Algebra of Complex Numbers

Welcome, students! 👋 In this lesson, you will learn how complex numbers can be added, subtracted, multiplied, divided, and simplified just like numbers in ordinary algebra, but with one important twist: the imaginary unit $i$, where $i^2=-1$. Complex algebra is the foundation for everything else in this topic, including modulus, argument, polar form, and roots.

Lesson objectives

By the end of this lesson, you should be able to:

explain the meaning of a complex number and the role of $i$
perform algebraic operations with complex numbers correctly
use conjugates to simplify expressions and divisions
connect algebraic form $a+bi$ to the complex plane 🌍
recognize how complex number algebra supports later topics such as polar form and roots

1. What is a complex number?

A complex number is written in the form $z=a+bi$, where $a$ and $b$ are real numbers and $i^2=-1$. The number $a$ is called the real part of $z$, written as $\Re(z)=a$, and $b$ is called the imaginary part, written as $\Im(z)=b$. Even though the word “imaginary” sounds like it means “not real,” imaginary numbers are very useful and are used in physics, engineering, and signal processing.

For example, $3+2i$ is a complex number. Its real part is $3$ and its imaginary part is $2$. The number $-5i$ is also complex, because it can be written as $0-5i$. The number $7$ is complex too, since it can be written as $7+0i$.

This algebraic form is called the rectangular form or Cartesian form. It is the main form used for basic calculations because it behaves like an expanded version of ordinary algebra.

2. Adding and subtracting complex numbers

To add or subtract complex numbers, combine the real parts with real parts and the imaginary parts with imaginary parts. This works because $i$ behaves like a symbol that stays attached to its coefficient.

If $z_1=a+bi$ and $z_2=c+di$, then

$$z_1+z_2=(a+c)+(b+d)i$$

and

$$z_1-z_2=(a-c)+(b-d)i.$$

Example 1

Let $z_1=4+3i$ and $z_2=2-5i$.

Then

$$z_1+z_2=(4+2)+(3+(-5))i=6-2i.$$

Also,

$$z_1-z_2=(4-2)+(3-(-5))i=2+8i.$$

This is very similar to combining like terms in algebra. The key is that $1$ and $i$ are different types of parts, so they cannot be combined into a single term like $5i$ unless they already match.

Real-world connection

If a complex number represents a quantity with two parts, such as a horizontal and vertical component, then addition can represent combining two effects. For example, if one force is $3+2i$ and another is $1-4i$, the combined effect is $4-2i$. The algebra keeps track of both directions at once.

3. Multiplying complex numbers

Multiplication works using the distributive property, just like multiplying binomials. The main extra rule is $i^2=-1$.

If $z_1=a+bi$ and $z_2=c+di$, then

$$z_1z_2=(a+bi)(c+di).$$

Expanding gives

$$z_1z_2=ac+adi+bci+bdi^2.$$

Since $i^2=-1$, this becomes

$$z_1z_2=(ac-bd)+(ad+bc)i.$$

This formula is one of the most important in complex algebra.

Example 2

Multiply $z_1=2+3i$ and $z_2=4-i$.

First expand:

$$ (2+3i)(4-i)=8-2i+12i-3i^2. $$

Now use $i^2=-1$:

$$8+10i+3=11+10i.$$

So the product is $11+10i$.

Why this matters

Multiplication of complex numbers does more than just scale values. Later, when you study polar form, multiplication will connect to stretching and rotating points in the plane. For now, in rectangular form, you should focus on careful expansion and replacement of $i^2$ with $-1$.

4. Powers of $i$

The powers of $i$ repeat in a cycle of length $4$:

$$i^1=i,$$

$$i^2=-1,$$

$$i^3=-i,$$

$$i^4=1.$$

Then the pattern repeats:

$$i^5=i,$$

$$i^6=-1,$$

$$i^7=-i,$$

$$i^8=1.$$

This cycle makes simplification fast. Any higher power of $i$ can be reduced by checking the remainder when the exponent is divided by $4$.

Example 3

Simplify $i^{17}$.

Because $17=4\cdot4+1$, the remainder is $1$, so

$$i^{17}=i.$$

Example 4

Simplify $-2i^3+5i^2$.

