Complex Numbers And Their Operations

A complex number $z = x + iy$ can be represented geometrically in the complex plane, called the Argand diagram. In the Argand diagram, $z = x + iy$ is represented by the point P($x$, $y$), where Re($z$) is on the x-axis (real axis) and Im($z$) is on the y-axis (imaginary axis).

The Argand diagram

If the discriminant in the general solution of the quadratic equation $ax^2 + bx + c = 0$ is negative, then the solution lies in the set of complex numbers.

Example 1

Find the solution of the equation $x^2 + 2x + 2 = 0$.

Solution:

Using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, with $a = 1$, $b = 2$, and $c = 2$:

$x = \frac{-2 \pm \sqrt{2^2 - 4(1)(2)}}{2(1)} = \frac{-2 \pm \sqrt{4 - 8}}{2} = \frac{-2 \pm \sqrt{-4}}{2}$

Since $\sqrt{-4} = \sqrt{4 \times -1} = \sqrt{4} \times \sqrt{-1} = 2i$, where $i = \sqrt{-1}$:

$x = \frac{-2 \pm 2i}{2} = -1 \pm i$

Therefore, the solutions are $x = -1 + i$ and $x = -1 - i$.

Example 2

Solve for $x$ in the equation $x^2 - 4x + 7 = 0$.

Solution:

Using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, with $a = 1$, $b = -4$, and $c = 7$:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(7)}}{2(1)} = \frac{4 \pm \sqrt{16 - 28}}{2} = \frac{4 \pm \sqrt{-12}}{2}$

Since $\sqrt{-12} = \sqrt{12 \times -1} = \sqrt{4 \times 3} \times \sqrt{-1} = 2\sqrt{3}i$:

$x = \frac{4 \pm 2\sqrt{3}i}{2} = 2 \pm i\sqrt{3}$

Therefore, the solutions are $x = 2 + i\sqrt{3}$ and $x = 2 - i\sqrt{3}$.

The powers of the imaginary unit $i$ are essential for multiplying and simplifying complex numbers.

Recall that $i = \sqrt{-1}$, so $i^2 = -1$.

If $i$ is raised to a non-negative integer $n$, the values cycle through a pattern:

(a) $i^3 = i^2 \times i = (-1)i = -i$

(b) $i^4 = i^2 \times i^2 = (-1)(-1) = 1$

(c) $i^5 = i^4 \times i = (1)i = i$

(d) $i^6 = i^4 \times i^2 = (1)(-1) = -1$

(e) $i^7 = i^6 \times i = (-1)i = -i$

The pattern repeats every four powers: $i, -1, -i, 1$.

Example 3

Simplify each of the following complex numbers:

(a) $i^{1776}$ (b) $i^{13} + i^{19}$ (c) $i^5 + i^3 + i + 1$ (d) $\sqrt{-48} - \sqrt{-27}$

Solution:

(a) $i^{1776}$. Since $1776$ is divisible by 4 ($1776 = 4 \times 444$), $i^{1776} = (i^4)^{444} = 1^{444} = 1$.

(b) $i^{13} + i^{19}$. $i^{13} = i^{12} \times i = (i^4)^3 \times i = 1^3 \times i = i$. $i^{19} = i^{16} \times i^3 = (i^4)^4 \times i^3 = 1^4 \times -i = -i$. Therefore, $i^{13} + i^{19} = i + (-i) = 0$.

(c) $i^5 + i^3 + i + 1$. $i^5 = i$, $i^3 = -i$. Therefore, $i^5 + i^3 + i + 1 = i + (-i) + i + 1 = i + 1$.

(d) $\sqrt{-48} - \sqrt{-27}$. $\sqrt{-48} = \sqrt{16 \times 3 \times -1} = 4i\sqrt{3}$. $\sqrt{-27} = \sqrt{9 \times 3 \times -1} = 3i\sqrt{3}$. Therefore, $\sqrt{-48} - \sqrt{-27} = 4i\sqrt{3} - 3i\sqrt{3} = i\sqrt{3}$.

Example 4

Find the complex number $z$ in the equation $4 + 5i = z - (1 - i)$.

