Indietro1.4 Number Systems and Complex Numbers: Foundations and Operations
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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, , where (includes fractions and terminating or repeating decimals).
Irrational Numbers: Numbers whose decimal representations are neither terminating nor repeating (e.g., , , ).
Real Numbers: The union of rational and irrational numbers.
Example: (since ), but is irrational.
Imaginary and Complex Numbers
Some equations, such as , require numbers beyond the real number system. This leads to the introduction of imaginary and complex numbers.
Imaginary Unit (): Defined as .
Imaginary Numbers: Numbers of the form , where is a real number.
Complex Numbers: Numbers of the form , where and are real numbers. The standard form is .
Example: because .
Operations with Complex Numbers
Addition and Subtraction
To add or subtract complex numbers, combine like terms (real with real, imaginary with imaginary).
Rule:
Examples:
Multiplication of Complex Numbers
Multiply complex numbers as binomials using the FOIL method. Remember that .
Rule:
Examples:
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 , the conjugate is .
Rule:
Examples:
Square Roots of Negative Numbers
Express all square roots of negative numbers in terms of and simplify.
Rule: for
Examples:
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 | , , | Non-terminating, non-repeating decimals |
Real Numbers | \( \mathbb{R} \) | All above | Union of rational and irrational numbers |
Imaginary Numbers | None | Multiples of | |
Complex Numbers | \( \mathbb{C} \) | Numbers of the form |
Practice Problems and Solutions
Addition/Subtraction
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Multiplication
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Division
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Simplifying Square Roots and Operations
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Key Properties and Formulas
Imaginary Unit: ,
Complex Conjugate: For , conjugate is
Standard Form:
Multiplication:
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.