Use $i^3=-i$ and $i^2=-1$:

$$-2(-i)+5(-1)=2i-5.$$

So the result is $-5+2i$.

This repeating cycle is a direct consequence of the rule $i^2=-1$, and it is a major reason complex algebra is manageable.

5. Complex conjugates and division

The complex conjugate of $z=a+bi$ is $\overline{z}=a-bi$. It changes the sign of the imaginary part while keeping the real part the same. Conjugates are extremely useful because multiplying a complex number by its conjugate removes the imaginary part.

For any $z=a+bi$,

$$z\overline{z}=(a+bi)(a-bi)=a^2+b^2.$$

Notice that the result is a real number and is always nonnegative.

Example 5

Let $z=3+4i$. Then its conjugate is $\overline{z}=3-4i$.

Multiply them:

$$ (3+4i)(3-4i)=9-12i+12i-16i^2. $$

Since $i^2=-1$,

$$9+16=25.$$

So $z\overline{z}=25$.

Division using conjugates

When dividing complex numbers, multiply top and bottom by the conjugate of the denominator. This clears the imaginary part from the denominator.

For example, simplify

$$\frac{2+3i}{1-2i}.$$

Multiply numerator and denominator by $1+2i$:

$$\frac{2+3i}{1-2i}\cdot\frac{1+2i}{1+2i}.$$

The denominator becomes

$$ (1-2i)(1+2i)=1+4=5. $$

The numerator becomes

$$ (2+3i)(1+2i)=2+4i+3i+6i^2=2+7i-6=-4+7i. $$

$$\frac{2+3i}{1-2i}=\frac{-4+7i}{5}=-\frac45+\frac75 i.$$

This method is standard in complex analysis because it keeps results in the form $a+bi$.

6. Algebra and the complex plane

Every complex number $z=a+bi$ can be shown as a point or vector in the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.

$a$ tells you how far to move horizontally
$b$ tells you how far to move vertically

So $3+2i$ is the point $(3,2)$.

This geometric view helps explain why algebra works the way it does. For example, the conjugate $a-bi$ is the reflection of $a+bi$ across the real axis. Also, the product with a conjugate gives $a^2+b^2$, which is linked to distance from the origin.

Example 6

If $z=-1+5i$, then the point is $(-1,5)$. Its conjugate is $-1-5i$, which is the reflected point $(-1,-5)$. These two points are mirror images across the horizontal axis.

Even though this lesson focuses on algebra, this plane picture is important because later modulus and argument will describe a complex number using distance and angle. The algebraic form $a+bi$ and the geometric form in the plane are two ways to describe the same object.

7. Common mistakes to avoid

Here are some frequent errors students make:

forgetting that $i^2=-1$ and leaving answers with $i^2$
combining real and imaginary parts incorrectly
multiplying without using the distributive property carefully
dividing by a complex number without using the conjugate
writing the conjugate as $-a-bi$ instead of $a-bi$

A good habit is to write each step clearly, especially when expanding products. This helps prevent sign errors.

Conclusion

Algebra of complex numbers is the starting point for understanding the whole topic of complex numbers and the complex plane. In rectangular form $a+bi$, complex numbers can be added, subtracted, multiplied, and divided using familiar algebra rules plus the special rule $i^2=-1$. The conjugate $a-bi$ makes division easier and reveals important structure. students, once you are comfortable with these operations, you will be ready to move on to modulus, argument, polar form, and roots 🚀

Study Notes

A complex number has the form $z=a+bi$, where $a, b\in\mathbb{R}$ and $i^2=-1$.
The real part is $\Re(z)=a$ and the imaginary part is $\Im(z)=b$.
Add and subtract complex numbers by combining real parts and imaginary parts separately.
Multiply using distributive expansion and replace $i^2$ with $-1$.
Powers of $i$ repeat in a cycle: $i, -1, -i, 1$.
The conjugate of $a+bi$ is $a-bi$.
Multiply by the conjugate to simplify divisions and remove imaginary parts from denominators.
Every complex number can be represented as a point $(a,b)$ in the complex plane.
Algebraic form is the foundation for modulus, argument, polar form, and roots.
Careful sign handling is essential for correct complex number calculations.