Complex Number¶

Definition¶

A complex number is of the form $a + \mathrm{i}b$$\mathrm{i}b$, where $a$ and $b$ are real numbers, and $\mathrm{i}$ is the imaginary unit. The number $a$ is the real part of the complex number $a + \mathrm{i}b$ and is denoted by $a = \operatorname{Re}(a+\mathrm{i}b)=\Re(a+\mathrm{i}b)$$\mathrm{i}b)$. Similarly, $b$ is the imaginary part and is represented as $b = \operatorname{Im}(a+\mathrm{i}b)=\Im(a+\mathrm{i}b).$ If the real part of a complex number $a+\mathrm{i}b$ is zero, we term the number $0+\mathrm{i}b$ as a purely imaginary number. The set of all complex numbers is denoted by $\mathbb{C}\text{.}$

Interpretation¶

Essentially, a complex number is an ordered pair of two real numbers, its real part $a$ and its imaginary part $b$. Writing it as $a+\mathrm{i}b$ lets one calculate as with ordinary algebraic expressions, using the extra rule $\mathrm{i}^2=-1$$-1$.

The point $z=a+\mathrm{i}b$ lies $a$ units along the horizontal axis and $b$ units up the vertical axis, so the two parts are its coordinates. The marked points $1$ and $\mathrm{i}$ are the units of the two axes: the real numbers fill the horizontal axis and the purely imaginary numbers $0+\mathrm{i}b$ the vertical one. This picture is the complex plane.

Both parts are real numbers: for $2-3\mathrm{i}$ the imaginary part is $-3$, not $-3\mathrm{i}$. A real number $a$ is the complex number $a+0\mathrm{i}$$0\mathrm{i}$, which is how $\mathbb{R}\subset\mathbb{C}$ should be read, and two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

What is gained is solvability: $x^2+1=0$ has no real solution but the two solutions $\pm\mathrm{i}$ in $\mathbb{C}$, and by the fundamental theorem of algebra every nonconstant polynomial has a root in $\mathbb{C}$. What is lost is order. No order on $\mathbb{C}$ follows the rules of the order on $\mathbb{R}$, since in such an order every square would be at least $0$, while $\mathrm{i}^2=-1$$-1$. Sizes are compared through the modulus $|z|$ instead.

Key Results¶

• Complex numbers do not have trichotomy, e.g., we cannot say $1-\mathrm{i}> 0$ or $1-\mathrm{i} < 0.$
• Complex numbers extend the real numbers by adding a solution of $x^2=-1$$-1$.

Examples¶

• $2-3\mathrm{i}$ is a complex number.
• Any real number is a special complex number, but not every complex number is real, i.e., $\mathbb{R}\subset \mathbb{C}$$\mathbb{C}$.

Prerequisites¶

• Real Number
• Imaginary Unit

Related Concepts¶

• Field
• Algebraically Closed Field
• De Moivre's Formula

Sources¶

• Math II: Linear Algebra · Week 1

Symbols: complex numbers, real numbers, subset

Last updated: 20 October 2026 03:02