Algebra of Complex Numbers

Complex numbers are an extension of the real number system. They are expressed in the form $a + ib$, where 'a' is the real part and 'b' is the imaginary part, and 'i' is the imaginary unit, defined as $i = \sqrt{-1}$. The imaginary unit 'i' has interesting properties: $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. This cycle of powers of 'i' repeats every four powers.

The set of all complex numbers is denoted by $\mathbb{C}$. A complex number can be represented as an ordered pair of real numbers $(a, b)$ corresponding to $a + ib$. The real numbers are a subset of complex numbers where the imaginary part is zero (e.g., $5 + 0i$ is the real number 5). Purely imaginary numbers have a real part of zero (e.g., $0 + 3i$ is $3i$).

Equality of Complex Numbers

Two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ are equal if and only if their real parts are equal and their imaginary parts are equal. That is, $z_1 = z_2$ if and only if $a = c$ and $b = d$.

For example, if $x + 3i = 5 + yi$, then for these two complex numbers to be equal, we must have $x = 5$ (equating the real parts) and $3 = y$ (equating the imaginary parts).

Operations on Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided, similar to algebraic expressions, keeping in mind that $i^2 = -1$.

Addition of Complex Numbers

The sum of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is obtained by adding their real parts and their imaginary parts separately. $z_1 + z_2 = (a + c) + i(b + d)$

Example: Let $z_1 = 2 + 3i$ and $z_2 = 4 + 5i$. $z_1 + z_2 = (2 + 4) + i(3 + 5) = 6 + 8i$.

Properties of addition:

• Commutative Law: $z_1 + z_2 = z_2 + z_1$
• Associative Law: $(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$
• Existence of Additive Identity: $0 + 0i$ (or 0) is the additive identity, such that $z + 0 = z$.
• Existence of Additive Inverse: For every complex number $z = a + ib$, there exists an additive inverse $-z = -a - ib$, such that $z + (-z) = 0$.

Subtraction of Complex Numbers

The difference between two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is given by: $z_1 - z_2 = (a - c) + i(b - d)$

Example: Let $z_1 = 7 + 2i$ and $z_2 = 3 + 8i$. $z_1 - z_2 = (7 - 3) + i(2 - 8) = 4 - 6i$.

Multiplication of Complex Numbers

The product of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is found by expanding the product as if they were binomials and then using $i^2 = -1$. $z_1 \cdot z_2 = (a + ib)(c + id)$ $z_1 \cdot z_2 = ac + i(ad) + i(bc) + i^2(bd)$ $z_1 \cdot z_2 = ac + i(ad + bc) - bd$ $z_1 \cdot z_2 = (ac - bd) + i(ad + bc)$

Example: Let $z_1 = 1 + 2i$ and $z_2 = 3 - 4i$. $z_1 \cdot z_2 = (1 \cdot 3 - 2 \cdot (-4)) + i(1 \cdot (-4) + 2 \cdot 3)$ $z_1 \cdot z_2 = (3 - (-8)) + i(-4 + 6)$ $z_1 \cdot z_2 = (3 + 8) + i(2) = 11 + 2i$.

Properties of multiplication:

• Commutative Law: $z_1 \cdot z_2 = z_2 \cdot z_1$
• Associative Law: $(z_1 \cdot z_2) \cdot z_3 = z_1 \cdot (z_2 \cdot z_3)$
• Existence of Multiplicative Identity: $1 + 0i$ (or 1) is the multiplicative identity, such that $z \cdot 1 = z$.
• Existence of Multiplicative Inverse: For every non-zero complex number $z$, there exists a multiplicative inverse $z^{-1}$ or $1/z$, such that $z \cdot z^{-1} = 1$.
• Distributive Law: $z_1 \cdot (z_2 + z_3) = z_1 \cdot z_2 + z_1 \cdot z_3$

