Complex Numbers and the Complex Plane

Definition The imaginary unit $i$ is a number with

The square of every real number is at least $0$, so $i$ is not a real number. The number $-i$ also squares to $-1$. For a real number $a > 0$, the principal square root of $-a$ is defined as

Powers of $i$ repeat in a cycle of four, since $i^1 = i$, $i^2 = -1$, $i^3 = -i$ and $i^4 = 1$. So $i^n$ depends only on the remainder when the whole number $n$ is divided by $4$.

Definition A complex number is a number of the form

This is its rectangular form, also called standard form. The real number $a$ is the real part, written $\operatorname{Re} z$, and the real number $b$ is the imaginary part, written $\operatorname{Im} z$. The set of all complex numbers is written $\mathbb{C}$. Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal. When $b = 0$ the number is the real number $a$, so every real number is complex. When $a = 0$ and $b \ne 0$ the number $bi$ is called purely imaginary. Unlike real numbers, complex numbers are not ordered, so an inequality such as $z < w$ has no meaning unless both numbers are real.

Definition For real numbers $a$, $b$, $c$ and $d$,

Addition is commutative and associative, $0$ is the additive identity and $-a - bi$ is the opposite of $a + bi$.

Definition For real numbers $a$, $b$, $c$ and $d$,

This is what you get by expanding the product with the distributive law and replacing $i^2$ with $-1$. Multiplication is commutative and associative, distributes over addition, and has $1$ as its identity. The rule $\sqrt{x}\sqrt{y} = \sqrt{xy}$ holds when $x \ge 0$ and $y \ge 0$, but it fails when $x$ and $y$ are both negative. So rewrite every square root of a negative number with $i$ before you multiply.

Definition The complex conjugate of $z = a + bi$, with $a$ and $b$ real, is

For all complex $z$ and $w$, the conjugate satisfies $\overline{z + w} = \overline{z} + \overline{w}$ and $\overline{zw} = \overline{z}\,\overline{w}$. Moreover

and both of these are real numbers, with $z\,\overline{z} > 0$ whenever $z \ne 0$.

Definition For complex numbers $z$ and $w$ with $w \ne 0$, the quotient $\frac{z}{w}$ is the unique complex number $u$ with $wu = z$. To find it in rectangular form, multiply the numerator and the denominator by $\overline{w}$, which makes the denominator the positive real number $w\,\overline{w}$. For $z = a + bi$ and $w = c + di \ne 0$ with $a$, $b$, $c$ and $d$ real, this gives

Division by $0$ is undefined, as for real numbers.

Definition The complex plane is a coordinate plane in which the complex number $a + bi$, with $a$ and $b$ real, is the point $(a, b)$. The horizontal axis is the real axis and the vertical axis is the imaginary axis. In this picture the sum $z + w$ is found by adding the arrows from $0$ to $z$ and to $w$ tip to tail, as with vectors. The conjugate $\overline{z}$ is the reflection of $z$ across the real axis, and $-z$ is the reflection of $z$ through the origin. Multiplying by $i$ turns every point a quarter turn counterclockwise about $0$, because

which sends the point $(a, b)$ to $(-b, a)$.

Definition The modulus, or absolute value, of $z = a + bi$ with $a$ and $b$ real is the real number

It is the distance from $0$ to $z$ in the complex plane, and $\lvert z - w \rvert$ is the distance between $z$ and $w$. Always $\lvert z \rvert \ge 0$, with $\lvert z \rvert = 0$ only for $z = 0$, and $\lvert zw \rvert = \lvert z \rvert \lvert w \rvert$. For a real number it is the usual absolute value.

Definition Let $a$, $b$ and $c$ be real with $a \ne 0$, and let $D = b^2 - 4ac$. If $D < 0$, the equation $ax^2 + bx + c = 0$ has no real solutions and exactly two complex solutions,

They are complex conjugates of each other. If $D \ge 0$ the usual quadratic formula gives real solutions.