Grade by Grade
B. Number · Number Sense: Scientific Notation, Real Numbers, Square Roots and Percents

The Real Number System

B1.2 A1About 40 minutes
Learning goals
  • I can describe natural numbers, whole numbers, integers, rational numbers and irrational numbers.
  • I can explain how these sets of numbers fit inside each other to make the real numbers.
  • I can classify a number into every set it belongs to.
  • I can show that a number is rational by writing it as a fraction of integers.

Words to know

natural numbers
The counting numbers: 1, 2, 3, 4, …
whole numbers
The natural numbers and zero: 0, 1, 2, 3, …
integers
The whole numbers and their opposites: …, −3, −2, −1, 0, 1, 2, 3, …
rational number
Any number that can be written as a fraction a/b, where a and b are integers and b is not 0. Examples: 3/4, −5, 0.6, 0.333…
irrational number
A number that cannot be written as a fraction of integers. Its decimal goes on forever without repeating. Examples: √2, π.
real numbers
All the rational and irrational numbers together. Every point on the number line is a real number.

Remember from Grade 7

In Grade 7 you learned that rational numbers include positive and negative fractions and decimals, like −3/4 and −4.5. You also found square roots of perfect squares, like √64 = 8, and noticed that √50 is not a whole number. This year you will find out what kind of number √50 really is.

Families of numbers

Think about your family. You might be part of your household, your extended family, your community and your country, all at once. Numbers are like that too. One number can belong to several sets at the same time.

Mathematicians sort numbers into sets that fit inside each other like nesting boxes. Together, they make up the real number system.

The sets, from smallest to largest

  • Natural numbers: the counting numbers 1, 2, 3, …
  • Whole numbers: add 0: 0, 1, 2, 3, …
  • Integers: add the negatives: …, −2, −1, 0, 1, 2, …
  • Rational numbers: any number you can write as a fraction a/b of two integers (b ≠ 0). This includes every integer (−5 = −5/1), every terminating decimal (0.75 = 3/4) and every repeating decimal (0.333… = 1/3).
  • Irrational numbers: numbers that cannot be written as a fraction of integers. Their decimals go on forever without repeating. Examples: √2 = 1.414 213 5…, π = 3.141 592 6…, √10.

Rational and irrational numbers together make the real numbers. A number is either rational or irrational, never both.

Real numbersRational numbersIntegers−7−1Whole numbers0Natural: 1, 2, 3, …3/4−2.50.333…Irrational√2π√10
The real number system. Natural numbers sit inside the whole numbers, which sit inside the integers, which sit inside the rational numbers. The irrational numbers are separate from the rational numbers. Together they make the real numbers.
NumberNaturalWholeIntegerRationalIrrationalReal
7✓✓✓✓✓
0✓✓✓✓
−5✓✓✓
0.75✓✓
−2 1/3✓✓
√5✓✓
Every number on this list is a real number. Each one is either rational or irrational.
Find the smallest set a number belongs toStart: simplify (√49 = 7)Can it be written as a/b with integers?(Its decimal ends or repeats.)NoIrrational: √5, πYesIs it an integer?(no fraction or decimal part)NoRational: 0.75, −2 1/3YesIs it 0 or greater?NoInteger: −5, −8YesIs it 1 or greater?NoWhole: 0YesNatural: 7, 12
Follow the arrows to find the smallest set a number belongs to. It also belongs to every bigger set around it: 7 is natural, so it is also whole, an integer, rational and real.
Worked example

Example 1: Classify a square root

Classify √49.

  1. First simplify: √49 = 7, because 7 × 7 = 49.
  2. 7 is a counting number, so it is natural. That means it is also whole, an integer, rational and real.
  3. It is not irrational.

√49 is a natural number, whole number, integer, rational number and real number.

Worked example

Example 2: Show a number is rational

Show that 1.6, −3 and 0.121 212… are rational.

