1.4: Complex Numbers

The square root of −1, called \(i\), is important in physics and mathematics for many reasons. Measurable physical quantities can always be described by real numbers. You never get a reading of \(i\) meters on your meter stick. However, we will see that when \(i\) is included along with real numbers and the usual arithmetic operations (addition, subtraction, multiplication and division), then algebra, trigonometry and calculus all become simpler. While complex numbers are not necessary to describe wave phenomena, they will allow us to discuss them in a simpler and more insightful way.

Some Definitions

An imaginary number is a number of the form \(i\) times a real number.

A complex number, \(z\), is a sum of a real number and an imaginary number: \(z = a + ib.\)

The real and “imaginary” parts, \(Re (z)\) and \(Im (z)\), of the complex number \(z = a+ib:\)

\[Re (z) = a , Im (z) = b.\]

Note that the imaginary part is actually a real number, the real coefficient of \(i\) in \(z = a + ib.\)

The complex conjugate, \(z^*\), of the complex number \(z\), is obtained by changing the sign of \(i\):

\[z^* = a − ib.\]

Note that \(Re (z) = (z + z^*)/2\) and \(Im (z) = (z − z^*)/2i.\)

The complex plane: Because a complex number \(z\) is specified by two real numbers, it can be thought of as a two-dimensional vector, with components \((a, b)\). The real part of \(z\), \(a = Re (z)\), is the \(x\) component and the imaginary part of \(z\), \(b = Im (z)\), is the \(y\) component. The diagrams in figures 1.5 and 1.6 show two vectors in the complex plane along with the corresponding complex numbers:

The absolute value, \(|z|\), of \(z\), is the length of the vector \((a, b)\):

\[|z| = √{a^2 + b^2} = √{z^* z}.\]

The absolute value \(|z|\) is always a real, non-negative number.

Figure 1.5: A vector with positive real part in the complex plane.

The argument or phase, arg\((z)\), of a nonzero complex number \(z\), is the angle, in radians, of the vector \((a, b)\) counterclockwise from the \(x\) axis:

\[arg(z) = { arctan(b/a) for a ≥ 0,\]

\[ { arctan(b/a) + π for a < 0.\]

Like any angle, \(arg(z)\) can be redefined by adding a multiple of \(2π\) radians or 360^◦ (see figure 1.5 and 1.6).

Figure 1.6: A vector with negative real part in the complex plane.

Arithmetic

The arithmetic operations addition, subtraction and multiplication on complex numbers are defined by just treating the \(i\) like a variable in algebra, using the distributive law and the relation \(i^2 = −1.\) Thus if \(z = a + ib\) and \(z' = a' + ib'\), then

\[z + z' = (a + a') + i(b + b'),\]

\[z − z' = (a − a') + i(b − b'),\]

\[zz' = (aa' − bb' ) + i(ab' + ba').\]

For example:

\[(3 + 4i) + (−2 + 7i) = (3 − 2) + (4 + 7)i = 1 + 11i ,\]

\[(3 + 4i) · (5 + 7i) = (3 · 5 − 4 · 7) + (3 · 7 + 4 · 5)i = −13 + 41i.\]

It is worth playing with complex multiplication and getting to know the complex plane. At this point, you should check out program 1-2.

Division is more complicated. To divide a complex number \(z\) by a real number \(r\) is easy, just divide both the real and the imaginary parts by \(r\) to get \(z/r = a/r + ib/r.\) To divide by a complex number, \(z'\), we can use the fact that \(z'^* z' = |z'|^2\) is real. If we multiply the numerator and the denominator of \(z/z' by z'^*\), we can write:

\[z/z' = z'^*z/|z'|^2 = (aa' + bb' )/(a'^2 + b'^2) + i(ba' − ab' )/(a'^2 + b'^2).\]

For example:

\[(3 + 4i)/(2 + i) = (3 + 4i) · (2 − i)/5 = (10 + 5i)/5 = 2 + i .\]

With these definitions for the arithmetic operations, the absolute value behaves in a very simple way under multiplication and division. Under multiplication, the absolute value of a product of two complex numbers is the product of the absolute values:

\[|z z' | = |z| |z' | .\]

Division works the same way so long as you don’t divide by zero:

\[|z/z' | = |z|/|z' | if z'=0 .\]

Mathematicians call a set of objects on which addition and multiplication are defined and for which there is an absolute value satisfying (1.51) and (1.52) a division algebra. It is a peculiar (although irrelevant, for us) mathematical fact that the complex numbers are one of only four division algebras, the others being the real numbers and more bizarre things called quaternions and octonians obtained by relaxing the requirements of commutativity and associativity (respectively) of the multiplication laws.

The wonderful thing about the complex numbers from the point of view of algebra is that all polynomial equations have solutions. For example, the equation \(x^2 − 2x + 5 = 0\) has no solutions in the real numbers, but has two complex solutions, \(x = 1 ± 2i.\) In general, an equation of the form \(p(x) = 0\), where \(p(x)\) is a polynomial of degree \(n\) with complex (or real) coefficients has \(n\) solutions if complex numbers are allowed, but it may not have any if \(x\) is restricted to be real.

