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College Algebra · Chapter 1

Where Does This Number Live?

The number sets are nested rooms — a natural number is automatically whole, an integer, rational, and real, all at once. Pick a number, then step through the reasoning that walks it inward until it hits a door it cannot open.

The nested sets

REAL NUMBERS every single point on the number line RATIONAL a ratio a/b of integers, b ≠ 0 IRRATIONAL never a ratio of two integers INTEGERS …, −2, −1, 0, 1, 2, … WHOLE 0, 1, 2, 3, … NATURAL 1, 2, 3, … (counting)

On the number line

Pick a number to see its decimal expansion.
Nine numbers, six setstap a number to begin

Choose a number

How to read the diagramThe rooms are nested, so membership flows outward for free: land in NATURAL and you are also whole, an integer, rational, and real. It never flows inward — being real does not make you an integer. IRRATIONAL sits inside REAL but completely outside RATIONAL: a real number is one or the other, never both.
Convention used here: natural numbers are 1, 2, 3, … and whole numbers are 0, 1, 2, 3, … — so zero is whole but not natural. A few books fold 0 into the naturals instead; whenever it matters, check which convention your course uses.