Division of Complex Numbers

To divide one complex number by another, $z_1 / z_2$, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $z = a + ib$ is denoted by $\bar{z}$ and is equal to $a - ib$. This process effectively removes the imaginary part from the denominator. $\frac{z_1}{z_2} = \frac{a + ib}{c + id}$ Multiply numerator and denominator by the conjugate of $c + id$, which is $c - id$: $\frac{z_1}{z_2} = \frac{(a + ib)(c - id)}{(c + id)(c - id)}$ Numerator: $(a + ib)(c - id) = ac - iad + ibc - i^2bd = (ac + bd) + i(bc - ad)$ Denominator: $(c + id)(c - id) = c^2 - (id)^2 = c^2 - i^2d^2 = c^2 - (-1)d^2 = c^2 + d^2$ So, $\frac{z_1}{z_2} = \frac{(ac + bd) + i(bc - ad)}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + i \frac{bc - ad}{c^2 + d^2}$

Example: Let $z_1 = 2 + 3i$ and $z_2 = 1 - i$. $\frac{z_1}{z_2} = \frac{2 + 3i}{1 - i}$ Multiply by the conjugate of $(1 - i)$, which is $(1 + i)$: $\frac{2 + 3i}{1 - i} \times \frac{1 + i}{1 + i} = \frac{(2 + 3i)(1 + i)}{(1 - i)(1 + i)}$ Numerator: $(2)(1) + (2)(i) + (3i)(1) + (3i)(i) = 2 + 2i + 3i + 3i^2 = 2 + 5i - 3 = -1 + 5i$ Denominator: $(1)^2 - (i)^2 = 1 - (-1) = 1 + 1 = 2$ So, $\frac{z_1}{z_2} = \frac{-1 + 5i}{2} = -\frac{1}{2} + \frac{5}{2}i$.

Conjugate of a Complex Number

The conjugate of a complex number $z = a + ib$ is denoted by $\bar{z}$ and is obtained by changing the sign of the imaginary part. If $z = a + ib$, then $\bar{z} = a - ib$.

Properties of Conjugates:

• $\overline{(\bar{z})} = z$
• $z + \bar{z} = (a + ib) + (a - ib) = 2a$ (a real number)
• $z - \bar{z} = (a + ib) - (a - ib) = 2ib$ (a purely imaginary number)
• $z \cdot \bar{z} = (a + ib)(a - ib) = a^2 - (ib)^2 = a^2 - i^2b^2 = a^2 + b^2$ (a real number, and also the square of the modulus)
• $\overline{(z_1 + z_2)} = \bar{z_1} + \bar{z_2}$
• $\overline{(z_1 - z_2)} = \bar{z_1} - \bar{z_2}$
• $\overline{(z_1 \cdot z_2)} = \bar{z_1} \cdot \bar{z_2}$
• $\overline{(z_1 / z_2)} = \bar{z_1} / \bar{z_2}$ (provided $z_2 \neq 0$)

Example: If $z = 3 - 4i$, then $\bar{z} = 3 + 4i$. $z + \bar{z} = (3 - 4i) + (3 + 4i) = 6$. $z \cdot \bar{z} = (3 - 4i)(3 + 4i) = 3^2 + 4^2 = 9 + 16 = 25$.

Modulus of a Complex Number

The modulus of a complex number $z = a + ib$ is its distance from the origin $(0, 0)$ in the complex plane (Argand plane). It is denoted by $|z|$. The formula for the modulus is derived from the Pythagorean theorem: $|z| = \sqrt{a^2 + b^2}$

Since $a^2 \ge 0$ and $b^2 \ge 0$, the modulus $|z|$ is always a non-negative real number. Also, note that $|z|^2 = a^2 + b^2 = z \cdot \bar{z}$.

Example: For the complex number $z = 3 + 4i$: $|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Example: For the complex number $z = -2 - 5i$: $|z| = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}$.

Example: For a purely imaginary number $z = 6i$ (which is $0 + 6i$): $|z| = \sqrt{0^2 + 6^2} = \sqrt{36} = 6$.

Example: For a real number $z = -7$ (which is $-7 + 0i$): $|z| = \sqrt{(-7)^2 + 0^2} = \sqrt{49} = 7$. This is the absolute value of the real number.

Properties of Modulus

Let $z_1$ and $z_2$ be two complex numbers.

• $|z| \ge 0$
• $|z| = 0$ if and only if $z = 0$
• $|z| = |\bar{z}|$
• $|z|^2 = z \bar{z}$
• $|z_1 \cdot z_2| = |z_1| \cdot |z_2|$
• $|z_1 / z_2| = |z_1| / |z_2|$ (provided $z_2 \neq 0$)
• $|z_1 + z_2| \le |z_1| + |z_2|$ (Triangle Inequality)
• $|z_1 - z_2| \ge ||z_1| - |z_2||$ (Reverse Triangle Inequality)
• $|z^n| = |z|^n$ for any integer n.

