Complex numbers Knowledge-Based Study

Study notes

1. The Imaginary Unit

   In the real number system, quadratic equations such as $x^2 + 1 = 0$ have no solution because the square of any real number cannot be negative. To expand our number system so that such equations can be solved, the imaginary unit, denoted by $i$, is defined with the fundamental property:

   $$i^2 = -1 \quad \text{or} \quad i = \sqrt{-1}$$

   Definition of a Complex Number

   A complex number is any number that can be expressed in standard form as:

   $$z = a + bi$$

   where $a$ and $b$ are real numbers.

   • $a = \text{Re}(z)$ is the real part of $z$.
   • $b = \text{Im}(z)$ is the imaginary part of $z$.

   If $b = 0$, then $z = a$, which represents a purely real number. If $a = 0$ and $b \neq 0$, then $z = bi$, which is called a pure imaginary number.

   Cyclic Powers of $i$

   The integer powers of $i$ repeat in a regular cycle of four:

   • $i^1 = i$
   • $i^2 = -1$
   • $i^3 = i^2 \cdot i = (-1) \cdot i = -i$
   • $i^4 = (i^2)^2 = (-1)^2 = 1$

   To evaluate $i^n$ for any integer $n$, divide the exponent by 4 and use the remainder: $i^{4k+r} = i^r$, where $r \in \{0, 1, 2, 3\}$.

   Addition and Subtraction

   To add or subtract two complex numbers, combine their respective real parts and their imaginary parts separately.

   Given $z_1 = a + bi$ and $z_2 = c + di$:

   $$(a + bi) + (c + di) = (a + c) + (b + d)i$$$$(a + bi) - (c + di) = (a - c) + (b - d)i$$Example:

   Let $z_1 = 4 + 7i$ and $z_2 = 2 - 3i$:

   • $z_1 + z_2 = (4 + 2) + (7 + (-3))i = 6 + 4i$
   • $z_1 - z_2 = (4 - 2) + (7 - (-3))i = 2 + 10i$

   Multiplication

   Multiplication is carried out using the standard distributive property (expanding binomials), followed by replacing $i^2$ with $-1$:

   $$(a + bi)(c + di) = ac + adi + bci + bdi^2$$

   Since $bdi^2 = bd(-1) = -bd$, we group the real and imaginary components:

   $$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$$Example:

   Calculate $(3 + 2i)(5 - 4i)$:

   $$(3 + 2i)(5 - 4i) = 15 - 12i + 10i - 8i^2$$$$= 15 - 2i - 8(-1) = 15 - 2i + 8 = 23 - 2i$$

   The Complex Conjugate

   For any complex number $z = a + bi$, its complex conjugate is denoted by $\overline{z}$ and is formed by reversing the sign of the imaginary part:

   $$\overline{z} = a - bi$$

   Key Property of the Conjugate

   Multiplying a complex number by its conjugate always produces a non-negative real number:

   $$z \cdot \overline{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2(-1) = a^2 + b^2$$

   Division of Complex Numbers

   To divide two complex numbers, eliminate the imaginary component from the denominator by multiplying both the numerator and the denominator by the complex conjugate of the denominator:

   $$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$$Example:

   Divide $5 + 2i$ by $3 - 4i$:

   $$\frac{5 + 2i}{3 - 4i} = \frac{(5 + 2i)(3 + 4i)}{(3 - 4i)(3 + 4i)} = \frac{15 + 20i + 6i + 8i^2}{3^2 - (4i)^2}$$$$= \frac{15 + 26i - 8}{9 - (-16)} = \frac{7 + 26i}{25} = \frac{7}{25} + \frac{26}{25}i$$

   The Complex Plane (Argand Diagram)

   Complex numbers can be represented geometrically as points or position vectors in a two-dimensional coordinate system known as the complex plane:

   • The horizontal axis represents the real axis ($\text{Re}$).
   • The vertical axis represents the imaginary axis ($\text{Im}$).

   The complex number $z = a + bi$ corresponds to the unique ordered pair $(a, b)$ in the Cartesian plane.

   The Modulus (Absolute Value)

   The modulus of a complex number $z = a + bi$, denoted by $|z|$, is the Euclidean distance from the origin $(0, 0)$ to the point $(a, b)$ in the complex plane.

   By the Pythagorean theorem:

   $$|z| = \sqrt{a^2 + b^2}$$Important Properties of the Modulus:

   • $|z| \geq 0$, and $|z| = 0$ if and only if $z = 0$
   • $|z|^2 = z \cdot \overline{z}$
   • $|z_1 \cdot z_2| = |z_1| \cdot |z_2|$
   • $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ (where $z_2 \neq 0$)

   Example:

   Find the modulus of $z = -3 + 4i$:

   $$|z| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$