Note that the complex conjugate of any sum, product, etc, of complex numbers can be obtained simply by changing the sign of \(i\) wherever it appears. This implies that if the polynomial \(p(z)\) has real coefficients, the solutions of \(p(z) = 0\) come in complex conjugate pairs. That is, if \(p(z) = 0\), then \(p(z^*) = 0\) as well.

Complex Exponentials

Consider a complex number \(z = a + ib\) with absolute value 1. Because \(|z| = 1\) implies \(a^2 + b^2 = 1\), we can write \(a\) and \(b\) as the cosine and sine of an angle \(θ\).

\[z = cos θ + isin θ for |z| = 1 .\]

Because

\[tan θ = \frac{sin θ}{cos θ} = \frac{b}{a}\]

the angle \(θ\) is the argument of \(z\):

\[arg(cos θ + isin θ) = θ .\]

Let us think about \(z\) as a function of \(θ\) and consider the calculus. The derivative with respect to \(θ\) is:

\[\frac{∂}{∂θ} (cos θ + isin θ) = − sin θ + i cos θ = i(cos θ + isin θ)\]

A function that goes into itself up to a constant under differentiation is an exponential. In particular, if we had a function of \(θ\), \(f(θ)\), that satisfied \(\frac{∂}{∂θ} f(θ) = kf(θ)\) for real \(k\), we would conclude that \(f(θ) = e^{kθ}.\) Thus if we want the calculus to work in the same way for complex numbers as for real numbers, we must conclude that

\[e^{iθ} = cos θ + isin θ.\]

We can check this relation by noting that the Taylor series expansions of the two sides are equal. The Taylor expansion of the exponential, cos, and sin functions are:

\[e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!} + …\]

\[cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{4!} + …\]

\[sin(x) = x - \frac{x^3}{3!} + …\]

Thus the Taylor expansion of the left side of (1.57) is

\[1 + iθ + (iθ)^2/2 + (iθ)^3/3! + ...\]

while the Taylor expansion of the right side is

\[(1 − θ^2/2 + ...) + i(θ − θ^3/6 +...)\]

The powers of \(i\) in (1.59) work in just the right way to reproduce the pattern of minus signs in (1.60).

Furthermore, the multiplication law works properly:

\[e^{iθ}e^{iθ'} = (cos θ + isin θ)(cos θ' + isin θ' )\]

\[= (cos θ cos θ' − sin θ sin θ' ) + i(sin θ cos θ' + cos θ sin θ' )\]

\[= cos(θ + θ' ) + isin(θ + θ' ) = e^{i(θ + θ')} .\]

Thus (1.57) makes sense in all respects. This connection between complex exponentials and trigonometric functions is called Euler’s Identity. It is extremely useful. For one thing, the logic can be reversed and the trigonometric functions can be “defined” algebraically in terms of complex exponentials:

\[cosθ = \frac{e^{iθ} + e^{-iθ}}{2}\]

\[sinθ = \frac{e^{iθ} - e^{-iθ}}{2i} = -i \frac{e^{iθ} - e^{-iθ}}{2}\]

Using (1.62), trigonometric identities can be derived very simply. For example:

\[cos 3θ = Re (e^{3iθ}) = Re ((e^{iθ})^3) = cos^3 θ − 3 cos θ sin^2 θ .\]

Another example that will be useful to us later is:

\[cos(θ + θ') + cos(θ - θ') = (e^{i(θ+θ')} + e^{−i(θ+θ')} + e^{i(θ-θ')} + e^{−i(θ-θ')}) / 2\]

\[= (e^{iθ} + e^{-iθ})(e^{iθ'} + e^{-iθ'}) / 2 = 2 cosθ cosθ'.\]

Every nonzero complex number can be written as the product of a positive real number (its absolute value) and a complex number with absolute value 1. Thus

\[z = x + iy = R e^{iθ} where R = |z| , and θ = arg(z).\]

In the complex plane, (1.65) expresses the fact that a two-dimensional vector can be written √ either in Cartesian coordinates, \((x, y)\), or in polar coordinates, \((R, θ)\). For example, \(√3+i = 2e^{iπ/6}; 1 + i = √2e^{iπ/4}; −8i = 8e^{3iπ/2} = 8e^{-iπ/2}\) Figure 1.7 shows the complex number \(1 + i = √2e^{iπ/4}.\)

The relation, (1.65), gives another useful way of thinking about multiplication of complex numbers. If

\[z_1 = R_1e^{iθ_1} and z_2 = R_2e^{iθ_2} ,\]

then

\[z_1z_2 = R_1R_2e^{i(θ_1+θ_2)} .\]

In words, to multiply two complex numbers, you multiply the absolute values and add the arguments. You should now go back and play with program 1-2 with this relation in mind.

Equation (1.57) yields a number of relations that may seem surprising until you get used to them. For example: \(e^{iπ} = −1; e^{iπ/2} = i; e^{2iπ} = 1.\) These have an interpretation in the complex plane where \(e^{iθ}\) is the unit vector \((cos θ,sin θ),\)