Overview
This lecture surveys the different types of numbers, their hierarchy, and their defining characteristics, emphasizing the distinction between real, imaginary, rational, and irrational numbers.
Types of Numbers
- All numbers can be categorized as either real or imaginary.
- Real numbers include all the 'ordinary' numbers you encounter daily.
- Imaginary numbers are defined by the square root of negative numbers, using the unit "i" (e.g., √-1 = i).
- The square root of negative four is 2i, since √-4 = √4 × √-1 = 2i.
Real Numbers: Rational and Irrational
- Real numbers are split into rational and irrational numbers.
- Rational numbers can be expressed as the ratio of two integers (e.g., 3/2, 93/604).
- Zero cannot be the denominator in a rational number.
- Decimals that repeat or terminate are rational (e.g., 1/3 = 0.333...).
- Irrational numbers cannot be written as a ratio of two integers.
- Irrational numbers have non-repeating, non-terminating decimals (e.g., √2, π).
- Examples: √2 and π are irrational but are real because they exist in nature (e.g., triangles, circles).
- Even with very close fractional approximations, irrational numbers can't be represented exactly by a ratio.
Repeating and Non-Repeating Decimals
- Repeating, non-terminating decimals (like 0.333... or 0.111...) are rational because they are fractions.
- 0.999... is exactly equal to 1, as there is no number between them.
Subcategories of Rational Numbers
- Rational numbers include fractions (numbers between integers).
- Integers are all whole numbers, positive, negative, and zero.
- Whole numbers are integers starting from zero upwards (0, 1, 2, 3...).
- Natural numbers are whole numbers starting from one (1, 2, 3...).
Key Terms & Definitions
- Real Numbers — Numbers that can be found on the number line, including both rational and irrational numbers.
- Imaginary Numbers — Numbers based on the imaginary unit "i," where i² = -1.
- Rational Numbers — Numbers that can be expressed as the ratio of two integers.
- Irrational Numbers — Numbers that cannot be written as a ratio of integers, with non-repeating, non-terminating decimals.
- Integer — Whole numbers including negatives, zero, and positives.
- Whole Numbers — Non-negative integers (0, 1, 2, ...).
- Natural Numbers — Positive integers (1, 2, 3, ...).
- Repeating Decimal — A decimal in which one or more digits repeat infinitely.
- Non-Terminating Decimal — A decimal that continues without end.
Action Items / Next Steps
- Review and be ready to describe differences between real/imaginary and rational/irrational numbers.
- Prepare for upcoming lessons on geometry, triangles, and circles.