Example of Triangle Inequality: Let $z_1 = 1 + i$ and $z_2 = 2 + 3i$. $|z_1| = \sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.414$ $|z_2| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.606$ $|z_1| + |z_2| \approx 1.414 + 3.606 = 5.020$ $z_1 + z_2 = (1+2) + i(1+3) = 3 + 4i$ $|z_1 + z_2| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ We see that $5 \le 5.020$, confirming $|z_1 + z_2| \le |z_1| + |z_2|$.

Argument of a Complex Number

The argument of a non-zero complex number $z = a + ib$ is the angle $\theta$ that the line segment joining the origin $(0, 0)$ to the point $(a, b)$ makes with the positive real axis in the complex plane. It is denoted by $\arg(z)$.

The argument is usually expressed in radians. For a complex number $z = a + ib$, the argument $\theta$ can be found using the relations: $\cos \theta = \frac{a}{|z|}$ and $\sin \theta = \frac{b}{|z|}$. Also, $\tan \theta = \frac{b}{a}$, but this needs to be used carefully as it only gives the reference angle. The quadrant in which the point $(a, b)$ lies must be considered to determine the correct value of $\theta$.

The argument is not unique. If $\theta$ is an argument of $z$, then $\theta + 2n\pi$, where $n$ is an integer, are also arguments of $z$. The set of all arguments is $\{\theta + 2n\pi \mid n \in \mathbb{Z}\}$.

Principal Argument

The principal argument of a complex number $z$, denoted by $\text{Arg}(z)$, is the unique value of the argument $\theta$ such that $-\pi < \theta \le \pi$.

Calculating the Argument

To find the argument of $z = a + ib$:

1. Calculate the modulus $|z| = \sqrt{a^2 + b^2}$.
2. Determine the reference angle $\alpha = \arctan \left| \frac{b}{a} \right|$. This is always a positive acute angle.
3. Identify the quadrant of the point $(a, b)$:
   • If $(a, b)$ is in Quadrant I ($a > 0, b > 0$), then $\text{Arg}(z) = \alpha$.
   • If $(a, b)$ is in Quadrant II ($a < 0, b > 0$), then $\text{Arg}(z) = \pi - \alpha$.
   • If $(a, b)$ is in Quadrant III ($a < 0, b < 0$), then $\text{Arg}(z) = -\pi + \alpha$ (or $\pi + \alpha$ if using the range $0 \le \theta < 2\pi$). For the principal argument, we use $-\pi + \alpha$.
   • If $(a, b)$ is in Quadrant IV ($a > 0, b < 0$), then $\text{Arg}(z) = -\alpha$.
4. Special Cases:
   • If $a = 0$ and $b > 0$ (positive imaginary axis), $\text{Arg}(z) = \pi/2$.
   • If $a = 0$ and $b < 0$ (negative imaginary axis), $\text{Arg}(z) = -\pi/2$.
   • If $a > 0$ and $b = 0$ (positive real axis), $\text{Arg}(z) = 0$.
   • If $a < 0$ and $b = 0$ (negative real axis), $\text{Arg}(z) = \pi$.
   • If $a = 0$ and $b = 0$, $z = 0$, and the argument is undefined.

Example 1: Find the principal argument of $z = 1 + i$. $a = 1, b = 1$. $|z| = \sqrt{1^2 + 1^2} = \sqrt{2}$. $\alpha = \arctan |1/1| = \arctan(1) = \pi/4$. The point $(1, 1)$ is in Quadrant I. So, $\text{Arg}(z) = \alpha = \pi/4$.

Example 2: Find the principal argument of $z = -1 + i$. $a = -1, b = 1$. $|z| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}$. $\alpha = \arctan |1/(-1)| = \arctan(1) = \pi/4$. The point $(-1, 1)$ is in Quadrant II. So, $\text{Arg}(z) = \pi - \alpha = \pi - \pi/4 = 3\pi/4$.

