Complex Numbers: Operations

Definition

A complex number is a number of the form $a + bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit, defined by $i^2 = -1$ .

How to read: “The value a plus b i, where i squared equals negative one.”
Meaning: A complex number extends reals with a perpendicular axis; $a$ is the real part, $bi$ the imaginary part.

It enables the mathematical description of 2D phenomena like electric circuits and fluid flow using a single, unified algebra.

Equality: $a + bi = c + di$ $a + bi = c + d i$ only if $a=c$ $a = c$ and $b=d$ $b = d$ .
- How to read: “The value a plus b i equals c plus d i only if a equals c and b equals d.”
- Meaning: Match real and imaginary parts separately—two complex numbers are equal iff both components agree.
Arithmetic:
- Addition/Subtraction: Combine real parts and imaginary parts separately.
- Multiplication: Use FOIL and replace $i^2$ with $-1$ .
- Division: Multiply numerator and denominator by the complex conjugate ( $\overline{z} = a - bi$ ) to rationalize the denominator. Note that $z\overline{z} = a^2 + b^2$ .
- How to read: “The value z bar equals a minus b i, and z times z bar equals a squared plus b squared.”
- Meaning: Conjugation eliminates $i$ from the denominator, leaving a real number $a^2+b^2$ .
Powers of $i$ : Cycles every four: $i^1=i, i^2=-1, i^3=-i, i^4=1$ $i^{1} = i, i^{2} = - 1, i^{3} = - i, i^{4} = 1$ .
- How to read: “The values i, negative one, negative i, and one, which then repeat.”
- Meaning: Reduce $i^n$ by dividing $n$ by 4 and using the remainder.
Principal Square Root of Negative Numbers: $\sqrt{-N} = i\sqrt{N}$ $- N = i N$ for $N > 0$ $N > 0$ .
- How to read: “The square root of negative N equals i times the square root of N.”
- Meaning: Extract $i$ from the radical; the remaining root is real.