B. Number · Number: Number Sets, Powers and Rational Numbers in Action

Number Sets: How the Real Number System Is Built

B1.2 A1About 70 minutes

Learning goals

I can define the natural numbers, whole numbers, integers, rational numbers, irrational numbers and real numbers.
I can use set notation, such as {1, 2, 3, ...} and the symbol ∈, to describe a set.
I can classify any real number into every set it belongs to.
I can describe similarities and differences between number sets, including which operations each set is closed under.

Words to know

set
A collection of objects, called elements. Sets are written with curly brackets: {2, 4, 6}.

element (∈)
A member of a set. 5 ∈ ℕ means "5 is an element of the natural numbers." 0 ∉ ℕ means 0 is not.

subset (⊂)
A set whose elements are all inside another set. ℕ ⊂ ℤ because every natural number is an integer.

natural numbers ℕ
The counting numbers {1, 2, 3, 4, ...}.

whole numbers 𝕎
The natural numbers and zero: {0, 1, 2, 3, ...}.

integers ℤ
The whole numbers and their opposites: {..., −2, −1, 0, 1, 2, ...}.

rational numbers ℚ
Numbers that can be written as a/b, where a and b are integers and b ≠ 0.

irrational numbers
Real numbers that cannot be written as a fraction of integers. Their decimals never end and never repeat.

real numbers ℝ
All the rational and irrational numbers together: every point on the number line.

closed (closure)
A set is closed under an operation if doing that operation on any two elements always gives an element of the same set.

Remember from Grade 8

In Grade 8 you met the real number system: natural numbers, whole numbers, integers, rational numbers and irrational numbers like √2 and π. In Grade 9 you will look more closely at how each set is defined, write sets with proper set notation, and compare the sets by asking which calculations keep you inside each set.

A family of sets

Think of the number sets as a family that grew over time. People first needed numbers for counting (1, 2, 3, ...). Then they needed zero for "none". Then they needed negatives for debts, temperatures below zero and positions below sea level. Then fractions for sharing and measuring. Finally they discovered numbers like √2 that are on the number line but cannot be written as fractions at all.

Set	Symbol	Definition	Roster (list) form
Natural numbers	ℕ	counting numbers	{1, 2, 3, 4, ...}
Whole numbers	𝕎	natural numbers and 0	{0, 1, 2, 3, ...}
Integers	ℤ	whole numbers and their opposites	{..., −2, −1, 0, 1, 2, ...}
Rational numbers	ℚ	a/b with a, b ∈ ℤ and b ≠ 0	too many to list; e.g. −3/4, 0.6, 5, 0.333…
Irrational numbers	(no single standard symbol)	real numbers that are not rational	e.g. √2, π, −√7
Real numbers	ℝ	all rational and irrational numbers	every point on the number line

The six number sets. The "..." means the pattern goes on forever. (Some books start the natural numbers at 0; in this course, ℕ starts at 1.)

Two ways to describe a set

Roster form lists the elements: A = {−2, −1, 0, 1}.
Set-builder form gives a rule: A = {x ∈ ℤ | −2 ≤ x ≤ 1}. Read it as "the set of all x in the integers such that x is between −2 and 1, inclusive." The vertical bar | means "such that".

The definition of the rational numbers is written in set-builder form: ℚ = {a/b | a, b ∈ ℤ, b ≠ 0}. Why must b ≠ 0? Because division by zero is undefined: there is no number that, multiplied by 0, gives 5.

The sets are nested

Every natural number is a whole number, every whole number is an integer, and every integer is rational (because 5 = 5/1 and −3 = −3/1). So ℕ ⊂ 𝕎 ⊂ ℤ ⊂ ℚ ⊂ ℝ. The irrational numbers sit beside the rationals, not inside them: no number can be both rational and irrational. Together, the rationals and irrationals make up all of ℝ.

The real number system. Each box is a subset of the box around it, so a number belongs to its own box and every box that contains it. For example, 7 is natural, whole, an integer, rational and real. The irrational numbers form a separate region of ℝ.

Worked example

Example 1: Classify each number

Name every set each number belongs to: −6, 0, 3/4, √25, √10, −2.75, 0.444…

−6: an integer (and so rational and real). It is not whole or natural, because it is negative.
0: whole, integer, rational, real. Not natural.
3/4: rational and real. It is not an integer.
√25 = 5: simplify first! 5 is natural, whole, integer, rational and real.
√10: 10 is not a perfect square, so √10 = 3.16227… never repeats. Irrational and real.
−2.75 = −11/4: rational and real.
0.444… = 4/9 (a repeating decimal): rational and real.

Simplify first, then work from the inside of the diagram outward: once a number is in a set, it is in every larger set that contains it.

A decision path for classifying a real number. Ask the first question first: only rational numbers move down the chain.

Similarities and differences: closure

One way to compare sets is to ask: if I add, subtract, multiply or divide two numbers from this set, is the answer always in the same set? If yes, the set is closed under that operation. One counterexample is enough to show a set is not closed.

