5.2 Complex Numbers and Operations

Complex Number Representation

Complex numbers extend the real number system so that every polynomial equation has a solution. Without them, an equation like $x^2 + 1 = 0$ simply has no answer. With complex numbers, it has two: $x = i$ and $x = -i$.

This section covers how complex numbers are written, how to perform arithmetic with them, and how they connect back to quadratic equations with negative discriminants.

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Definition and Components

A complex number takes the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, defined by the property $i^2 = -1$.

• The real part is $a$
• The imaginary part is $b$ (just the coefficient, not $bi$)
• Examples: $3 + 2i$, $-4 - 7i$, $6i$ (here $a = 0$), and $5$ (here $b = 0$, so every real number is also a complex number)

Rectangular Form and the Complex Plane

In rectangular form $(a + bi)$, you can plot any complex number as a point on the complex plane (also called the Argand plane). The horizontal axis is the real axis, and the vertical axis is the imaginary axis. So $3 + 2i$ corresponds to the point $(3, 2)$.

The absolute value (or modulus) of a complex number is its distance from the origin, calculated with the Pythagorean theorem:

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

For example, $|3 + 4i| = \sqrt{9 + 16} = 5$.

Polar Form

In polar form, a complex number is written as $r(\cos\theta + i\sin\theta)$, where:

• $r$ is the modulus (distance from the origin)
• $\theta$ is the argument, the angle measured counterclockwise from the positive real axis

You can find $\theta$ using $\theta = \arctan(b/a)$, but you need to adjust based on which quadrant the point falls in. For instance, if both $a$ and $b$ are negative, the point is in Quadrant III, and you'd add $\pi$ (or 180°) to the arctangent result.

Complex Number Arithmetic

Addition and Subtraction

Add or subtract the real and imaginary parts separately:

$(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i$

Example: $(3 + 2i) + (4 - 5i) = (3 + 4) + (2 - 5)i = 7 - 3i$

Think of it like combining like terms: real with real, imaginary with imaginary.

Multiplication

Multiply complex numbers using the distributive property (FOIL), then apply $i^2 = -1$:

$(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$

Example: $(2 + 3i)(4 - i)$

1. Multiply: $2 \cdot 4 + 2 \cdot (-i) + 3i \cdot 4 + 3i \cdot (-i) = 8 - 2i + 12i - 3i^2$

2. Replace $i^2$ with $-1$: $8 - 2i + 12i - 3(-1) = 8 + 3 + 10i$

3. Result: $11 + 10i$

Conjugates

The conjugate of $a + bi$ is $a - bi$. You flip the sign of the imaginary part. The key property: multiplying a complex number by its conjugate always gives a real number.

$(a + bi)(a - bi) = a^2 + b^2$

This fact is what makes division possible.

Division

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. This eliminates the imaginary part from the denominator.

Example: $(3 + 4i) \div (1 - 2i)$

1. Multiply top and bottom by the conjugate of the denominator: $\frac{(3 + 4i)(1 + 2i)}{(1 - 2i)(1 + 2i)}$

2. Expand the numerator: $3 + 6i + 4i + 8i^2 = 3 + 10i - 8 = -5 + 10i$

3. Expand the denominator: $1 + 4 = 5$

4. Simplify: $\frac{-5 + 10i}{5} = -1 + 2i$

Complex Solutions to Quadratics

Quadratic Formula and Discriminant

The quadratic formula solves any equation of the form $ax^2 + bx + c = 0$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The discriminant is the expression under the square root: $b^2 - 4ac$. It tells you what kind of solutions to expect:

• Positive discriminant: two distinct real solutions
• Zero discriminant: one repeated real solution
• Negative discriminant: two complex conjugate solutions

When the discriminant is negative, you'll factor out $i$ from the square root of the negative number.

Example: Solve $x^2 + 2x + 5 = 0$

1. Identify $a = 1$, $b = 2$, $c = 5$

2. Compute the discriminant: $2^2 - 4(1)(5) = 4 - 20 = -16$

3. Apply the formula: $x = \frac{-2 \pm \sqrt{-16}}{2}$

4. Simplify $\sqrt{-16} = 4i$: $x = \frac{-2 \pm 4i}{2} = -1 \pm 2i$

The two solutions are $-1 + 2i$ and $-1 - 2i$. Notice they're conjugates of each other. This always happens when the coefficients of the quadratic are real numbers.

Sum and Product of Complex Solutions

For $ax^2 + bx + c = 0$, regardless of whether the solutions are real or complex:

• Sum of solutions $= -b/a$
• Product of solutions $= c/a$

These relationships come from Vieta's formulas and give you a quick way to check your work.

Example: For $x^2 + 2x + 5 = 0$, the solutions are $-1 + 2i$ and $-1 - 2i$.

• Sum: $(-1 + 2i) + (-1 - 2i) = -2$, which matches $-b/a = -2/1 = -2$ ✓
• Product: $(-1 + 2i)(-1 - 2i) = 1 + 4 = 5$, which matches $c/a = 5/1 = 5$ ✓

Geometric Interpretation of Complex Numbers

The Complex Plane (Argand Plane)

Every complex number $a + bi$ maps to a unique point $(a, b)$ on the complex plane. This gives you a visual way to think about complex arithmetic:

• The modulus $|a + bi|$ is the point's distance from the origin
• The argument $\theta$ is the angle from the positive real axis to the line connecting the origin to the point

Adding two complex numbers works just like vector addition: you're combining their horizontal and vertical components.

Geometric Transformations

Operations on complex numbers have clean geometric meanings:

• Multiplying by $i$ rotates a point 90° counterclockwise. For example, $i \cdot (2 + 3i) = 2i + 3i^2 = -3 + 2i$. The point $(2, 3)$ moves to $(-3, 2)$.
• Multiplying by $-i$ rotates a point 90° clockwise.
• Taking the conjugate reflects a point across the real axis. The conjugate of $2 + 3i$ is $2 - 3i$, so $(2, 3)$ reflects to $(2, -3)$.
• Multiplying by a real number $k$ scales the distance from the origin by a factor of $|k|$ (dilation). If $k$ is negative, it also rotates 180°.