Solution:

$z = (4 + 5i) + (1 - i) = 4 + 1 + 5i - i = 5 + 4i$

Example 5

Plot the complex numbers $z = 1 + 2i$ and $w = 1 - 2i$ on the same Argand diagram. (Argand diagram not included, but would show points (1,2) and (1,-2))

Solution:

The Argand diagram for the given complex numbers is as shown in the following figure.

Example 6

Find the imaginary and real parts in each of the following complex numbers:

(a) $(3 + i)(3 - i)$ (b) $(2 + i)^2$ (c) $(-2 - 3i)^3$ (d) $\frac{2 - i}{2 + 3i}$

Solution:

(a) $(3 + i)(3 - i) = 9 - i^2 = 9 - (-1) = 10$. Re($z$) = 10, Im($z$) = 0.

(b) $(2 + i)^2 = 4 + 4i + i^2 = 4 + 4i - 1 = 3 + 4i$. Re($w$) = 3, Im($w$) = 4.

(c) $(-2 - 3i)^3 = (-1)^3(2 + 3i)^3 = -(8 + 36i + 54i^2 + 27i^3) = -(8 + 36i - 54 - 27i) = -(-46 + 9i) = 46 - 9i$. Re($z$) = 46, Im($z$) = -9.

(d) $\frac{2 - i}{2 + 3i} = \frac{(2 - i)(2 - 3i)}{(2 + 3i)(2 - 3i)} = \frac{4 - 6i - 2i + 3i^2}{4 - 9i^2} = \frac{4 - 8i - 3}{4 + 9} = \frac{1 - 8i}{13} = \frac{1}{13} - \frac{8}{13}i$. Re($z$) = $\frac{1}{13}$, Im($z$) = $-\frac{8}{13}$.

If $z = x + iy$ is a complex number, the modulus of $z$, denoted by $|z|$ or Mod($z$), is the distance of the point representing $z$ on the Argand plane from the origin. It is calculated as:

$|z| = \sqrt{x^2 + y^2}$

Geometrical representation of a complex number

Basic properties of the modulus:

1. $|z| = 0$ if and only if $z = 0$ (i.e., Re($z$) = 0 and Im($z$) = 0)
2. -$|z|$ ≤ Re($z$) ≤ $|z|$ and -$|z|$ ≤ Im($z$) ≤ $|z|$
3. $|z_1z_2| = |z_1||z_2|$
4. $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$, $z_2 \ne 0$
5. $|z_1 + z_2| \le |z_1| + |z_2|$ (Triangle Inequality)
6. $|z_1 - z_2| \ge ||z_1| - |z_2||$
7. $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2)$

Example 7

Find the modulus of each complex number:

(a) $z = -3 + i$ (b) $w = 2 + 7i$ (c) $u = 5 + 2i$

Solution:

(a) $|z| = \sqrt{(-3)^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}$

(b) $|w| = \sqrt{2^2 + 7^2} = \sqrt{4 + 49} = \sqrt{53}$

(c) $|u| = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29}$

Example 8

Find the modulus of $z$ if:

(a) $z = -2\sqrt{6} - 3i$ (b) $z = (a + b) + i(a - b)$

Solution:

(a) $|z| = \sqrt{(-2\sqrt{6})^2 + (-3)^2} = \sqrt{24 + 9} = \sqrt{33}$

(b) $|z| = \sqrt{(a + b)^2 + (a - b)^2} = \sqrt{a^2 + 2ab + b^2 + a^2 - 2ab + b^2} = \sqrt{2a^2 + 2b^2} = \sqrt{2(a^2 + b^2)}$

Example 9

If $z_1 = 2 + 3i$ and $z_2 = -3 + 4i$, verify that $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2)$.

Solution:

Left-hand side:

$z_1 + z_2 = (2 + 3i) + (-3 + 4i) = -1 + 7i$. $|z_1 + z_2|^2 = (-1)^2 + 7^2 = 1 + 49 = 50$

$z_1 - z_2 = (2 + 3i) - (-3 + 4i) = 5 - i$. $|z_1 - z_2|^2 = 5^2 + (-1)^2 = 25 + 1 = 26$

$|z_1 + z_2|^2 + |z_1 - z_2|^2 = 50 + 26 = 76$

Right-hand side:

$|z_1|^2 = 2^2 + 3^2 = 4 + 9 = 13$

$|z_2|^2 = (-3)^2 + 4^2 = 9 + 16 = 25$

$2(|z_1|^2 + |z_2|^2) = 2(13 + 25) = 2(38) = 76$

Since both sides equal 76, the property is verified.

