Module 12 of 45Lesson 1 of 28 · Unit 12 · The Number Families
Number · Unit N12
The Number Families
√16 is not irrational. 0.4161616… is not irrational. π is, and so is √20, and knowing which is which is worth a mark every time it is asked. This unit gives the five families of number their proper names and symbols, shows you why they sit inside one another, and hands you a routine that classifies any number you will meet.
By the end of this unit you can
- Name the five families — ℕ, ℤ, ℚ, ℚ′ and ℝ — and say what each one holds
- Read and write ∈, ∉ and ⊂ correctly, and know which joins a number to a set and which joins two sets
- Explain why ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ, and why ℚ′ sits beside ℚ rather than inside it
- Show that any terminating or recurring decimal is rational by producing the fraction
- Test a number for irrationality properly — never ends AND never repeats
- Classify any number into every family it belongs to, and answer “find a number such that” questions with a counterexample
The one idea behind all of it
There is only one question, and everything in this unit is an answer to it: can this number be written as a fraction of two integers? If it can, it is rational. If it cannot, it is irrational. Every symbol, every oval in the diagram and every trap in this unit is that one question wearing different clothes.
ab, with a and b integers and b ≠ 0
Can it be written this way? Yes → ℚ · No → ℚ′
Everything else follows. Whole and not negative is ℕ; whole is ℤ; a fraction of integers is ℚ; anything real that is not is ℚ′; and all of them together are ℝ.
Where this unit sits
This unit opens the Standard band of the Number course. It is the vocabulary unit: almost nothing in it is new arithmetic, and almost all of it is new language. That is exactly why it is worth doing carefully — the e-assessment gives away one-mark naming questions, and they are only free if you know the words.
It leans on two earlier units and repeats neither. Turning a recurring decimal into a fraction is N05 Repeating Forever, and it is used here as evidence, not re-taught from scratch. Simplifying a square root is N10 Squares, Roots and Powers; here the point is what the simplified root tells you about which family the number lives in.
Surds themselves are Algebra Unit 19. This unit only needs to know that √2 is irrational and why. Manipulating surds — simplifying √45, adding like surds, rationalising a denominator — is taught once, in Algebra, and the Standard-band bridge page after N17 links straight to it.
1
The five families
ℕ, ℤ, ℚ, ℚ′ and ℝ — what each one holds
Mathematics sorts numbers into five named families, and each one has a symbol you are expected to recognise on sight. They are not five separate boxes: four of them nest, each sitting wholly inside the next, and the fifth — the irrational numbers — fills in what is left over.
ℕ the positive integers and zero · ℤ the integers · ℚ the rational numbers · ℚ′ the irrational numbers · ℝ the real numbers. Each oval is drawn wholly inside the next, because every member of the smaller family really is a member of the larger one. ℚ′ is drawn beside ℚ, never inside it.
| Symbol | Name | What is in it | Examples |
|---|---|---|---|
| ℕ | positive integers and zero | counting numbers, starting at 0 | 0, 1, 2, 3, 47 |
| ℤ | integers | whole numbers, negative as well as positive | −12, −1, 0, 5 |
| ℚ | rational numbers | anything writable as ab with a, b integers and b ≠ 0 | 34, −2.5, 7, 0.3 with 3 recurring |
| ℚ′ | irrational numbers | real numbers that are not rational | √2, √20, π |
| ℝ | real numbers | every number on the number line | all of the above |
EXAMPLEA shop till, a bank statement, a tape measure and the circumference of a round kraal need four different families between them. The till counts in ℕ; the bank statement goes negative, so it needs ℤ; the tape measure reads 1.35 m, which is ℚ; and the distance round the kraal is 24π metres, which is in ℚ′ and can never be written exactly as a decimal.
Worked example 1 · Name the smallest family holding each of 7, −4, 23 and √5
1
Start at the smallest family and work outwards
ℕ → ℤ → ℚ, and if none of those, ℚ′
2
7 is whole and not negative, so the first family already holds it
7 ∈ ℕ
3
−4 is whole but negative, so ℕ is too small
−4 ∈ ℤ
4
23 is not whole, but it IS a fraction of two integers
23 ∈ ℚ
5
√5 = 2.236067977… never ends and never repeats
√5 ∈ ℚ′
“Smallest family” and “every family” are different questions with different answers. 7 is in four families and its smallest is ℕ; √5 is in two and its smallest is ℚ′. Read which one the question asked for before you write anything down.
Practice · The five families · 10 questions
P1Write down the symbol used for the set of integers.Fluency
Answerℤ
P2Write down whether 5 belongs to ℕ.Fluency
Answeryes — 5 ∈ ℕ
P3Write down the smallest of the five families that contains −4.Fluency
Answerℤ
P4Write down what the symbol ℚ′ stands for.Fluency
Answerthe set of irrational numbers
P5Write down the smallest family that contains 34.Applied
Answerℚ34 is not a whole number, so ℕ and ℤ are both too small. It is a fraction of two integers with a non-zero denominator, which is exactly the definition of a rational number, so the smallest family is ℚ.
P6Determine the smallest family that contains 0.Applied
Answerℕℕ in this course is the set of positive integers and zero, so 0 is already inside the smallest family. It is also in ℤ, ℚ and ℝ — but the question asks for the smallest.
P7Determine the smallest family that contains √2.Applied
Answerℚ′√2 = 1.414213562… The decimal never stops and no block of digits ever repeats, so it cannot be written as a fraction of two integers. That puts it in ℚ′, and ℚ′ is not inside ℚ — it sits beside it.
