Number System — Formula Sheet

Irrational numbers were first discovered by the Pythagoreans around 400 BC. Hippacus of Croton discovered that √2 is irrational.

Examples of irrational numbers

√2, √3, √5, √7, √15, π, 0.10110111011110...

√4 = 2 (rational), √9 = 3 (rational) — perfect square roots are rational, non-perfect square roots are irrational.

TERMINATING DECIMAL

The decimal expansion ends (remainder becomes 0). Example: 7/8 = 0.875

NON-TERMINATING RECURRING (REPEATING)

The decimal goes on forever but a block of digits repeats. Example: 1/7 = 0.142857142857... = 0.̄1̄4̄2̄8̄5̄7̄

NON-TERMINATING NON-RECURRING

The decimal goes on forever and NO block of digits repeats. Example: √2 = 1.4142135..., π = 3.14159...

Rational + Irrational = Irrational (e.g., 2 + √3 is irrational)

Rational × Irrational (non-zero) = Irrational (e.g., 2√3 is irrational)

Irrational ± Irrational = May be Rational or Irrational (e.g., √6 + (-√6) = 0, which is rational; but √2 + √3 is irrational)

Examples include √2, √3, √5, π, and 0.