Example 3: Find the principal argument of $z = -1 - i$. $a = -1, b = -1$. $|z| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}$. $\alpha = \arctan |-1/(-1)| = \arctan(1) = \pi/4$. The point $(-1, -1)$ is in Quadrant III. So, $\text{Arg}(z) = -\pi + \alpha = -\pi + \pi/4 = -3\pi/4$.

Example 4: Find the principal argument of $z = 1 - i$. $a = 1, b = -1$. $|z| = \sqrt{1^2 + (-1)^2} = \sqrt{2}$. $\alpha = \arctan |-1/1| = \arctan(1) = \pi/4$. The point $(1, -1)$ is in Quadrant IV. So, $\text{Arg}(z) = -\alpha = -\pi/4$.

Properties of Argument

Let $z_1$ and $z_2$ be non-zero complex numbers.

• $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2) + 2k\pi$ for some integer $k$.
• $\arg(z_1 / z_2) = \arg(z_1) - \arg(z_2) + 2k\pi$ for some integer $k$.
• $\arg(z^n) = n \arg(z) + 2k\pi$ for some integer $k$, for any integer $n$.
• $\arg(\bar{z}) = -\arg(z) + 2k\pi$ for some integer $k$.

When dealing with principal arguments, these relations might need adjustments by adding or subtracting $2\pi$ to keep the result within the $(-\pi, \pi]$ range.

Example: Let $z_1 = 1 + i$ and $z_2 = 1 - i$. $\arg(z_1) = \pi/4$. $\arg(z_2) = -\pi/4$. $z_1 z_2 = (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 2$. $z_1 z_2 = 2 + 0i$. This is on the positive real axis. $\arg(z_1 z_2) = 0$. Using the property: $\arg(z_1) + \arg(z_2) = \pi/4 + (-\pi/4) = 0$. This matches.

Example: Let $z = 1 + i$. $n = 2$. $\arg(z) = \pi/4$. $z^2 = (1+i)^2 = 1^2 + 2i + i^2 = 1 + 2i - 1 = 2i$. $z^2 = 0 + 2i$. This is on the positive imaginary axis. $\arg(z^2) = \pi/2$. Using the property: $n \arg(z) = 2 \times (\pi/4) = \pi/2$. This matches.

Polar Form of a Complex Number

A complex number $z = a + ib$ can also be represented in polar form using its modulus $r = |z|$ and its argument $\theta = \arg(z)$. From trigonometry, in a right-angled triangle with hypotenuse $r$, adjacent side $a$, and opposite side $b$: $a = r \cos \theta$ $b = r \sin \theta$ Substituting these into $z = a + ib$: $z = (r \cos \theta) + i (r \sin \theta)$ $z = r (\cos \theta + i \sin \theta)$

This is the polar form of a complex number. It is often abbreviated using Euler's notation: $z = r e^{i\theta}$ where $e^{i\theta} = \cos \theta + i \sin \theta$ (Euler's formula).

The modulus $r$ is the distance from the origin, and the argument $\theta$ is the angle from the positive real axis.

Example: Express $z = 1 + i$ in polar form. We found $|z| = \sqrt{2}$ and $\arg(z) = \pi/4$. So, $r = \sqrt{2}$ and $\theta = \pi/4$. The polar form is $z = \sqrt{2} (\cos(\pi/4) + i \sin(\pi/4))$. Using Euler's notation: $z = \sqrt{2} e^{i\pi/4}$.

Example: Express $z = 2(\cos(2\pi/3) + i \sin(2\pi/3))$ in the form $a + ib$. Here $r = 2$ and $\theta = 2\pi/3$. $\cos(2\pi/3) = -1/2$ $\sin(2\pi/3) = \sqrt{3}/2$ $z = 2 (-1/2 + i \sqrt{3}/2) = -1 + i\sqrt{3}$.

Operations in Polar Form

The polar form is particularly useful for multiplication, division, and exponentiation of complex numbers.

Multiplication in Polar Form

Let $z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)$. $z_1 z_2 = [r_1 (\cos \theta_1 + i \sin \theta_1)] \cdot [r_2 (\cos \theta_2 + i \sin \theta_2)]$ $z_1 z_2 = r_1 r_2 [(\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2) + i (\sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2)]$ Using trigonometric sum identities: $z_1 z_2 = r_1 r_2 [\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]$

In words: To multiply two complex numbers in polar form, multiply their moduli and add their arguments.

