Complex Numbers

A complex number is a number of the form $a + bi$ , where a and b are real numbers and $i
   = \sqrt{-1}$ (so $i^2 = -1$ ). For example, here are some complex numbers:

$$2 + 3i, \quad -77.5i, \quad 13\sqrt{7}, \quad \sqrt{-54}, \quad \dfrac{1 + i}{2}.$$

Notice that real numbers are special kinds of complex numbers --- namely, those that don't have an i-term.

Complex numbers are often called imaginary numbers, though there is nothing "imaginary" about them. It's unfortunate terminology, but it's very common.

For instance, in a complex number $a +
   bi$ , a is called the real part and b is called the imaginary part. Thus, in $4 + 7i$ , the real part is 4 and the imaginary part is 7.


Example. Express $\sqrt{-36}$ in the form $a + bi$ , where a and b are real numbers.

$$\sqrt{-36} = \sqrt{36}\sqrt{-1} = 6i.\quad\halmos$$


Example. Express $\sqrt{-125}$ in the form $a + bi$ , where a and b are real numbers.

$$\sqrt{-125} = \sqrt{25}\sqrt{5}\sqrt{-1} = 5i\sqrt{5}.\quad\halmos$$


Example. Express $\sqrt{-72}$ in the form $a + bi$ , where a and b are real numbers.

$$\sqrt{-72} = \sqrt{36}\sqrt{2}\sqrt{-1} = 6i\sqrt{2}.\quad\halmos$$


Example. (a) Express $i^{37}$ in the form $a + bi$ , where a and b are real numbers.

Since $i^2 = -1$ , I'm going to find the largest even number less than 37 and use that as the basis for breaking up the power.

$$i^{37} = i^{36}\cdot i = (i^2)^{18}\cdot i = (-1)^{18}\cdot i = 1\cdot i = i.\quad\halmos$$

(b) Express $(-2i)^{7}$ in the form $a + bi$ , where a and b are real numbers.

$$(-2i)^7 = (-2)^7\cdot i^7 = -128\cdot i^6\cdot i = -128\cdot (i^2)^3\cdot i = -128\cdot (-1)^3\cdot i = -128\cdot (-1)\cdot i = 128i.\quad\halmos$$

(c) Express $i^{34}$ in the form $a + bi$ , where a and b are real numbers.

$$i^{34} = (i^2)^{17} = (-1)^{17} = -1.\quad\halmos$$


Example. You can add or subtract complex numbers by adding or subtracting the real parts and the imaginary parts:

$$(2 + 17i) + (5 + 6i) = (2 + 5) + (17i + 6i) = 7 + 23i.$$

$$(3 - 7i) + 6(4 + 3i) = (3 - 7i) + (24 + 18i) = (3 + 24) + (-7i + 18i) = 27 + 11i.\quad\halmos$$


Example. Complex numbers can be represented by points in the plane. Use the real part for the x-direction and the imaginary part for the y-direction. Here are some examples:

$$\hbox{\epsfysize=2in \epsffile{complex1.eps}}\quad\halmos$$


Example. Multiply complex numbers by using the distributive law:

$$3i(2 + 4i) = 6i + 12i^2 = 6i + 12(-1) = -12 + 6i.$$

$$(2 - i)^2 = 4 - 4i + i^2 = 4 - 4i - 1 = 3 - 4i.$$

$$(5 + 6i)(5 - 6i) = 25 - 36i^2 = 25 - 36(-1) = 61.$$

$$(3 + 2i)(-1 + 6i) = (3)(-1) + (3)(6i) + (2i)(-1) + (2i)(6i) = -3 + 18i - 2i + 12i^2 = -3 + 18i - 2i - 12 = -15 + 16i.$$

Standard rules for exponents apply. For example,

$$i^{16} = (i^2)^8 = (-1)^8 = 1,$$

$$i^{55} = i^{54}i = (i^2)^{27}i = (-1)^{27}i = -i.$$

I used the fact that -1 raised to an even power is 1 and -1 raised to an odd power is -1.


To divide one complex number by another, or to compute reciprocals of complex numbers, use the technique of multiplying the top and bottom by the conjugate.

What's the conjugate of a complex number? The conjugate of $2 + 3i$ is $2 - 3i$ ; the conjugate of $-6 - 7i$ is $-6 + 7i$ . In others words, find the conjugate by flipping the sign of the imaginary part.


Example. Express $\dfrac{1}{3 + 4i}$ in the form $a + bi$ .

Multiply the top and bottom by $3 -
   4i$ :

$$\dfrac{1}{3 + 4i} = \dfrac{1}{3 + 4i}\cdot \dfrac{3 - 4i}{3 - 4i} = \dfrac{3 - 4i}{25}.\quad\halmos$$


Example. Express $\dfrac{2 - i}{3 - 2i}$ in the form $a + bi$ .

Multiply the top and bottom by $3 +
   2i$ :

$$\dfrac{2 - i}{3 - 2i} = \dfrac{2 - i}{3 - 2i}\cdot \dfrac{3 + 2i}{3 + 2i} = \dfrac{(2 - i)(3 + 2i)}{13} = \dfrac{8 + i}{13}.\quad\halmos$$


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