For a complex number $z = x + iy$, the complex conjugate, denoted by $\bar{z}$, is $x - iy$. The conjugate is obtained by changing the sign of the imaginary part.

Basic properties of complex conjugates:

1. $\bar{z} = z$ if and only if $z$ is a real number (i.e., $z \in \mathbb{R}$).
2. If $z$ is a complex number, then $z\bar{z}$ is a non-negative real number.
3. The conjugate of a sum is the sum of the conjugates: $\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$.
4. The conjugate of a product is the product of the conjugates: $\overline{z_1z_2} = \bar{z_1}\bar{z_2}$.
5. The conjugate of a quotient is the quotient of the conjugates: $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}$, for $z_2 \ne 0$.
6. Re($z$) = $\frac{z + \bar{z}}{2}$ and Im($z$) = $\frac{z - \bar{z}}{2i}$.
7. $z\bar{z} = |z|^2$.
8. The multiplicative inverse (reciprocal) of a complex number $z = a + ib$ ($b \ne 0$) is $\frac{1}{z} = \frac{\bar{z}}{|z|^2} = \frac{a - ib}{a^2 + b^2}$.

Example 10

Find the complex conjugate of each complex number:

(a) $3 + 6i$ (b) $\frac{1}{2} - 5i$ (c) $a + ib$

Solution:

(a) If $z = 3 + 6i$, then $\bar{z} = 3 - 6i$.

(b) If $w = \frac{1}{2} - 5i$, then $\bar{w} = \frac{1}{2} + 5i$.

(c) If $z = a + ib$, then $\bar{z} = a - ib$.

Example 11

Show that $\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$.

Solution:

Let $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$. Then:

$z_1 + z_2 = (x_1 + iy_1) + (x_2 + iy_2) = (x_1 + x_2) + i(y_1 + y_2)$

$\overline{z_1 + z_2} = (x_1 + x_2) - i(y_1 + y_2)$

Also:

$\bar{z_1} = x_1 - iy_1$

$\bar{z_2} = x_2 - iy_2$

$\bar{z_1} + \bar{z_2} = (x_1 - iy_1) + (x_2 - iy_2) = (x_1 + x_2) - i(y_1 + y_2)$

Therefore, $\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$.

If P(x, y) is a point in an Argand diagram representing a complex number $z = x + iy$, the modulus $r$ is the same as $|z|$. The angle $\theta$ is called the argument of $z$ and is denoted by arg($z$). The principal argument lies in the range $(-\pi, \pi]$ or $(-180^\circ, 180^\circ]$. The argument is undefined when $z = 0$.

Modulus and argument in Argand diagram

From Figure $\tan(\theta) = \frac{y}{x}$, so $\theta = \arctan\left(\frac{y}{x}\right)$. However, care must be taken to choose the correct quadrant for $\theta$ based on the signs of $x$ and $y$. The argument of a complex number $z = x + iy$ is given by:

arg($z$) = $\arctan\left(\frac{y}{x}\right)$, with quadrant adjustments as needed.

Example 12

Find the argument of each complex number:

(a) $z = 2 + 3i$ (b) $w = 1 + i$ (c) $p = 5 - 2i$

Solution:

(a) $z = 2 + 3i$. $x = 2$, $y = 3$. $\theta = \arctan\left(\frac{3}{2}\right) \approx 56.3^\circ$. Since $x$ and $y$ are positive, $z$ is in the first quadrant, so arg($z$) $\approx 56.3^\circ$.

(b) $w = 1 + i$. $x = 1$, $y = 1$. $\theta = \arctan\left(\frac{1}{1}\right) = \arctan(1) = 45^\circ$. Since $x$ and $y$ are positive, $w$ is in the first quadrant, so arg($w$) = $45^\circ$.

