Complex Numbers in Precalculus: Definitions, Operations, and Examples

Complex Numbers

Introduction to Complex Numbers

Complex numbers extend the real number system to include solutions to equations that do not have real solutions, such as $x^2 + 1 = 0$. They are essential in precalculus for understanding the roots of quadratic equations and for performing various algebraic operations.

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

• Real Part: In $a + bi$, the real part is $a$.

• Imaginary Part: In $a + bi$, the imaginary part is $b$.

• Pure Imaginary Number: A complex number of the form $a + bi$ where $a = 0$ and $b \neq 0$.

• General Complex Number: $a + bi$ where $a \neq 0$ and $b \neq 0$.

The Complex-Number System

Some functions have zeros that are not real numbers. The complex-number system is used to find zeros of functions that are not real numbers. For example, if the graph of a function does not cross the x-axis, it has no real-number zeros, but it may have complex zeros.

• Imaginary Unit: $i$ is defined such that $i^2 = -1$.

• Square Roots of Negative Numbers: $\sqrt{-a} = i\sqrt{a}$ for $a > 0$.

Expressing Numbers in Terms of $i$

Negative square roots can be rewritten using the imaginary unit $i$.

• Example 1:

  • $\sqrt{-7} = \sqrt{-1} \cdot \sqrt{7} = i\sqrt{7}$

  • $\sqrt{-16} = \sqrt{-1} \cdot \sqrt{16} = i \cdot 4 = 4i$

  • $-\sqrt{-13} = -\sqrt{-1} \cdot \sqrt{13} = -i\sqrt{13}$

  • $\sqrt{-64} = \sqrt{-1} \cdot \sqrt{64} = i \cdot 8 = 8i$

  • $\sqrt{-48} = \sqrt{-1} \cdot \sqrt{48} = i \cdot \sqrt{16 \cdot 3} = i \cdot 4 \sqrt{3} = 4i\sqrt{3}$

Operations with Complex Numbers

Addition and Subtraction

Complex numbers obey the commutative, associative, and distributive laws. To add or subtract complex numbers, combine the real parts and the imaginary parts separately, similar to combining like terms in binomials.

• General Rule: $(a + bi) + (c + di) = (a + c) + (b + d)i$

• Example 2:

  • $(8 + 6i) + (3 + 2i) = (8 + 3) + (6i + 2i) = 11 + 8i$

  • $(4 + 5i) - (6 - 3i) = (4 - 6) + [5i - (-3i)] = -2 + 8i$

Multiplication

Multiplication of complex numbers uses the distributive property and the fact that $i^2 = -1$. Note that the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ only holds for non-negative real numbers.

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

• Example 3:

  • $\sqrt{-16} \cdot \sqrt{-25} = (i \cdot 4) \cdot (i \cdot 5) = i^2 \cdot 20 = -20$

  • $(1 + 2i)(1 + 3i) = 1 + 3i + 2i + 6i^2 = 1 + 5i - 6 = -5 + 5i$

  • $(3 - 7i)^2 = 3^2 - 2 \cdot 3 \cdot 7i + (7i)^2 = 9 - 42i + 49i^2 = 9 - 42i - 49 = -40 - 42i$

Simplifying Powers of $i$

Powers of $i$ cycle through four values: $i$, $-1$, $-i$, and $1$.

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

• Example: $i^7 = i^{4 \cdot 1 + 3} = (i^4)^1 \cdot i^3 = 1 \cdot (-i) = -i$

Complex Conjugates

Definition and Properties

The conjugate of a complex number $a + bi$ is $a - bi$. The product of a complex number and its conjugate is always a real number.

• Examples of Conjugate Pairs:

  • $3 + 7i$ and $3 - 7i$

  • $14 - 5i$ and $14 + 5i$

  • $8i$ and $-8i$

• Product Formula: $(a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2 i^2 = a^2 + b^2$

• Example 5:

  • $(5 + 7i)(5 - 7i) = 25 - 49i^2 = 25 + 49 = 74$

  • $(8i)(-8i) = -64i^2 = 64$

Division of Complex Numbers

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

• General Formula: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$

• Example 6:

  • Divide $2 - 5i$ by $1 - 6i$:

  • $\frac{2 - 5i}{1 - 6i} \cdot \frac{1 + 6i}{1 + 6i} = \frac{(2 - 5i)(1 + 6i)}{(1 - 6i)(1 + 6i)}$

  • Numerator: $(2 - 5i)(1 + 6i) = 2 \cdot 1 + 2 \cdot 6i - 5i \cdot 1 - 5i \cdot 6i = 2 + 12i - 5i - 30i^2 = 2 + 7i + 30$ (since $i^2 = -1$)

  • Denominator: $(1 - 6i)(1 + 6i) = 1^2 - (6i)^2 = 1 - 36i^2 = 1 + 36 = 37$

  • Final answer: $\frac{32 + 7i}{37}$

Summary Table: Types of Complex Numbers

Type	Form	Condition
General Complex Number	$a + bi$	$a \neq 0$, $b \neq 0$
Pure Imaginary Number	$a + bi$	$a = 0$, $b \neq 0$
Real Number	$a + 0i$	$b = 0$

Additional info: These notes cover the foundational concepts of complex numbers as required for precalculus, including their definition, arithmetic operations, powers, conjugates, and division. Mastery of these topics is essential for solving quadratic equations with non-real solutions and for further study in mathematics.