us the sum of their arguments, we have 𝑒⁽ᵝ⁺ᵞ⁾ⁱ = 𝑒ᵝⁱ × 𝑒ᵞⁱ, which can be written as cos(𝛽 + 𝛾) + sin(𝛽 + 𝛾) 𝑖 = (cos(𝛽) + sin(𝛽) 𝑖) × (cos(𝛾) + sin(𝛾) 𝑖). The right side expands into (cos(𝛽) cos(𝛾) − sin(𝛽) sin(𝛾)) + (cos(𝛽) sin(𝛾) + sin(𝛽) cos(𝛾)) 𝑖, giving us the horizontal component cos(𝛽 + 𝛾) = cos(𝛽) cos(𝛾) − sin(𝛽) sin(𝛾) and the vertical component sin(𝛽 + 𝛾) = cos(𝛽) sin(𝛾) + sin(𝛽) cos(𝛾). A similar process can be used to derive other identities, such as the double-angle identities (by squaring) and the angle-subtraction identities (by dividing).
Just as the complex numbers extend the real numbers into two dimensions, we can keep going into even higher dimensions — specifically, powers of two. Quaternions, first described in 1843 by Irish mathematician William Rowan Hamilton, extend into four dimensions with two additional “imaginary” units 𝑗 and 𝑘 in the form 𝑎 + 𝑏 𝑖 + 𝑐 𝑗 + 𝑑 𝑘. Just as complex numbers describe rotation and scaling in two dimensions, quaternions describe rotation and scaling in three dimensions, making them very useful in computer graphics and 3D modelling/animation. Multiplication of quaternions is not commutative, meaning that the order of operations matters: A × B is not the same as B × A. This reflects the fact that when rotating an object in three dimensions, the order in which you perform the rotations affects the final orientation: a 45° rotation about the x-axis and then about the y-axis leaves you pointing in a different direction than a 45° rotation about the y-axis and then about the x-axis. For more on quaternions, see this interactive lesson by Grant Sanderson (3blue1brown) and Ben Eater.
We can keep constructing number systems with higher powers of two, like the 8-dimensional octonions and the 16-dimensional sedenions, but these systems are rarely used — and much more deserving of the name “complex” than the simple 2-dimensional complex numbers.
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