Using Euler's notation: If $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$, then $z_1 z_2 = (r_1 e^{i\theta_1})(r_2 e^{i\theta_2}) = r_1 r_2 e^{i(\theta_1 + \theta_2)}$.

Division in Polar Form

Let $z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)$. $\frac{z_1}{z_2} = \frac{r_1 (\cos \theta_1 + i \sin \theta_1)}{r_2 (\cos \theta_2 + i \sin \theta_2)}$ Multiplying numerator and denominator by the conjugate of the denominator, which is $r_2 (\cos \theta_2 - i \sin \theta_2) = r_2 (\cos(-\theta_2) + i \sin(-\theta_2))$: $\frac{z_1}{z_2} = \frac{r_1}{r_2} \frac{(\cos \theta_1 + i \sin \theta_1)(\cos \theta_2 - i \sin \theta_2)}{(\cos \theta_2 + i \sin \theta_2)(\cos \theta_2 - i \sin \theta_2)}$ Numerator: $r_1 [(\cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2) + i (\sin \theta_1 \cos \theta_2 - \cos \theta_1 \sin \theta_2)]$ Using trigonometric difference identities: $r_1 [\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)]$ Denominator: $r_2 (\cos^2 \theta_2 + \sin^2 \theta_2) = r_2(1) = r_2$. So, $\frac{z_1}{z_2} = \frac{r_1}{r_2} [\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)]$

In words: To divide two complex numbers in polar form, divide their moduli and subtract their arguments.

Using Euler's notation: If $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$, then $\frac{z_1}{z_2} = \frac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$.

De Moivre's Theorem

De Moivre's theorem provides a way to calculate powers of complex numbers in polar form. It states that for any complex number $z = r(\cos \theta + i \sin \theta)$ and any integer $n$: $z^n = [r(\cos \theta + i \sin \theta)]^n = r^n (\cos(n\theta) + i \sin(n\theta))$

Using Euler's notation: If $z = r e^{i\theta}$, then $z^n = (r e^{i\theta})^n = r^n e^{in\theta}$.

Example: Calculate $(1 + i)^4$. First, convert $1 + i$ to polar form. We found $r = \sqrt{2}$ and $\theta = \pi/4$. So, $z = \sqrt{2} (\cos(\pi/4) + i \sin(\pi/4))$. Using De Moivre's theorem with $n = 4$: $z^4 = (\sqrt{2})^4 (\cos(4 \cdot \pi/4) + i \sin(4 \cdot \pi/4))$ $z^4 = (2^2) (\cos(\pi) + i \sin(\pi))$ $z^4 = 4 (-1 + i \cdot 0)$ $z^4 = -4$.

This is much simpler than expanding $(1+i)^4$ directly.

Roots of a Complex Number

De Moivre's theorem can also be used to find the $n$-th roots of a complex number. If $z = r(\cos \theta + i \sin \theta)$, then its $n$ distinct $n$-th roots are given by: $z_k = \sqrt[n]{r} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right)$ for $k = 0, 1, 2, \dots, n-1$.

Here, $\sqrt[n]{r}$ denotes the positive real $n$-th root of $r$.

Example: Find the cube roots of $1$. The complex number $1$ can be written as $1 + 0i$. In polar form, $r = |1| = 1$. The argument is $\theta = 0$. So, $z = 1 (\cos 0 + i \sin 0)$. We want to find the cube roots, so $n = 3$. The roots are given by: $z_k = \sqrt[3]{1} \left( \cos\left(\frac{0 + 2k\pi}{3}\right) + i \sin\left(\frac{0 + 2k\pi}{3}\right) \right)$ $z_k = 1 \left( \cos\left(\frac{2k\pi}{3}\right) + i \sin\left(\frac{2k\pi}{3}\right) \right)$ for $k = 0, 1, 2$. For $k=0$: $z_0 = \cos(0) + i \sin(0) = 1 + 0i = 1$. For $k=1$: $z_1 = \cos(2\pi/3) + i \sin(2\pi/3) = -1/2 + i\sqrt{3}/2$. For $k=2$: $z_2 = \cos(4\pi/3) + i \sin(4\pi/3) = -1/2 - i\sqrt{3}/2$. These are the three cube roots of unity.