## 8.1. Review of Complex Numbers

For a more detailed treatment of complex numbers, see ch. 3 of James Nearing's free book at http://www.physics.miami.edu/nearing/mathmethods/.

Figure A. Visualizing complex numbers as points in a plane.

Figure B. Addition of complex numbers is just like addition of vectors, although the real and imaginary axes don't actually represent directions in space.

Figure C. A complex number and its conjugate.

We assume there is a number, i, such that i2 = -1. The square roots of -1 are then i and -i. (In electrical engineering work, where i stands for current, j is sometimes used instead.) This gives rise to a number system, called the complex numbers, containing the real numbers as a subset. Any complex number z can be written in the form z = a + bi, where a and b are real, and a and b are then referred to as the real and imaginary parts of z. A number with a zero real part is called an imaginary number. The complex numbers can be visualized as a plane, Figure A, with the real number line placed horizontally like the x axis of the familiar x-y plane, and the imaginary numbers running along the y axis. The complex numbers are complete in a way that the real numbers aren't: every nonzero complex number has two square roots. For example, 1 is a real number, so it is also a member of the complex numbers, and its square roots are -1 and 1. Likewise, -1 has square roots i and -i, and the number i has square roots $$1/\sqrt{2}+i/\sqrt{2}$$ and $$-1/\sqrt{2}-i/\sqrt{2}$$.

Complex numbers can be added and subtracted by adding or subtracting their real and imaginary parts, Figure B. Geometrically, this is the same as vector addition.

The complex numbers a+bi and a-bi, lying at equal distances above and below the real axis, are called complex conjugates. The results of the quadratic formula are either both real, or complex conjugates of each other. The complex conjugate of a number z is notated as $$\bar{z}$$ or $$z^*$$.

The complex numbers obey all the same rules of arithmetic as the reals, except that they can't be ordered along a single line. That is, it's not possible to say whether one complex number is greater than another. We can compare them in terms of their magnitudes (their distances from the origin), but two distinct complex numbers may have the same magnitude, so, for example, we can't say whether 1 is greater than i or i is greater than 1.

#### Example 88

◊ Prove that $$1/\sqrt{2}+i/\sqrt{2}$$ is a square root of i.

◊ Our proof can use any ordinary rules of arithmetic, except for ordering.

\begin{align} (\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}})^2 &= \frac{1}{\sqrt{2}}\cdot\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}\cdot\frac{i}{\sqrt{2}} \\ &+ \frac{i}{\sqrt{2}}\cdot\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}\cdot\frac{i}{\sqrt{2}} \\ &= \frac{1}{2}(1+i+i-1) \\ &= i \end{align}

Example 88 showed one method of multiplying complex numbers. However, there is another nice interpretation of complex multiplication. We define the argument of a complex number, Figure D, as its angle in the complex plane, measured counterclockwise from the positive real axis. Multiplying two complex numbers then corresponds to multiplying their magnitudes, and adding their arguments, Figure E.

Figure D. A complex number can be described in terms of its magnitude and argument.

Figure E. The argument of uv is the sum of the arguments of u and v.
##### Self-Check:

Using this interpretation of multiplication, how could you find the square roots of a complex number?
The magnitude |z| of a complex number z obeys the identity $$|z|^2=z\bar{z}$$. To prove this, we first note that $$\bar{z}$$ has the same magnitude as z, since flipping it to the other side of the real axis doesn't change its distance from the origin. Multiplying z by $$\bar{z}$$ gives a result whose magnitude is found by multiplying their magnitudes, so the magnitude of $$z\bar{z}$$ must therefore equal |z|2. Now we just have to prove that $$z\bar{z}$$ is a positive real number. But if, for example, z lies counterclockwise from the real axis, then $$\bar{z}$$ lies clockwise from it. If z has a positive argument, then $$\bar{z}$$ has a negative one, or vice-versa. The sum of their arguments is therefore zero, so the result has an argument of zero, and is on the positive real axis [1].
[1] I cheated a little. If z's argument is 30 degrees, then we could say $$\bar{z}$$'s was -30, but we could also call it 330. That's OK, because 330+30 gives 360, and an argument of 360 is the same as an argument of zero.