Real numbers
Classify real numbers into natural, whole, integer, rational and irrational sets.
Objective
Classify real numbers into natural, whole, integer, rational and irrational sets.
What it means
The real numbers are built from nested sets: natural numbers (1,2,3,...) sit inside whole numbers (0,1,2,...), which sit inside integers (...,-1,0,1,...), which sit inside rational numbers (ratios of integers). Irrational numbers cannot be written as a ratio of integers and have non-repeating, non-terminating decimals.
Key terms
- Natural numbers
- Counting numbers 1, 2, 3, ...
- Whole numbers
- Natural numbers plus 0
- Integers
- Whole numbers and their negatives
- Rational number
- Can be written as a/b with integers a, b and b not 0
- Irrational number
- Cannot be written as a ratio of integers
Rules to remember
- Every natural number is also whole, integer, and rational.
- A number is rational if its decimal terminates or repeats.
- A square root of a non-perfect square is irrational.
- Pick the MOST SPECIFIC set that a number belongs to when asked to classify it.
Step by step
- 1Check for a perfect square or clean divisionIf a root simplifies to a whole number, it may be natural or whole.
- 2Look at the decimalTerminating or repeating means rational; non-repeating forever means irrational.
- 3Move up the chainNatural -> whole -> integer -> rational -> real, and stop at the smallest set that fits.
Common mistakes
- Calling every root irrational without checking for a perfect square.
- Forgetting that negative numbers can still be whole-number-based integers, not whole numbers themselves.
Tips
- Always simplify roots first.
- Remember 0 is whole but not natural.