The Imaginary Unit and Complex Numbers

This lesson introduces complex numbers, one of the most important extensions of the number system you will meet in Further Mathematics. Complex numbers arise naturally when we try to solve equations that have no real solutions, and they have far-reaching applications in pure mathematics, physics, and engineering.

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Why Do We Need Complex Numbers?

Consider the equation:

$x^2 + 1 = 0$x2+1=0

Rearranging gives $x^2 = -1$x2=−1. There is no real number whose square is negative, so this equation has no real roots.

To resolve this, mathematicians introduced the imaginary unit $i$i, defined by:

$i^2 = -1 \quad \text{or equivalently} \quad i = \sqrt{-1}$i2=−1or equivalentlyi=−1​

Key Point: The symbol $i$i is not a real number. It is a new kind of number that, when squared, gives $-1$−1.

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Definition of a Complex Number

A complex number $z$z is a number of the form:

$z = a + bi$z=a+bi

where:

• $a$a is the real part, written $\text{Re}(z) = a$Re(z)=a
• $b$b is the imaginary part, written $\text{Im}(z) = b$Im(z)=b
• Both $a$a and $b$b are real numbers

Example	Real part	Imaginary part
$3 + 2i$3+2i	3	2
$-1 + 4i$−1+4i	$-1$−1	4
$5$5	5	0
$-3i$−3i	0	$-3$−3

Important: The imaginary part is the real coefficient of $i$i. For $z = 3 + 2i$z=3+2i, the imaginary part is $2$2, not $2i$2i.

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The Set of Complex Numbers

The set of all complex numbers is denoted $\mathbb{C}$C. Every real number is a complex number (with imaginary part 0), so $\mathbb{R} \subset \mathbb{C}$R⊂C.

The number system hierarchy is:

$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$N⊂Z⊂Q⊂R⊂C

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Powers of i

The powers of $i$i cycle with period 4:

Power	Value
$i^0$i0	1
$i^1$i1	$i$i
$i^2$i2	$-1$−1
$i^3$i3	$-i$−i
$i^4$i4	1
$i^5$i5	$i$i

To find $i^n$in for any positive integer $n$n, divide $n$n by 4 and look at the remainder:

• Remainder 0: $i^n = 1$in=1
• Remainder 1: $i^n = i$in=i
• Remainder 2: $i^n = -1$in=−1
• Remainder 3: $i^n = -i$in=−i

Worked Example: Find $i^{23}$i23.

$23 = 4 \times 5 + 3$23=4×5+3, so the remainder is 3.

$i^{23} = i^3 = -i$i23=i3=−i

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Square Roots of Negative Numbers

Using $i$i, we can write the square root of any negative number:

$\sqrt{-k} = \sqrt{k} \cdot i \quad \text{for } k > 0$−k​=k​⋅ifor k>0

Examples:

• $\sqrt{-9} = 3i$−9​=3i
• $\sqrt{-2} = \sqrt{2}\, i$−2​=2​i
• $\sqrt{-25} = 5i$−25​=5i

Exam Tip: Be careful with notation. Always write $\sqrt{2}\, i$2​i rather than $\sqrt{2i}$2i​ to avoid ambiguity.

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Equality of Complex Numbers

Two complex numbers are equal if and only if their real parts are equal AND their imaginary parts are equal:

$a + bi = c + di \quad \Leftrightarrow \quad a = c \text{ and } b = d$a+bi=c+di⇔a=c and b=d

This is called equating real and imaginary parts and is a fundamental technique in complex number problems.

Worked Example: Find real numbers $x$x and $y$y such that $(x + 2) + (3y - 1)i = 5 + 8i$(x+2)+(3y−1)i=5+8i.

Equating real parts: $x + 2 = 5$x+2=5, so $x = 3$x=3.

Equating imaginary parts: $3y - 1 = 8$3y−1=8, so $3y = 9$3y=9 and $y = 3$y=3.

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Solving Quadratic Equations with Complex Roots

For a quadratic $ax^2 + bx + c = 0$ax2+bx+c=0 with discriminant $\Delta = b^2 - 4ac < 0$Δ=b2−4ac<0:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-b \pm \sqrt{-(4ac - b^2)}}{2a} = \frac{-b \pm i\sqrt{4ac - b^2}}{2a}$x=2a−b±b2−4ac​​=2a−b±−(4ac−b2)​​=2a−b±i4ac−b2​​

Worked Example: Solve $z^2 + 2z + 5 = 0$z2+2z+5=0.

$\Delta = 4 - 20 = -16$Δ=4−20=−16

$z = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i$z=2−2±−16​​=2−2±4i​=−1±2i

The roots are $z = -1 + 2i$z=−1+2i and $z = -1 - 2i$z=−1−2i.

Notice the roots come in a conjugate pair — this always happens when the coefficients of the quadratic are real.

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Purely Real and Purely Imaginary Numbers

• A complex number with $b = 0$b=0 is purely real: $z = a$z=a
• A complex number with $a = 0$a=0 (and $b \neq 0$b=0) is purely imaginary: $z = bi$z=bi
• The number $0 = 0 + 0i$0=0+0i is both purely real and purely imaginary.

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Practice Problems

1. Express $\sqrt{-50}$−50​ in terms of $i$i.
2. Find $i^{102}$i102.
3. Solve $z^2 - 4z + 13 = 0$z2−4z+13=0.
4. Find real numbers $p$p and $q$q such that $(2p + 1) + (q - 3)i = 7 - i$(2p+1)+(q−3)i=7−i.
5. Determine whether $z = 3i$z=3i is purely real, purely imaginary, or neither.

Solutions:

1. $\sqrt{-50} = \sqrt{50} \cdot i = 5\sqrt{2}\, i$−50​=50​⋅i=52​i
2. $102 = 4 \times 25 + 2$102=4×25+2, so $i^{102} = i^2 = -1$i102=i2=−1
3. $\Delta = 16 - 52 = -36$Δ=16−52=−36, so $z = \frac{4 \pm 6i}{2} = 2 \pm 3i$z=24±6i​=2±3i
4. $2p + 1 = 7$2p+1=7 gives $p = 3$p=3; $q - 3 = -1$q−3=−1 gives $q = 2$q=2
5. Purely imaginary (real part is 0, imaginary part is 3)

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Summary

• The imaginary unit $i$i satisfies $i^2 = -1$i2=−1.
• A complex number has the form $z = a + bi$z=a+bi where $a, b \in \mathbb{R}$a,b∈R.
• $\text{Re}(z) = a$Re(z)=a and $\text{Im}(z) = b$Im(z)=b (both real numbers).
• Powers of $i$i repeat with period 4: $1, i, -1, -i, 1, \ldots$1,i,−1,−i,1,…
• Two complex numbers are equal iff their real and imaginary parts are respectively equal.
• Quadratics with real coefficients and negative discriminant have complex conjugate roots.

Exam Tip: Equating real and imaginary parts is one of the most frequently used techniques. Practise it until it becomes automatic.