The real number system organizes the numbers used to measure, count, compare, and calculate in everyday math. It includes whole-number counting values, negative numbers, fractions, decimals, square roots, and many constants from geometry and science. Seeing these numbers as nested sets helps you classify them quickly and understand which properties they share.
This matters because algebra, geometry, statistics, and calculus all rely on choosing the right kind of number for a situation.
The largest set is the real numbers, written ℝ, which contains every number that can be placed on a number line. Real numbers split into rational numbers, written ℚ, and irrational numbers, which cannot be written as a ratio of two integers. Inside the rational numbers are integers, written ℤ, and inside the integers are whole numbers and natural numbers, depending on the convention used.
A Venn or nesting diagram shows that every natural number is an integer, every integer is rational, and every rational number is real, but the reverse statements are not always true.
Key Facts
- Real numbers ℝ are all numbers that can be located on a number line.
- Rational numbers ℚ can be written as a/b, where a and b are integers and b ≠ 0.
- Integers ℤ include ..., -3, -2, -1, 0, 1, 2, 3, ...
- Natural numbers are the counting numbers 1, 2, 3, 4, ... and sometimes whole-number sets include 0.
- Irrational numbers cannot be written as a/b and have nonterminating, nonrepeating decimal forms.
- Set nesting: Natural numbers ⊂ Integers ℤ ⊂ Rational numbers ℚ ⊂ Real numbers ℝ.
Vocabulary
- Real number
- A real number is any number that can be represented as a point on the number line.
- Rational number
- A rational number is any number that can be written as a fraction a/b with integers a and b and b not equal to zero.
- Irrational number
- An irrational number is a real number that cannot be written as a ratio of two integers.
- Integer
- An integer is a whole number that may be positive, negative, or zero.
- Natural number
- A natural number is a counting number such as 1, 2, 3, and so on.
Common Mistakes to Avoid
- Calling every decimal irrational is wrong because terminating decimals and repeating decimals are rational. For example, 0.75 = 3/4 and 0.333... = 1/3.
- Assuming negative numbers cannot be rational is wrong because rational numbers can be negative. For example, -5 = -5/1 and -2.4 = -12/5.
- Putting integers outside the rational numbers is wrong because every integer can be written with denominator 1. For example, 7 = 7/1.
- Treating π as equal to 3.14 is wrong because 3.14 is only a rational approximation. The exact value of π is irrational and its decimal never terminates or repeats.
Practice Questions
- 1 Classify each number as natural, integer, rational, irrational, and real where applicable: -8, 0, 5/6, √49, √2, 3.14159.
- 2 Write each number as a ratio of two integers if possible: 0.6, -4, 1.25, 0.272727..., √5.
- 3 Explain why every integer is rational, but not every rational number is an integer. Give one example that supports each part of the statement.