Complex Numbers

Intuition and core definition

A complex number has form $a+bi$, where $a,b$ are real and $i^2=-1$. The real part is $a$ and imaginary part is $b$. Complex numbers extend the real numbers so every nonconstant degree-$n$ polynomial has exactly $n$ complex roots when multiplicity is counted, including equations such as $x^2+1=0$.

Notation, language, and conditions

$\mathbb C$ denotes the complex numbers. $\operatorname{Re}(z)=a$, $\operatorname{Im}(z)=b$, and the conjugate of $a+bi$ is $a-bi$. Two complex numbers are equal exactly when their real parts and imaginary parts match. Powers of $i$ repeat with period four: $i,i^2=-1,i^3=-i,i^4=1$.

Why this idea matters

Complex numbers extend algebra to equations with negative discriminants and encode two-dimensional magnitude and direction in a single number.

A dependable method

1. For addition or subtraction, combine real parts and imaginary parts separately.
2. For multiplication, distribute and replace every $i^2$ by $-1$.
3. Simplify powers of $i$ using exponent modulo $4$.
4. For division, multiply numerator and denominator by the denominator’s conjugate.
5. Write the result in standard $a+bi$ form and check by reversing the operation.

Worked example

Common mistakes