Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 9: write 3/8 as a decimal. Does the decimal end or repeat?
Answer
0.375. It ends, because 8 = 2 × 2 × 2 has only 2s as prime factors. - From Year 8: list the integers that are greater than −3 and less than 2.
Answer
−2, −1, 0 and 1. Neither −3 nor 2 is included, because both inequalities are strict. - From Year 9: is √50 rational or irrational?
Answer
Irrational. 50 is not a square number (7² = 49, 8² = 64), so √50 = 7.071… never ends or repeats.
Words used in this lesson
- set
- a collection of things, called its elements, written in curly brackets. For example: {1, 2, 3, 6}.
- element
- one member of a set. For example: 5 is an element of {1, 3, 5, 7}.
- subset
- a set whose elements are all inside another set, written ⊆; ⊂ means a subset that is not the whole set. For example: {1, 3} ⊆ {1, 3, 5}.
- empty set
- the set with no elements, written ∅. For example: {x | x ∈ ℕ, x < 0} = ∅.
- integer
- a whole number, positive, negative or zero. For example: −3, 0, 7.
- rational number
- a number that can be written as one integer over another, not over 0. For example: −2/3, 0.25, 5.
- irrational number
- a real number that cannot be written as a fraction of two integers. For example: √5 and π.
Sorting the numbers in a day
A weather app in Ulaanbaatar shows −4 °C at dawn. Twelve students in Nairobi share pizzas, a quarter each, so each gets 1/4 of a pizza. A square floor tile with 1 m sides has a diagonal of √2 m, and a round table with a 1 m radius has a top of area π m².
Predict first: which of −4, 12, 1/4, √2 and π can be written as a fraction of two integers? Which cannot, and how could you tell? Then use the first figure to see where numbers like these belong.
x = −3: a whole number below zero. It is in ℤ, and in ℚ (−3 = −3/1) and ℝ, but not in ℕ.
Show me the steps
- x = −3: a whole number below zero. It is in ℤ, and in ℚ (−3 = −3/1) and ℝ, but not in ℕ, which has no negative numbers.
- Slide to √9. The root sign does not make it irrational: √9 = 3, so it lands in ℕ and is in all four sets.
- Slide to −2/3. It is a fraction of two integers, so it is in ℚ and ℝ, but it is not whole, so it is not in ℤ or ℕ.
- Slide to √5. 5 is not a square number, so √5 = 2.236… never ends or repeats. It is irrational: in ℝ but outside ℚ.
The number sets fit inside each other: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ.
- ℕ, the natural numbers: {0, 1, 2, 3, …}. In IB mathematics ℕ starts at 0.
- ℤ, the integers: {…, −2, −1, 0, 1, 2, …}.
- ℚ, the rational numbers: every number p/q where p and q are integers and q ≠ 0. Its decimal ends or repeats.
- ℝ, the real numbers: every point on the number line. The irrational numbers are the real numbers that are not in ℚ, such as √2 and π.
In set notation, x ∈ A means x is an element of A, x ∉ A means it is not, B ⊆ A means every element of B is also in A (B is a subset of A), B ⊂ A means B ⊆ A and B ≠ A (a proper subset), n(A) is the number of elements in A, and ∅ is the empty set. Set-builder notation describes a set by a rule: {x | x ∈ ℤ, −2 ≤ x < 3} is read “the set of x such that x is an integer and −2 ≤ x < 3”.
A = {x | x ∈ ℤ, −2 ≤ x < 3} = {−2, −1, 0, 1, 2}, so n(A) = 5.
Show me the steps
- x ∈ ℤ with −2 ≤ x < 3: A = {−2, −1, 0, 1, 2} and n(A) = 5. −2 is in, because of ≤; 3 is not, because of <.
- Press x ∈ ℕ with the same limits. The negative integers drop out: A = {0, 1, 2} and n(A) = 3.
- Press x ∈ ℝ. Every number from −2 up to 3 is in, including 0.5 and √2, so A has infinitely many elements and cannot be listed.
- Press x ∈ ℤ and set a = 2 and b = −1. No number is at least 2 and also below −1, so A = ∅, the empty set, and n(A) = 0.
Two common mistakes. A root sign does not make a number irrational: √9 = 3 is a natural number. Only the root of a number that is not a perfect square is irrational. And ∈ is not ⊂: 3 ∈ ℕ says the number 3 is an element, while {3} ⊂ ℕ says the set containing 3 sits inside ℕ. Writing 3 ⊂ ℕ mixes the two up.
Worked example: reading set-builder notation
E = {x | x ∈ ℤ, −1 < x ≤ 4}. List E, find n(E), and decide whether 2.5 ∈ E.
- −1 < x is strict, so −1 is not in E. The first integer above −1 is 0.
- x ≤ 4 includes 4. So E = {0, 1, 2, 3, 4}.
- Count the elements: n(E) = 5.
- 2.5 is between −1 and 4, but it is not an integer, so 2.5 ∉ E.
Your go. The first step is given. F = {x | x ∈ ℕ, x ≤ 3}. List F and find n(F).
- ℕ starts at 0, so the smallest element of F is 0.
- Now decide whether 3 is in F, list the elements, and count them.
Check the answer
x ≤ 3 includes 3, so F = {0, 1, 2, 3} and n(F) = 4. A common slip is to start at 1 and get n(F) = 3.
Check before you go on
1. Which of these numbers is irrational?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A = {x | x ∈ ℤ, −6 ≤ x < 4}. How many elements does A have? That is, what is n(A)?
3. How many of these six numbers are rational: −7, √2, 0.25, π, 3/8, √36?
Level 2 · Building
4. A = {1, 2, 3, 6}, the factors of 6. Which statement is true?
5. C = {x | x ∈ ℤ, x² ≤ 16}. What is n(C)?
6. Show that 0.185185… (the block 185 repeats) is in ℚ by writing it as a fraction in its simplest form. What is the denominator?
Level 3 · Stretch
7. D = {x | x ∈ ℕ, 3x − 1 < 20}. What is n(D)?
8. Which statement is true?
Something to think about
Can a set have no elements at all? Write one in set-builder notation, then decide: is the empty set ∅ a subset of every set?
One way to think about it
{x | x ∈ ℕ, x < 0} is empty, because no natural number is below 0, so it is ∅ and n = 0. And yes: B ⊆ A means every element of B is in A. ∅ has no elements, so there is nothing that could fail to be in A. So ∅ ⊆ A for every set A, and ∅ ⊂ A for every set A that is not empty.
Exit ticket
9. x = −√49. Which list gives every set that x belongs to?