Introduction to Complex Numbers

students, imagine trying to solve a square root problem like $x^2=-1$. In the real-number system, no number works because every real square is $0$ or positive. Yet mathematics does not stop there 😊. Complex numbers extend the number system so that equations like this can be solved. This lesson introduces the key ideas, notation, and basic operations for complex numbers, which are essential in IB Mathematics: Analysis and Approaches HL.

What is a Complex Number?

A complex number is written in the form $z=a+bi$, where $a$ and $b$ are real numbers and $i$ is defined by $i^2=-1$. The number $a$ is called the real part of $z$, written as $\operatorname{Re}(z)=a$, and $b$ is called the imaginary part of $z$, written as $\operatorname{Im}(z)=b$. Even though the term “imaginary” may sound like it means “not real,” complex numbers are fully valid mathematical objects and are used in engineering, physics, and computer science.

If $b=0$, then $z=a+0i=a$, so real numbers are a subset of complex numbers. If $a=0$, then $z=bi$ is called a purely imaginary number. For example, $3-2i$ is a complex number with $\operatorname{Re}(3-2i)=3$ and $\operatorname{Im}(3-2i)=-2$.

The symbol $i$ is the key new idea. Since $i^2=-1$, the powers of $i$ repeat in a pattern:

$$i^1=i, \quad i^2=-1, \quad i^3=-i, \quad i^4=1,$$

and then the cycle repeats every $4$ powers. This pattern makes simplifying expressions with $i$ much easier.

Why Complex Numbers Are Needed

In the real-number system, some equations have no solutions. For example, $x^2+1=0$ can be rewritten as $x^2=-1$. No real number squared gives $-1$. Complex numbers solve this problem by introducing $i$, since $i^2=-1$.

This idea is not just about one equation. Many quadratic equations that seem to have no real solutions do have complex solutions. For example, using the quadratic formula on $x^2+4x+13=0$ gives

$$x=\frac{-4\pm\sqrt{4^2-4(1)(13)}}{2(1)}=\frac{-4\pm\sqrt{-36}}{2}=-2\pm 3i.$$

So the solutions are complex numbers. This is important because it shows that the algebraic tools students already know still work, even when the solutions are not real.

Complex numbers also help complete the picture of polynomial equations. In IB Mathematics, you often study how equations can have two, three, or more roots. Complex numbers allow a richer and more complete understanding of these roots.

Arithmetic with Complex Numbers

To add or subtract complex numbers, combine real parts and imaginary parts separately. If $z_1=a+bi$ and $z_2=c+di$, then

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

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

For example,

$$ (3-2i)+(5+7i)=8+5i,$$

and

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

Multiplication uses distribution and the rule $i^2=-1$. For instance,

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

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

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

A common mistake is to forget that $i^2=-1$. Always simplify powers of $i$ carefully.

Division is done by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of $a+bi$ is $a-bi$. For example,

$$\frac{3+2i}{1-4i}$$

is simplified by multiplying top and bottom by $1+4i$:

$$\frac{3+2i}{1-4i}\cdot\frac{1+4i}{1+4i}=\frac{(3+2i)(1+4i)}{1+16}.$$

Expanding the numerator gives

$$3+12i+2i+8i^2=3+14i-8=-5+14i,$$

so the result is

$$\frac{-5+14i}{17}=-\frac{5}{17}+\frac{14}{17}i.$$

This method is powerful because it removes complex numbers from the denominator.

The Complex Plane and Modulus

Complex numbers can be represented on a plane called the Argand diagram. The horizontal axis shows the real part and the vertical axis shows the imaginary part. The complex number $z=a+bi$ is represented by the point $(a,b)$.

For example, $z=2-3i$ is the point $(2,-3)$. This geometric view helps students see complex numbers as more than symbols. It also makes operations like addition easier to visualize: adding complex numbers is like adding vectors.

The modulus of $z=a+bi$ is the distance from the origin to the point $(a,b)$, given by

$$|z|=\sqrt{a^2+b^2}.$$

So for $z=3-4i$,

$$|z|=\sqrt{3^2+(-4)^2}=\sqrt{25}=5.$$

The modulus is always non-negative. It measures how far the complex number is from the origin.

