Natural Numbers, Integers, Rational and Irrational Numbers

Learning outcome

Classify numbers correctly, recognise rational decimals and avoid false rules about irrational arithmetic.

Concepts and reasons

In this module, natural numbers mean 1, 2, 3, …; zero is excluded. Whole numbers are 0, 1, 2, 3, …. Integers include these and their negatives: …, -2, -1, 0, 1, 2, ….

A rational number can be written as p/q, where p and q are integers and q is not 0. Every integer is rational because n = n/1. A denominator of 0 is never allowed.

A terminating decimal is rational: its decimal places give a denominator of 10, 100, and so on. An eventually repeating decimal is also rational; multiplying by suitable powers of 10 and subtracting removes its repeating tail.

An irrational number cannot be written as such a fraction. Its decimal expansion neither terminates nor eventually repeats. Examples include √2 and π. Rational and irrational numbers together form the real numbers. An approximate decimal such as 1.414 is rational, even when used to estimate an irrational number.

Adding or subtracting rational numbers stays rational: a common denominator produces another fraction. However, sums and products of two irrational numbers need separate checking.

Worked examples

Example 1. Classify -7, 0, 4 and 3/5. The first three are integers and rational. Of these, 0 and 4 are whole numbers, but only 4 is natural. The number 3/5 is rational, not an integer. None is irrational.

Example 2. Let x = 0.272727…, with 27 repeating. Then 100x = 27.272727…. Subtracting gives 99x = 27, so x = 3/11.

Example 3. Irrational numbers can give rational results: √2 + (-√2) = 0 and √2 × √2 = 2. Other choices give irrational results: √2 + √2 = 2√2 and √2 × √3 = √6. No universal “always irrational” rule works.

Common traps

Do not classify every square root as irrational: √9 = 3. A minus sign does not make a number irrational. Different categories can overlap; being natural also means being an integer and rational.

Practice questions

1. List every applicable category introduced above for 1, -5/2 and √9.
2. Express 0.363636…, with 36 repeating, as a fraction.
3. Is 5 + √2 rational or irrational? Explain.

Answers and explanations

1. Both 1 and √9 = 3 are natural, whole, integer and rational. The number -5/2 is rational only among these categories. All three are real.
2. For x = 0.363636…, subtract x from 100x: 99x = 36. Therefore x = 36/99 = 4/11.
3. It is irrational. If 5 + √2 were rational, subtracting rational number 5 would make √2 rational, a contradiction.