Complex Numbers

Read the standard form

A complex number has the form

The real part is $\operatorname{Re}(z)=a$ and the imaginary part is $\operatorname{Im}(z)=b$. The imaginary part is a real coefficient; the term $bi$ is the imaginary term. For $z=3-2i$, the real part is 3 and the imaginary part is $-2$, not $-2i$.

If $b=0$, the number is real: $-7=-7+0i$. If $a=0$ and $b\ne0$, it is non-zero purely imaginary, such as $4i$. Zero is $0+0i$ and lies on both coordinate axes. The component formulas agree with the standard definitions of real and imaginary parts.

Interpret the imaginary unit without changing its rule

The letter $i$ denotes a fixed number, not an unknown to be solved for. From its defining equation,

The cycle repeats every four integer powers. Both $i$ and $-i$ square to $-1$, but the notation $i$ chooses one of the two and fixes it consistently.

The term “imaginary” is conventional. It does not mean that algebraic rules are optional or that complex numbers can be replaced by arbitrary symbols. Their arithmetic is defined precisely, using ordinary distribution together with $i^2=-1$.

Compare equality component by component

Two complex numbers are equal exactly when their real parts match and their imaginary parts match:

To solve $(2x+1)+(y-3)i=7-2i$ for real $x,y$, compare components:

Thus $x=3$ and $y=1$. Substitution reproduces $7-2i$.

The requirement that the components be real ensures the representation is unique. Treating $a$ and $b$ themselves as unrestricted complex quantities would no longer provide unique “real” and “imaginary” parts.

Solve a quadratic beyond the real line

Consider

Complete the square to obtain $(x+2)^2=-9$. The two complex numbers with square $-9$ are $3i$ and $-3i$, so

Both solutions are required. Their substitution is easiest in the completed-square form: $(\pm3i)^2+9=-9+9=0$.

Over the real numbers, $x^2+1$ cannot be factored into real linear factors. Over the complex numbers,

Factoring therefore depends on the number system in which the factors are allowed. OpenStax’s complex-number lesson connects this extension to quadratic equations.

Separate a principal root from both solutions

For a positive real $c$, the principal square root of $-c$ is written $i\sqrt c$. Thus

The equation $z^2=-20$, however, has two solutions $z=\pm2i\sqrt5$. The radical notation selects a value; solving the equation asks for every value that works.

Real radical rules do not extend unchanged to arbitrary complex inputs. For example,

Thus $\sqrt a\sqrt b=\sqrt{ab}$ cannot be used indiscriminately when negative or complex inputs occur. The familiar identity is safe for non-negative real $a,b$.

Introduce conjugates and magnitude

The conjugate of $z=a+bi$ is $\overline z=a-bi$. Its modulus is

For $z=3+4i$, the conjugate is $3-4i$ and the modulus is 5. The identity $z\overline z=a^2+b^2$ turns a conjugate product into a non-negative real number. It will provide the denominator in complex division.

The modulus measures distance from zero. It does not declare that $3+4i$ equals 5, and equal moduli do not imply equal complex numbers. For example, 5, $-5$ and $5i$ all have modulus 5 but are distinct.

Know which real-number properties do not extend

Complex numbers retain addition, multiplication, distribution and non-zero division. There is no order on all complex numbers compatible with the real ordered-field rules. In such an order, a non-zero square would have to be positive; but $i^2=-1$ would force a negative real number to be positive.

A comparison such as $|z|<3$ is meaningful because moduli are real. An inequality such as $z<3$ has no ordinary complex-number meaning without some separately defined comparison convention.

Related topics

Operations with Complex Numbers develops calculations and checking methods. The Complex Plane interprets those operations as geometry.