Table of Contents
Complex numbers extend the familiar real numbers by including solutions to equations like $x^2 + 1 = 0$, which have no real solution. They form a number system where every quadratic equation has a solution, and they play a central role in higher algebra, geometry, and many applications.
In this chapter, we introduce the basic objects, notation, and simple operations with complex numbers. Later subsections will discuss the imaginary unit and operations in more detail; here we focus on the overall picture of what complex numbers are and how they are represented.
The basic idea of complex numbers
A complex number is written in the form
$$
z = a + bi,
$$
where
- $a$ and $b$ are real numbers,
- $i$ is a special symbol called the imaginary unit (covered in the “Imaginary unit” subsection),
- $a$ is called the real part of $z$, written $\Re(z)$,
- $b$ is called the imaginary part of $z$, written $\Im(z)$.
For example:
- $3 + 4i$ is a complex number with $\Re(3 + 4i) = 3$ and $\Im(3 + 4i) = 4$.
- $-2 - 5i$ has $\Re(-2 - 5i) = -2$ and $\Im(-2 - 5i) = -5$.
- $7$ can be thought of as $7 + 0i$, so $\Re(7) = 7$ and $\Im(7) = 0$.
This shows that every real number is also a complex number (with imaginary part $0$).
The set of all complex numbers is usually denoted by $\mathbb{C}$.
Equality of complex numbers
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
If
$$
z_1 = a + bi \quad\text{and}\quad z_2 = c + di,
$$
then
$$
z_1 = z_2 \quad\Longleftrightarrow\quad a = c \text{ and } b = d.
$$
For instance:
- $3 + 5i = 3 + 5i$ (same real parts, same imaginary parts).
- $3 + 5i \neq 3 + 4i$ (same real part, different imaginary parts).
- $3 + 5i \neq 4 + 5i$ (different real parts, same imaginary parts).
This definition of equality is very useful for solving equations involving complex numbers, because it allows you to compare real and imaginary parts separately.
The complex plane (geometric view)
Complex numbers can be visualized as points in a plane, called the complex plane or Argand plane.
To plot $z = a + bi$:
- Use the horizontal axis (called the real axis) for the real part $a$.
- Use the vertical axis (called the imaginary axis) for the imaginary part $b$.
So the complex number $a + bi$ corresponds to the point $(a, b)$ in the plane.
Examples:
- $3 + 4i$ is plotted at the point $(3, 4)$.
- $-2 + i$ is plotted at $(-2, 1)$.
- $-5i$ is plotted at $(0, -5)$.
Thinking of complex numbers as points (or arrows from the origin) is very helpful later, for understanding magnitudes, angles, and more advanced operations.
Standard (rectangular) form
The expression $a + bi$ is called the standard form or rectangular form of a complex number, because it uses horizontal and vertical components, like coordinates in a rectangle grid.
When working with complex numbers in Algebra II, you will often be asked to:
- Write an expression as a complex number in standard form.
- Simplify an expression until it has the shape $a + bi$.
For instance, if an expression contains $i^2$, you typically rewrite $i^2$ as $-1$ (as covered in the “Imaginary unit” subsection) and then collect real and imaginary parts.
Example of the kind of simplification you aim for (details of the $i$-rules will be handled later):
- An expression like $5 - 2i + 3i$ would be simplified to $5 + i$.
- An expression like $4 + 6i - 7$ would be simplified to $-3 + 6i$.
The goal is always to combine all real parts into a single real number $a$, and all imaginary parts into a single coefficient $b$ in $bi$.
Basic operations: overview
Complex numbers can be added, subtracted, multiplied, and divided, forming a number system that behaves much like the real numbers, but richer.
The detailed rules and examples of these operations are developed in the “Operations with complex numbers” subsection. Here is the high-level overview of what those operations look like.
Addition and subtraction (component-wise)
If
$$
z_1 = a + bi, \quad z_2 = c + di,
$$
then their sum and difference are obtained by combining real parts with real parts, and imaginary parts with imaginary parts:
- $z_1 + z_2$ has real part $a + c$ and imaginary part $b + d$,
- $z_1 - z_2$ has real part $a - c$ and imaginary part $b - d$.
