Key Points

A number of the form $z = a + ib$, where $a$ and $b$ are real numbers, is a complex number. The term $a$ is called the real part, denoted $\text{Re}(z)$, and $b$ is called the imaginary part, denoted $\text{Im}(z)$.

The symbol $i$ represents the principal square root of -1, defined as $i = \sqrt{-1}$. This leads to the fundamental property $i^2 = -1$.

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

The sum of two complex numbers is found by adding their real and imaginary parts separately: $(a + ib) + (c + id) = (a+c) + i(b+d)$. Subtraction follows the same principle.

The product of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is defined as $z_1 z_2 = (ac - bd) + i(ad + bc)$.

The powers of $i$ follow a cycle of four: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. In general, for any integer $k$, $i^{4k}=1$, $i^{4k+1}=i$, $i^{4k+2}=-1$, and $i^{4k+3}=-i$.

For any positive real number $a$, the square root of its negative is given by $\sqrt{-a} = \sqrt{a} \times \sqrt{-1} = i\sqrt{a}$. The property $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ is not valid if both $a$ and $b$ are negative.

The modulus of a complex number $z = a + ib$, denoted by $|z|$, is the non-negative real number defined as $|z| = \sqrt{a^2 + b^2}$. It represents the distance of the point $(a,b)$ from the origin in the Argand plane.

The conjugate of a complex number $z = a + ib$, denoted by $\bar{z}$, is defined as $\bar{z} = a - ib$. An important property is that the product of a complex number and its conjugate is the square of its modulus: $z \bar{z} = |z|^2 = a^2 + b^2$.

For a non-zero complex number $z$, its multiplicative inverse $z^{-1}$ is given by the formula $z^{-1} = \frac{\bar{z}}{|z|^2}$. This is equivalent to $z^{-1} = \frac{a}{a^2+b^2} - i\frac{b}{a^2+b^2}$.

To divide a complex number $z_1$ by a non-zero complex number $z_2$, multiply the numerator and denominator by the conjugate of the denominator: $\frac{z_1}{z_2} = \frac{z_1 \times \bar{z_2}}{z_2 \times \bar{z_2}} = \frac{z_1 \bar{z_2}}{|z_2|^2}$.

For any two complex numbers $z_1$ and $z_2$: $|z_1 z_2| = |z_1| |z_2|$, $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$ for $z_2 \neq 0$, $\overline{z_1 \pm z_2} = \bar{z_1} \pm \bar{z_2}$, and $\overline{z_1 z_2} = \bar{z_1} \bar{z_2}$.

A complex number $z = x + iy$ can be represented as a unique point $P(x, y)$ on a two-dimensional plane called the Argand plane or complex plane. The x-axis is the real axis, and the y-axis is the imaginary axis.

For a quadratic equation $ax^2 + bx + c = 0$ with real coefficients, if the discriminant $D = b^2 - 4ac < 0$, the solutions are complex numbers given by $x = \frac{-b \pm i\sqrt{4ac - b^2}}{2a}$.