Number Systems

Real, rational and irrational numbers

The Real Number System

Real numbers include all rational and irrational numbers. • Natural numbers (ℕ): 1, 2, 3, ... • Whole numbers (ℕ₀): 0, 1, 2, 3, ... • Integers (ℤ): ..., −2, −1, 0, 1, 2, ... • Rational (ℚ): fractions a/b • Irrational: √2, π (non-repeating, non-terminating decimals)

Example

Classifying Numbers

Classify each number: • 7 → Natural, Whole, Integer, Rational, Real • −3 → Integer, Rational, Real • 2/5 → Rational, Real • √5 → Irrational, Real • 0.333... → Rational, Real (= 1/3)

Between Rational and Irrational

A rational number can be expressed as a/b. Its decimal either terminates (0.25) or recurs (0.333...). An irrational number CANNOT be written as a fraction — its decimal never terminates or repeats.

Note

Remember

ℕ ⊂ ℕ₀ ⊂ ℤ ⊂ ℚ ⊂ ℝ. Every natural number is also rational and real. √ of a non-perfect square is always irrational. π is irrational.

Key Vocabulary

Natural numbersCounting numbers: 1, 2, 3, ...

IntegerPositive and negative whole numbers including zero

Rational numberAny number expressible as a fraction a/b (b≠0)

Irrational numberA number that cannot be written as a fraction

Real numberAll rational and irrational numbers

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Natural numbers

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