This mirrors vector addition in the plane and fits well with the complex plane picture.
Multiplication (using distributive law)
Multiplication of complex numbers uses the distributive law (FOIL pattern) and the rule for $i^2$.
If $z_1 = a + bi$ and $z_2 = c + di$, then $z_1 z_2$ is obtained by expanding $(a + bi)(c + di)$ and then collecting real and imaginary parts.
Because $i^2 = -1$, multiplication of complex numbers is not just a matter of combining coefficients; the cross terms involving $i$ interact in a specific way that leads to rotation-like behavior in the complex plane (an idea that appears in more advanced study of complex numbers).
Division (when the denominator is nonzero)
Every nonzero complex number has a multiplicative inverse (a number you can multiply by to get $1$). Division of complex numbers is defined using this inverse.
If $z_2 \neq 0$, then
$$
\frac{z_1}{z_2}
$$
is another complex number. In practice, to express this in standard form $a + bi$, one uses a method that removes $i$ from the denominator. The detailed procedure and examples appear in the “Operations with complex numbers” subsection.
Division is one of the reasons complex numbers form such a convenient system: unlike the real numbers extended with just $i$ in an ad hoc way, $\mathbb{C}$ is closed under all four basic operations.
Real and imaginary parts as functions
For any complex number $z$, you can talk about its real part and imaginary part as separate quantities.
If
$$
z = a + bi,
$$
then:
- $\Re(z) = a$,
- $\Im(z) = b$.
These two functions, $\Re$ and $\Im$, simply “pick out” the horizontal and vertical components in the complex plane. They are useful when equations or expressions involve both parts and you want to isolate or compare them.
For example, if an equation states that
$$
\Re(z) = 2, \quad \Im(z) = -3,
$$
then $z$ must be $2 - 3i$.
Complex conjugate: basic idea
For a complex number
$$
z = a + bi,
$$
its complex conjugate, usually written $\overline{z}$, is defined by
$$
\overline{z} = a - bi.
$$
So you keep the real part the same and change the sign of the imaginary part.
Examples:
- If $z = 3 + 4i$, then $\overline{z} = 3 - 4i$.
- If $z = -2 - 5i$, then $\overline{z} = -2 + 5i$.
- If $z$ is real (say $z = 7$), then $\overline{z} = 7$ as well.
The complex conjugate has important algebraic uses:
- It pairs naturally with $z$ in expressions like $z \overline{z}$.
- It is used to carry out division and simplify fractions with complex numbers in denominators.
These uses are handled in more depth in the “Operations with complex numbers” subsection, but knowing the basic definition of $\overline{z}$ is essential for working with complex numbers at all levels.
Magnitude (modulus) of a complex number: preview
When you view $z = a + bi$ as a point $(a, b)$ in the plane, you can talk about its distance from the origin. This length is called the magnitude or modulus of $z$, denoted $|z|$.
Geometrically, $|z|$ is the length of the arrow from $0$ to $z$ in the complex plane. It can be computed using the Pythagorean theorem, relating real and imaginary parts.
Although a detailed treatment of the modulus and its algebraic properties goes beyond this introductory chapter, you should be aware that:
- The modulus measures “size” of a complex number, in a geometric sense.
- It interacts nicely with multiplication (for example, the modulus of a product is the product of the moduli).
Understanding $|z|$ becomes important later in trigonometry, complex analysis, and in many applications to physics and engineering.
Why complex numbers are useful in algebra
From the Algebra II perspective, complex numbers are especially useful because:
- Every quadratic equation with real coefficients has its solutions in $\mathbb{C}$.
- Many polynomial equations that have no real solutions do have complex solutions.
- Certain algebraic factorizations and identities become more natural when complex numbers are allowed.
For example, the equation
$$
x^2 + 1 = 0
$$
has no real solutions, but in $\mathbb{C}$ it has two solutions, $x = i$ and $x = -i$.
By enlarging the number system from $\mathbb{R}$ (the real numbers) to $\mathbb{C}$ (the complex numbers), we gain a framework where algebraic equations behave more predictably and often more symmetrically.
Later chapters—especially those on polynomial functions, trigonometry, and beyond—will rely on these basic ideas about complex numbers.