Lesson 2.4: Complex Numbers

New Concept

Complex numbers extend the real number system by introducing the imaginary unit $i$, where $i^2 = -1$. This allows us to solve new equations and perform arithmetic—addition, subtraction, multiplication, and division—on numbers in the form $a+bi$.

What’s next

Next, you'll master these operations through a series of interactive examples and practice cards, starting with how to plot complex numbers visually.

Property

A complex number is a number of the form $a + bi$ where $a$ is the real part of the complex number, and $b$ is the imaginary part of the complex number.
If $b = 0$, then $a + bi$ is a real number. If $a = 0$ and $b$ is not equal to $0$, the complex number is called a pure imaginary number.
An imaginary number is an even root of a negative number. The imaginary number $i$ is defined as the square root of $-1$, so $\sqrt{-1} = i$ and $i^2 = -1$.
To express an imaginary number in standard form, write $\sqrt{-a}$ as $\sqrt{a}\sqrt{-1}$, which simplifies to $i\sqrt{a}$.

Examples

• Express $\sqrt{-16}$ in standard form. This is written as $\sqrt{16}\sqrt{-1}$, which simplifies to $4i$. In standard form, this is $0 + 4i$.

• Express $\sqrt{-75}$ in standard form. This becomes $\sqrt{75}\sqrt{-1} = \sqrt{25 \cdot 3}\sqrt{-1} = 5\sqrt{3}i$, or $0 + 5i\sqrt{3}$.

Property

In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis.
A complex number $a+bi$ is represented by the ordered pair $(a, b)$.
To plot the point, move $a$ units along the horizontal (real) axis and $b$ units along the vertical (imaginary) axis.

Examples

• Plot the complex number $5 + 3i$ on the complex plane. The real part is $5$ and the imaginary part is $3$, so we plot the ordered pair $(5, 3)$.

• Plot the complex number $-4 - 2i$ on the complex plane. The real part is $-4$ and the imaginary part is $-2$, so we plot the point $(-4, -2)$.

Property

To add or subtract complex numbers, combine the real parts and combine the imaginary parts.
Adding complex numbers:

Subtracting complex numbers:

Examples

• Add $(4 - 3i) + (1 + 6i)$. We combine the real parts $(4+1)$ and the imaginary parts $(-3+6)i$, which results in $5 + 3i$.

• Subtract $(9 + 2i) - (5 - 4i)$. We subtract the real parts $(9-5)$ and the imaginary parts $(2 - (-4))i$, which results in $4 + 6i$.

Property

To multiply two complex numbers, use the distributive property or the FOIL method, just as with binomials. The key difference is to replace $i^2$ with $-1$. The general form is:

Examples

• Multiply $3(4 + 7i)$. Distribute the real number to get $(3 \cdot 4) + (3 \cdot 7i)$, which simplifies to $12 + 21i$.

• Multiply $(3 + 2i)(5 - 4i)$. Using FOIL, we get $15 - 12i + 10i - 8i^2$. Since $i^2 = -1$, this becomes $15 - 2i - 8(-1) = 15 - 2i + 8 = 23 - 2i$.

Property

The complex conjugate of a complex number $a + bi$ is $a - bi$.
To divide complex numbers, write the problem as a fraction.
Then, multiply the numerator and denominator by the complex conjugate of the denominator to eliminate the imaginary number in the denominator.
Simplify the result into standard form $a+bi$.

Examples

• Find the complex conjugate of $-5 + 2i$. The real part stays the same and the sign of the imaginary part is changed, so the conjugate is $-5 - 2i$.

• Divide $(3 + 4i)$ by $(2 - i)$. Multiply the numerator and denominator by the conjugate $2+i$: $\frac{3+4i}{2-i} \cdot \frac{2+i}{2+i} = \frac{6+3i+8i+4i^2}{4-i^2} = \frac{2+11i}{5} = \frac{2}{5} + \frac{11}{5}i$.

Property

The powers of $i$ are cyclic, repeating every four powers:
$i^1 = i$
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$
To simplify $i^n$, divide the exponent $n$ by 4 and find the remainder $r$. The simplified form is $i^r$. If the remainder is 0, the result is $i^4=1$.

Examples

• Evaluate $i^{22}$. Since $22 \div 4 = 5$ with a remainder of $2$, we have $i^{22} = i^2 = -1$.

• Evaluate $i^{51}$. Since $51 \div 4 = 12$ with a remainder of $3$, we have $i^{51} = i^3 = -i$.