(c) $p = 5 - 2i$. $x = 5$, $y = -2$. $\theta = \arctan\left(\frac{-2}{5}\right) \approx -21.8^\circ$. Since $x$ is positive and $y$ is negative, $p$ is in the fourth quadrant, so arg($p$) $\approx -21.8^\circ$.

Example 13

If $z_1 = -5 + 6i$ and $z_2 = -1 - 2i$, find the argument of $z_1 + z_2$.

Solution:

$z_1 + z_2 = (-5 + 6i) + (-1 - 2i) = -6 + 4i$

Let $z = z_1 + z_2 = -6 + 4i$. Here $x = -6$ and $y = 4$. Because x is negative and y is positive, the complex number lies in the second quadrant.

$\theta = \arctan\left(\frac{4}{-6}\right) \approx -33.7^\circ$. This is the reference angle.

Since the complex number lies in the second quadrant, we need to add 180° to the reference angle:

arg($z$) = -33.7° + 180° = 146.3°

Or in radians:

$\theta = \arctan\left(\frac{4}{-6}\right) \approx -0.59$ radians.

Since it's in the second quadrant, we add $\pi$:

arg($z$) = $-0.59 + \pi \approx 2.55$ radians.

The fundamental operations on complex numbers are addition, subtraction, multiplication, and division. These operations are performed similarly to those on polynomials.

Given complex numbers $z = a + ib$ and $w = c + id$:

Addition

Addition of $z$ and $w$ is done by adding the real and imaginary parts separately:

$z + w = (a + ib) + (c + id) = (a + c) + i(b + d)$

Subtraction

Subtraction is done by subtracting the real and imaginary parts separately:

$z - w = (a + ib) - (c + id) = (a - c) + i(b - d)$

Multiplication

Multiplication is similar to multiplying polynomial factors:

$zw = (a + ib)(c + id) = ac + iad + ibc + i^2bd = ac + i(ad + bc) - bd = (ac - bd) + i(ad + bc)$

Division

Division is done by multiplying the numerator and denominator by the complex conjugate of the denominator:

$\frac{z}{w} = \frac{a + ib}{c + id} = \frac{(a + ib)(c - id)}{(c + id)(c - id)} = \frac{ac - iad + ibc - i^2bd}{c^2 - i^2d^2} = \frac{ac + bd + i(bc - ad)}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + i\frac{bc - ad}{c^2 + d^2}$

Example 15

Simplify each of the following expressions:

(a) $(3 + 4i)(-3 - i)$ (b) $(-5 - 10i)(3 + i)$ (c) $\frac{1 + 3i}{3 - i}$

Solution:

(a) $(3 + 4i)(-3 - i) = -9 - 3i - 12i - 4i^2 = -9 - 15i + 4 = -5 - 15i$

(b) $(-5 - 10i)(3 + i) = -15 - 5i - 30i - 10i^2 = -15 - 35i + 10 = -5 - 35i$

(c) $\frac{1 + 3i}{3 - i} = \frac{(1 + 3i)(3 + i)}{(3 - i)(3 + i)} = \frac{3 + i + 9i + 3i^2}{9 - i^2} = \frac{3 + 10i - 3}{9 + 1} = \frac{10i}{10} = i$

Example 16

Solve for $z$ and $w$ in the following system of equations:

$2z + iw = 8$ $(i)$

$iz - w = 4 + 2i$ $(ii)$

Solution:

Multiply equation (i) by 3 and equation (ii) by $i$:

$6z + 3iw = 24$ $(iii)$

$-z - iw = 4i - 2$ $(iv)$

Add equations (iii) and (iv):

$5z = 22 + 4i$

$z = \frac{22}{5} + \frac{4}{5}i$

Substitute the value of $z$ into equation (i):

$2\left(\frac{22}{5} + \frac{4}{5}i\right) + iw = 8$

$\frac{44}{5} + \frac{8}{5}i + iw = 8$

$iw = 8 - \frac{44}{5} - \frac{8}{5}i$

$iw = \frac{40 - 44}{5} - \frac{8}{5}i$

$iw = -\frac{4}{5} - \frac{8}{5}i$

$w = -\frac{4}{5}i^{-1} - \frac{8}{5} = \frac{4i}{5}-\frac{8}{5}$

$w = -\frac{8}{5} + \frac{4}{5}i$

Therefore, $z = \frac{22}{5} + \frac{4}{5}i$ and $w = -\frac{8}{5} + \frac{4}{5}i$.

