Classifying numbers: simplify first, then use the definitions
Before you start: Read negative numbers, fractions, square roots and decimal notation.
The nested number families
This activity uses natural for the positive counting numbers 1, 2, 3, … . Some textbooks include zero in that word, so state the convention. Whole numbers add zero. Integers also include negative whole numbers, such as −3. Every natural number is whole, and every whole number is an integer.
Rational and irrational numbers
A rational number can be written a/b with integer a and b and b ≠ 0. Thus −3 = −3/1 is rational, 0.75 = 3/4 is rational, and the repeating decimal 0.333… = 1/3 is rational. An irrational number, such as √2 or π, cannot be written that way. Its decimal neither terminates nor eventually repeats. Both rational and irrational numbers are real numbers.
Worked example: appearances can mislead
Classify √49. Since 7 × 7 = 49 and the square-root symbol denotes the nonnegative root, √49 = 7. Its most specific family here is natural, even though it was written with a radical. It also belongs to the whole, integer, rational and real sets.
Now simplify −12/4 = −3. Its smallest matching family is integer. It is also rational and real. The fraction's written form does not prevent it from being an integer.
Why a repeating decimal is rational
Let x = 0.272727… . Then 100x = 27.272727… . Subtract x to cancel the repeating tail: 99x = 27, so x = 27/99 = 3/11. In contrast, “does not end” alone does not imply irrationality: the repeating pattern matters.
A denominator of zero is excluded from the definition: 5/0 is undefined, not an irrational number. Also, √(−1) is not a real number and falls outside this activity's real-number classification.
Try it yourself
Give the most specific family for 0, √36, −8/2 and 0.121212… using this activity's convention.
Show answer and explanation
0: whole; √36 = 6: natural; −8/2 = −4: integer; 0.121212…: rational. For the last number, x = 0.121212… gives 99x = 12, so x = 4/33.
Use the activity to check your understanding
Say both the smallest matching family and one larger family. Simplify fractions and roots before making a selection.