Number Systems

5 minQuiz at the end

The Number System Hierarchy

Numbers are organised into nested sets, each building on the previous:

Real Numbers
 β”œβ”€β”€ Rational Numbers (fractions, decimals that terminate or repeat)
 β”‚    β”œβ”€β”€ Integers (…-3, -2, -1, 0, 1, 2, 3…)
 β”‚    β”‚    β”œβ”€β”€ Whole Numbers (0, 1, 2, 3…)
 β”‚    β”‚    β”‚    └── Natural Numbers (1, 2, 3…)
 └── Irrational Numbers (√2, Ο€, e…)

Definitions

Natural numbers (β„•): positive counting numbers: 1, 2, 3, 4… Whole numbers (W): natural numbers + zero: 0, 1, 2, 3… Integers (β„€): whole numbers + negatives: …-2, -1, 0, 1, 2… Rational numbers (β„š): any number expressible as p/q (where p, q are integers, qβ‰ 0)

  • Includes: fractions (3/4), terminating decimals (0.75), repeating decimals (0.333…) Irrational numbers: cannot be expressed as p/q
  • Examples: √2, √5, Ο€, e
  • Their decimal expansions never terminate or repeat

Real numbers (ℝ): all rational and irrational numbers together

Key Points

  • Every natural number is also a whole number, integer, rational, and real number.
  • -3 = -3/1, so integers are rational numbers.
  • √9 = 3 (rational), but √5 β‰ˆ 2.2360679… (irrational β€” never repeats).