The conjugate of $z=a+bi$ is $\bar{z}=a-bi$. On the Argand diagram, the conjugate is the reflection of $z$ across the real axis. Also,

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

This identity is very useful in simplifying fractions and proving relationships.

Powers of $i$ and Simplifying Expressions

Since powers of $i$ repeat every $4$, you can simplify any power by looking at the remainder when the exponent is divided by $4$. For example, to find $i^{37}$, notice that $37=4\cdot 9+1$, so

$$i^{37}=i^{4\cdot 9+1}=(i^4)^9i=1^9i=i.$$

Similarly,

$$i^{202}$$

can be simplified by noting that $202=4\cdot 50+2$, so

$$i^{202}=i^2=-1.$$

This pattern appears often in IB questions and saves time.

You may also need to simplify expressions such as

$$\frac{i^7-2i^5}{i^2}.$$

Using the cycle,

$$i^7=i^4i^3=-i,$$

$$i^5=i^4i=i,$$

and $i^2=-1$. Therefore,

$$\frac{i^7-2i^5}{i^2}=\frac{-i-2i}{-1}=3i.$$

Careful organization of the powers is a strong exam skill.

Complex Numbers in Solving Equations

Complex numbers are often used to solve quadratic equations with negative discriminants. If a quadratic equation has a discriminant $b^2-4ac<0$, then the solutions are complex.

For example, consider

$$x^2-2x+5=0.$$

Using the quadratic formula,

$$x=\frac{2\pm\sqrt{(-2)^2-4(1)(5)}}{2}=\frac{2\pm\sqrt{-16}}{2}=1\pm 2i.$$

These are two distinct complex roots. In IB Mathematics, this links algebraic methods to number systems: even when there are no real solutions, the equation still has solutions in the complex number system.

Another important pattern is that if a polynomial has real coefficients and one root is $a+bi$, then $a-bi$ is also a root. These are called complex conjugate roots. For example, if $1+2i$ is a root of a real polynomial, then $1-2i$ must also be a root. This happens because the coefficients are real, so complex roots come in conjugate pairs.

Why This Topic Matters in Number and Algebra

Introduction to complex numbers fits naturally into Number and Algebra because it extends the idea of number systems. In earlier study, real numbers are used for arithmetic, algebra, and graphs. Complex numbers build on that foundation and allow you to solve more equations, understand more polynomial behavior, and work with more advanced symbolic expressions.

This topic also supports later work in sequences, series, and proof. For example, repeating patterns in powers of $i$ are a simple example of modular thinking and cyclic behavior. In proof, the identity $z\bar{z}=|z|^2$ can be shown algebraically, and the conjugate root theorem can be reasoned from polynomial coefficients. These habits of careful reasoning are central to IB Mathematics: Analysis and Approaches HL.

Complex numbers also strengthen algebraic fluency. You will often need to simplify expressions, factor polynomials, or solve equations accurately. The structure of complex numbers gives more tools for these tasks.

Conclusion

students, complex numbers expand mathematics beyond the real number system. A complex number has the form $a+bi$, where $i^2=-1$. You have learned how to add, subtract, multiply, and divide complex numbers, how to represent them on the Argand diagram, and how to find modulus and conjugates. You have also seen why complex numbers are needed to solve equations such as $x^2+1=0$ and how they fit into the broader study of Number and Algebra. This topic is a foundation for more advanced algebraic reasoning and a powerful example of how mathematics creates useful systems to solve problems.

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 $\operatorname{Re}(z)=a$ and the imaginary part is $\operatorname{Im}(z)=b$.
Real numbers are a subset of complex numbers: $a=a+0i$.
Powers of $i$ repeat in the cycle $i, -1, -i, 1$.
Add and subtract complex numbers by combining real and imaginary parts separately.
Multiply using distribution and simplify with $i^2=-1$.
Divide by multiplying top and bottom by the conjugate of the denominator.
The conjugate of $a+bi$ is $a-bi$.
The modulus is $|z|=\sqrt{a^2+b^2}$.
On the Argand diagram, $a$ is the horizontal coordinate and $b$ is the vertical coordinate.
Quadratic equations with negative discriminants have complex solutions.
If a polynomial has real coefficients, non-real complex roots occur in conjugate pairs.
Complex numbers extend Number and Algebra by making the system of solutions more complete.