Example 17

Given $z = 3 + i$ and $w = 3i$, determine the simplified expressions for $zw$ and $\frac{z}{w}$, giving the answers in the form $x + iy$, where $x, y \in \mathbb{R}$.

Solution:

$zw = (3 + i)(3i) = 9i + 3i^2 = 9i - 3 = -3 + 9i$

$\frac{z}{w} = \frac{3 + i}{3i} = \frac{(3 + i)(-3i)}{(3i)(-3i)} = \frac{-9i - 3i^2}{-9i^2} = \frac{-9i + 3}{9} = \frac{1}{3} - i$

Equations can be solved using different methods, including factorization and the general quadratic formula. Similarly, solutions of equations involving complex numbers can be obtained by these methods.

If $z$ is a complex number, the general polynomial equation involving complex numbers takes the form:

$f(z) = a_nz^n + a_{n-1}z^{n-1} + ... + a_1z + a_0 = 0$, where $a_0, a_1, ..., a_n$ are real coefficients.

If $z$ is a root of the equation $f(z) = 0$, then the complex conjugate of $z$ is also a root of the equation. This means that complex roots of polynomial equations with real coefficients occur in complex conjugate pairs.

Example 18

Find the solution of the polynomial equation $z^3 - z^2 + 25z - 25 = 0$.

Solution:

Let $f(z) = z^3 - z^2 + 25z - 25$. The constant term is -25. Factors of 25 are ±1, ±5, and ±25.

Testing these values:

$f(1) = 1 - 1 + 25 - 25 = 0$. Thus, $z = 1$ is a root.

Since $z = 1$ is a root, $z - 1$ is a factor. Dividing the polynomial by $z - 1$ gives:

$\frac{z^3 - z^2 + 25z - 25}{z - 1} = z^2 + 25$

The quadratic equation $z^2 + 25 = 0$ can be solved as follows:

$z^2 = -25$

$z = \pm\sqrt{-25} = \pm 5i$

Therefore, the solutions are $z = 1$, $z = 5i$, and $z = -5i$.

Example 19

Find the roots of the polynomial equation $9z^4 - 35z^2 + 4 = 0$.

Solution:

Let $u = z^2$. The equation becomes:

$9u^2 - 35u + 4 = 0$

Using the quadratic formula:

$u = \frac{35 \pm \sqrt{(-35)^2 - 4(9)(4)}}{2(9)} = \frac{35 \pm \sqrt{1225 - 144}}{18} = \frac{35 \pm \sqrt{1081}}{18} = \frac{35 \pm 33}{18}$

$u = \frac{68}{18} = \frac{34}{9}$ or $u = \frac{2}{18} = \frac{1}{9}$

Since $u = z^2$:

$z^2 = \frac{34}{9}$ or $z^2 = \frac{1}{9}$

$z = \pm\frac{\sqrt{34}}{3}$ or $z = \pm\frac{1}{3}$

Therefore, the roots are $z = \frac{\sqrt{34}}{3}$, $z = -\frac{\sqrt{34}}{3}$, $z = \frac{1}{3}$, and $z = -\frac{1}{3}$.

Example 20

If $2 + i$ is one of the roots of the equation $z^3 - 5z^2 + 9z - 5 = 0$, find the other roots.

Solution:

Since $2 + i$ is a root, its conjugate $2 - i$ is also a root.

The quadratic factor corresponding to these roots is $(z - (2 + i))(z - (2 - i)) = (z - 2 - i)(z - 2 + i) = (z - 2)^2 - i^2 = z^2 - 4z + 4 + 1 = z^2 - 4z + 5$.

Dividing the original cubic by this quadratic factor:

$\frac{z^3 - 5z^2 + 9z - 5}{z^2 - 4z + 5} = z - 1$

Therefore, $z - 1 = 0$ which means $z = 1$ is the remaining root.

The other roots are $2 - i$ and $1$.