1.4 Number Systems and Complex Numbers: Foundations and Operations

Number Systems in Algebra

Introduction to Number Sets

Understanding the different sets of numbers is fundamental in algebra. Each set expands upon the previous to accommodate new mathematical needs.

• Natural Numbers: The set of counting numbers: {1, 2, 3, 4, 5, ...}.

• Whole Numbers: Natural numbers plus zero: {0, 1, 2, 3, ...}.

• Integers: Whole numbers and their negatives: {..., -3, -2, -1, 0, 1, 2, 3, ...}.

• Rational Numbers: Numbers that can be expressed as a quotient of two integers, $\frac{a}{b}$, where $b \neq 0$ (includes fractions and terminating or repeating decimals).

• Irrational Numbers: Numbers whose decimal representations are neither terminating nor repeating (e.g., $\pi$, $e$, $\sqrt{3}$).

• Real Numbers: The union of rational and irrational numbers.

Example: $\sqrt{49} = 7$ (since $7^2 = 49$), but $\sqrt{3} \approx 1.73205...$ is irrational.

Imaginary and Complex Numbers

Some equations, such as $\sqrt{-4}$, require numbers beyond the real number system. This leads to the introduction of imaginary and complex numbers.

• Imaginary Unit ($i$): Defined as $i = \sqrt{-1}$.

• Imaginary Numbers: Numbers of the form $bi$, where $b$ is a real number.

• Complex Numbers: Numbers of the form $a + bi$, where $a$ and $b$ are real numbers. The standard form is $a + bi$.

Example: $\sqrt{-4} = 2i$ because $\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i$.

Operations with Complex Numbers

Addition and Subtraction

To add or subtract complex numbers, combine like terms (real with real, imaginary with imaginary).

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

Examples:

• $(5 - 2i) + (3 + 3i) = (5 + 3) + (-2i + 3i) = 8 + i$

• $(2 + 6i) - (12 - i) = (2 - 12) + (6i - (-i)) = -10 + 7i$

Multiplication of Complex Numbers

Multiply complex numbers as binomials using the FOIL method. Remember that $i^2 = -1$.

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

Examples:

• $7i(2 - 9i) = 14i - 63i^2 = 14i + 63 = 63 + 14i$

• $(5 + 4i)(6 - 7i) = 30 + 24i - 35i - 28i^2 = 30 - 11i + 28 = 58 - 11i$

Division of Complex Numbers

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

• Complex Conjugate: For $a + bi$, the conjugate is $a - bi$.

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

Examples:

• $\frac{5i}{7 + i} \times \frac{7 - i}{7 - i} = \frac{35i - 5i^2}{49 - i^2} = \frac{35i + 5}{50} = \frac{1}{10} + \frac{7}{10}i$

• $\frac{5 + 4i}{4 - i} \times \frac{4 + i}{4 + i} = \frac{20 + 5i + 16i + 4i^2}{16 + 1} = \frac{16 + 21i}{17}$

Square Roots of Negative Numbers

Express all square roots of negative numbers in terms of $i$ and simplify.

• Rule: $\sqrt{-a} = i\sqrt{a}$ for $a > 0$

Examples:

• $\sqrt{-27} + \sqrt{-48} = i\sqrt{27} + i\sqrt{48} = (3i + 4i)\sqrt{3} = 7i\sqrt{3}$

• $(-2 + \sqrt{-3})^2 = (-2 + i\sqrt{3})^2 = 4 - 4i\sqrt{3} + 3i^2 = 1 - 4i\sqrt{3}$

• $\sqrt{-25} \cdot \sqrt{-4} = (5i)(2i) = 10i^2 = -10$

Summary Table: Number Sets

Set Name	Symbol	Examples	Description
Natural Numbers	\( \mathbb{N} \)	1, 2, 3, ...	Counting numbers
Whole Numbers	\( \mathbb{W} \)	0, 1, 2, 3, ...	Natural numbers plus zero
Integers	\( \mathbb{Z} \)	..., -2, -1, 0, 1, 2, ...	Whole numbers and negatives
Rational Numbers	\( \mathbb{Q} \)	\( \frac{1}{2}, -3, 0.75 \)	Fractions, terminating/repeating decimals
Irrational Numbers	None	$\pi$, $e$, $\sqrt{2}$	Non-terminating, non-repeating decimals
Real Numbers	\( \mathbb{R} \)	All above	Union of rational and irrational numbers
Imaginary Numbers	None	$i, 2i, -5i$	Multiples of $i$
Complex Numbers	\( \mathbb{C} \)	$2 + 3i, -1 - 4i$	Numbers of the form $a + bi$

Practice Problems and Solutions

Addition/Subtraction

• #2: $(-2 + 6i) + (4 - i) = 2 + 5i$

• #4: $(-7 + 5i) - (-9 - 11i) = 2 + 16i$

• #6: $7 - (-9 + 2i) - (-17 - i) = 33 - i$

• #8: $15i - (12 - 11i) = -12 + 26i$

Multiplication

• #10: $-8i(2i - 7) = 16 + 56i$

• #12: $(-4 - 8i)(3 + i) = -4 - 28i$

• #14: $(8 - 4i)(-3 + 9i) = 12 - 84i$

• #16: $(2 + 7i)(2 - 7i) = 53$

• #18: $(-7 - i)(-7 + i) = 50$

• #20: $(5 - 2i)^2 = 21 - 20i$

Division

• #22: $\frac{3}{4 + i} \times \frac{4 - i}{4 - i} = \frac{12 - 3i}{17}$

• #24: $\frac{5i}{2 - i} \times \frac{2 + i}{2 + i} = \frac{-1 + 2i}{5}$

• #28: $\frac{3 - 4i}{4 + 3i} \times \frac{4 - 3i}{4 - 3i} = -i$

Simplifying Square Roots and Operations

• #32: $\sqrt{-54} = 3i\sqrt{6}$

• #38: $\sqrt{-81} - \sqrt{-144} = -3i$

• #40: $5\sqrt{8} + 3\sqrt{-18} = (10 + 9i)\sqrt{2}$

• #42: $(-5 - \sqrt{-9})^2 = 16 + 30i$

• #44: $(-2 + \sqrt{-11})^2 = -7 - 4i\sqrt{11}$

• #46: $\frac{-12 + \sqrt{-28}}{32} = -\frac{3}{8} + \frac{i\sqrt{7}}{16}$

• #50: $\sqrt{-12}(\sqrt{-4} - \sqrt{2}) = -4 - 2i\sqrt{6}$

• #52: $3\sqrt{-7}[2\sqrt{-8}] = -12\sqrt{14}$

Key Properties and Formulas

• Imaginary Unit: $i = \sqrt{-1}$, $i^2 = -1$

• Complex Conjugate: For $a + bi$, conjugate is $a - bi$

• Standard Form: $a + bi$

• Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

• Division: Multiply numerator and denominator by the conjugate of the denominator

Additional info:

• Practice problems and worksheets are available in Canvas Modules for further study.

• For more help, use MyMathLab resources, tutoring centers, or contact your instructor.