Set	+	−	×	÷ (not by 0)	Counterexample
ℕ	yes	no	yes	no	3 − 5 = −2 ∉ ℕ; 3 ÷ 5 = 0.6 ∉ ℕ
𝕎	yes	no	yes	no	2 − 7 = −5 ∉ 𝕎
ℤ	yes	yes	yes	no	1 ÷ 4 = 0.25 ∉ ℤ
ℚ	yes	yes	yes	yes	none: ℚ is closed under all four
Irrationals	no	no	no	no	√2 × √2 = 2, which is rational
ℝ	yes	yes	yes	yes	none

Closure of each set under the four operations. Each new set was "invented" partly to fix a closure problem: negatives make subtraction work, and fractions make division work.

Worked example

Example 2: Use a counterexample

Is the set of integers closed under division? Is the set of irrational numbers closed under multiplication?

Integers: try 3 ÷ 4 = 0.75. Both 3 and 4 are integers, but 0.75 is not. So ℤ is not closed under division.
Irrationals: √2 and √8 are both irrational. √2 × √8 = √16 = 4, which is rational. So the irrationals are not closed under multiplication.
Note: some products of irrationals are irrational (√2 × √3 = √6), but one counterexample is enough to show "not always".

No and no, shown by counterexamples.

Worked example

Example 3: Roster and set-builder form

a) Write {x ∈ ℤ | −3 ≤ x < 2} in roster form. b) Write {2, 4, 6, 8, ...} in set-builder form.

a) The integers from −3 up to, but not including, 2: −3, −2, −1, 0, 1.
b) These are the even natural numbers. Each one is 2 times a natural number.

a) {−3, −2, −1, 0, 1} b) {2n | n ∈ ℕ}

Why are they called "real" numbers?

Mathematicians later invented numbers, such as a number whose square is −1, that are not on the number line. These were nicknamed imaginary numbers, and the numbers on the line became the real numbers. Every number you work with in Grade 9 is a real number.

Hands-on activity

Sticky-note number sort

You need: a large sheet of paper (or the back of a poster), a pencil, 16 sticky notes or small paper squares.

Draw the nested diagram: ℝ around everything, ℚ and the irrationals side by side inside it, then ℤ, 𝕎 and ℕ nested inside ℚ.
Write one number on each note: √36, −√9, 22/7, π, 0, 1.010010001…, 4/2, −0.5, 1 ¼, √3, 100, −15, 0.27272727…, √0.25, 2π, −1/3.
Place each note in the smallest box it belongs to. Explain each choice to someone at home.

Model answers: ℕ: √36 = 6, 4/2 = 2, 100. 𝕎 only: 0. ℤ only: −√9 = −3, −15. ℚ only: 22/7, −0.5, 1 ¼, 0.272727… (= 3/11), √0.25 = 0.5, −1/3. Irrational: π, 1.010010001…, √3, 2π.

Watch for the trap: 22/7 is a rational approximation of π, not π itself.

Just for fun

Why was the irrational number so hard to talk to?

It just went on and on and never repeated itself.

Your turn

List every set that −9 belongs to.

List every set that √49 belongs to.

Is −√49 natural? Explain.

Classify 5.121212… and 5.121121112…

Write {x ∈ 𝕎 | x < 4} in roster form.

Give a counterexample to show that the whole numbers are not closed under subtraction.

True or false: every integer is a rational number. Explain.

Why does the definition of ℚ say b ≠ 0?

More practice: warm-up

What is the smallest natural number?

What is the smallest whole number?

Is 0.75 rational?

Is π rational or irrational?

Fill in the symbol ∈ or ∉: −2 __ 𝕎

More practice: on your own

Write the set of integers greater than −4 and less than or equal to 1 in set-builder and roster form.

Is ℚ closed under subtraction? Give a reason.

Give two irrational numbers whose sum is rational.

Name a number that is rational but not an integer, and one that is an integer but not whole.

Sam says, "√12 is rational because 12 is a whole number." Is he right?

Challenge

Is the set of even integers closed under addition? Under division? Explain.

Explain why no number can be both rational and irrational.

Find all values in {x ∈ ℤ | x² < 10}.

Check your understanding

Quiz

Which number is whole but not natural?

Which number is irrational?

Which statement is true?

Which set is closed under subtraction but not division?

What is {x ∈ ℤ | −1 < x ≤ 2} in roster form?

Which pair shows that the irrational numbers are not closed under multiplication?

For parents and teachers

Big idea: Expectation B1.2 asks students to describe how subsets of the real number system are defined and to describe similarities and differences between them. Closure ("does the answer stay in the set?") is a powerful way to compare sets, and it explains why each new set was needed.

Common difficulties:

Forgetting to simplify first. √25, 4/2 and −√9 are all integers.
Thinking a number belongs to only one set. 7 belongs to five sets.
Mixing up roster form inequalities. Check whether the endpoints use < or ≤.
Believing any long decimal on a calculator is irrational. A calculator shows only about 10 digits; 1/7 looks long but repeats.

How to help: Name a number (your age, the temperature outside, a price, π) and ask your teen to list every set it belongs to. Ask "How do you know?" and listen for a definition, not just a guess.