  1. 1.6 = 16/10 = 8/5.
  2. −3 = −3/1.
  3. 0.121 212… repeats the block 12. Use a calculator to check that 12 ÷ 99 = 0.121 212…, so it equals 12/99 = 4/33.

Each can be written as a fraction of two integers, so all three are rational.

Worked example

Example 3: Rational or irrational?

Classify √5 and √0.25.

  1. 5 is not a perfect square. √5 = 2.236 067 9… The decimal never ends and never repeats, so √5 is irrational (and real).
  2. 0.25 = 0.5 × 0.5, so √0.25 = 0.5 = 1/2. That is rational (and real).

√5 is irrational. √0.25 is rational.

Watch out!

  • A number that looks messy is not always irrational. 0.121 212… keeps going, but it repeats, so it is rational.
  • A square root is not always irrational. √49 = 7 is rational. The square root of a whole number is irrational when the number is not a perfect square.
  • 0 is a whole number, but it is not a natural number.
Just for fun

Why can't the natural numbers ever finish a rocket countdown?

"5, 4, 3, 2, 1..." They don't have a 0 to blast off with. They need the whole numbers for that!

Hands-on activity

Number sorting mat

You need: a large sheet of paper (or tape on a table), a marker, 16 small scraps of paper.

  1. Draw the nested boxes from the diagram: Real numbers on the outside; inside it, Rational (with Integers, Whole and Natural inside) and Irrational.
  2. Write one number on each scrap: 12, 0, −8, 3/5, −0.4, 0.666…, √2, √81, π, −√16, 2.5, −1 1/2, √7, 100, √0.09, 0.101 001 000 1… (one more 0 each time).
  3. Place each scrap in the smallest box it fits in. Explain your choice out loud.

Answers: Natural: 12, √81 (= 9), 100. Whole only: 0. Integer only: −8, −√16 (= −4). Rational only: 3/5, −0.4, 0.666…, 2.5, −1 1/2, √0.09 (= 0.3). Irrational: √2, π, √7, 0.101 001 000 1… (no block repeats).

Math talk: "Which numbers surprised you? Why?"

Your turn

1

Is −8 a whole number? Name every set it belongs to.

2

Classify 2/5.

3

Classify √36.

4

Classify √7.

5

Show that 1.6 is rational.

6

Show that −3 is rational.

7

Is 0 a natural number? Name every set it belongs to.

More practice: warm-up

1

Name three natural numbers.

2

Is 0.5 rational?

3

Is 12 an integer?

4

Is −0.25 rational?

5

Is π rational or irrational?

More practice: on your own

1

Classify −√16.

2

Classify 0.454 545… and write it as a fraction.

3

True or false: every integer is a whole number.

4

True or false: every whole number is an integer.

5

Write one number that is rational but not an integer, and one that is real but not rational.

Challenge

1

Which of √1, √2, √3, …, √20 are rational? How many are irrational?

2

Is √(9/4) rational? Explain.

3

A number is an integer but not a whole number. What must be true about it? Give two examples.

Check your understanding

Quiz

1.

Which number is irrational?

2.

Which is the smallest set that −6 belongs to?

3.

Which number is not rational?

4.

The number 0.777… (the 7 repeats forever) is:

5.

Which statement is true?

6.

√64 is:

For parents and teachers

Big idea: B1.2 asks Grade 8 students to describe, compare and order numbers in the real number system (rational and irrational numbers). This lesson gives the vocabulary and the "nesting boxes" picture. Irrational numbers are new in Grade 8.

Common difficulties:

  • Thinking every never-ending decimal is irrational. Repeating decimals are rational.
  • Thinking every square root is irrational. √49 = 7 is rational; only square roots of non-perfect squares (of whole numbers) are irrational.
  • Forgetting a number belongs to several sets. 7 is natural, whole, an integer, rational and real.
  • Is 0 natural? In this course, natural numbers start at 1.

How to help: Do the sorting mat together and ask your child to justify each placement. Explaining and defending a classification builds the A1 processes of reasoning and communicating.