L1 · The real number sets: natural, whole, integer, rational, irrational

Learning goal

“I can name the number sets, describe each one, and place any real number in the smallest set it belongs to - and justify the placement.”

Set — a collection of elements (objects). A set is written in braces: A = {2, 4, 6, 8}.

Rational number — a number that can be written as a/b, where a and b are integers and b ≠ 0. Every terminating or repeating decimal is rational. The set of rationals is Q.

Irrational number — a number that CANNOT be written as a ratio of two integers. Its decimal never terminates and never repeats. The set of irrationals is Q′.

Real number — every number on the number line: rationals and irrationals together. The set of reals is R.

Worked example

Build the chain with one number per region. Step 1: 4 is a natural number (N = {1, 2, 3, ...}), so 4 ∈ N. Step 2: 0 is not natural in our convention, but 0 joins the whole numbers - the natural numbers together with 0. Every natural number is also whole. Step 3: -6 is an integer (Z = {..., -2, -1, 0, 1, 2, ...}). Every whole number is an integer. Step 4: 3/4 is a rational number - it is already written as a ratio of integers. Every integer is rational too, because n = n/1. Step 5: √2 cannot be written as a ratio of integers, so √2 is irrational (Q′). Step 6: All five numbers are real numbers (R): R is Q and Q′ together. Say it plainly: the sets nest - N ⊆ whole ⊆ Z ⊆ Q ⊆ R, and Q′ ⊆ R. Q and Q′ share no numbers at all; together they fill R.

Check yourself — explain it

Why is every natural number automatically an integer, but not every integer a natural number?

Paired problem — example still visible

Does -11 belong to N? To Z? To R? -> No; yes; yes (-11 is a negative integer, and every integer is real).

Faded completion — last step is yours

"The number 0 belongs to these sets: whole numbers, ______, ______ and ______, but not to ______ or ______." -> integers (Z); rationals (Q); reals (R); naturals (N); irrationals (Q′)

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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