P8Determine the smallest family that contains −2.5.Applied
Answerℚ−2.5 is negative, so it is not in ℕ, and it is not whole, so it is not in ℤ. It stops after one decimal place, so it is a fraction: −2.5 = −52 ∈ ℚ.
P9Explain why ℝ is the largest of the five families, and write down the two families whose union is ℝ.Challenge
Answerℚ and ℚ′; ℚ ∪ ℚ′ = ℝEvery number in this course is real. A real number either can be written as a fraction of two integers or it cannot: there is no third possibility. The ones that can are ℚ, the ones that cannot are ℚ′, so together they are the whole of ℝ. Nothing is left over and nothing is counted twice.
P10Determine one number that is in ℤ but not in ℕ, and one number that is in ℚ but not in ℤ.Challenge
Answer−3 ∈ ℤ but −3 ∉ ℕ; 12 ∈ ℚ but 12 ∉ ℤAny negative whole number does the first job, because ℕ stops at 0. Any fraction that does not cancel to a whole number does the second. Both answers show that the nesting is strict: each family is genuinely bigger than the one inside it.
2
Reading the notation
∈ is an element of · ∉ is not an element of · ⊂ is a subset of
Three symbols carry almost all of the notation in this unit. ∈ says a number is in a family. ∉ says it is not. ⊂ says a whole family sits inside another family. The first two join a number to a set; the third joins a set to a set, and mixing them up is a notation error the e-assessment marks.
Read a membership statement straight off the picture. ∈ asks “is this number inside that oval?” and ∉ asks the opposite. ⊂ is a different question altogether: it asks whether one whole oval sits inside another.
| Written | Read aloud | True or false |
|---|---|---|
| 5 ∈ ℕ | 5 is an element of ℕ | true |
| −5 ∈ ℕ | −5 is an element of ℕ | false |
| −5 ∉ ℕ | −5 is not an element of ℕ | true |
| ℤ ⊂ ℚ | ℤ is a subset of ℚ | true |
| ℚ ⊂ ℤ | ℚ is a subset of ℤ | false |
| √2 ∈ ℚ′ | √2 is an element of ℚ′ | true |
Statements that parse
- 7 ∈ ℤ — a number, then ∈, then a family.
- ℕ ⊂ ℤ — a family, then ⊂, then a family.
- π ∉ ℚ — the negative form of the first.
Statements that do not
- 7 ⊂ ℤ — 7 is a number, not a family.
- ℕ ∈ ℤ — ℕ is a family, not a member of one.
- ℚ ⊂ ℚ′ — parses, but is false: the two families share nothing.
Worked example 2 · Determine whether each statement is true or false
1
−3 ∈ ℕ — is −3 a positive integer or zero?
false
2
−3 ∈ ℤ — is −3 a whole number?
true
3
ℤ ⊂ ℚ — is every integer a fraction of two integers?
true, because n = n1
4
√2 ∈ ℚ — can √2 be written as a fraction?
false
5
0 ∈ ℕ — does this course count zero as natural?
true
∈ joins a number to a family; ⊂ joins a family to a family. Writing 3 ⊂ ℤ or ℤ ∈ ℚ is not a wrong answer, it is a sentence that does not parse — and the e-assessment marks notation.
Practice · Reading the notation · 10 questions
P11Write down how the symbol ∈ is read.Fluency
Answeris an element of
P12Write down whether −3 ∈ ℕ is true or false.Fluency
Answerfalse
P13Write down whether 7 ∈ ℚ is true or false.Fluency
Answertrue
P14Write down how ℤ ⊂ ℚ is read.Fluency
Answerℤ is a subset of ℚ
P15Determine whether √9 ∈ ℕ is true or false.Applied
AnswertrueSimplify before you classify: √9 = 3. Three is a whole number and it is not negative, so 3 ∈ ℕ and the statement is true. The root sign is not part of the number — it is an instruction.
P16Determine whether 0.5 ∉ ℤ is true or false.Applied
Answertrue — 0.5 ∉ ℤ∉ means “is not an element of”, so the statement claims 0.5 is not an integer. It is not: 0.5 = 12 sits halfway between 0 and 1. The statement is therefore true.
P17Determine whether ℚ ⊂ ℝ is true or false.Applied
Answertrue⊂ asks whether every member of the first set is a member of the second. Every rational number is a real number, so yes. Note that ⊂ joins two sets while ∈ joins a number to a set — 3 ⊂ ℤ is not a statement, it is a spelling mistake.
P18Determine whether π ∈ ℚ is true or false.Applied
Answerfalseπ = 3.141592653… and its decimal neither terminates nor settles into a repeating block, so it is not a fraction of two integers. π ∉ ℚ; π ∈ ℚ′.
P19Write down the statement “227 is not irrational” in set notation.Challenge
Answer227 ∉ ℚ′“Irrational” is the family ℚ′, and “is not” is ∉. You could equally write 227 ∈ ℚ, which says the same thing, because a real number that is not irrational must be rational.
P20Explain why ℕ ⊂ ℤ is true but ℤ ⊂ ℕ is false.Challenge
Answerℕ ⊂ ℤ is true; ℤ ⊂ ℕ is false because −1 ∈ ℤ but −1 ∉ ℕEvery member of ℕ is a whole number that is not negative, so every one of them is an integer — the first statement holds with no exceptions. The second statement needs every integer to be in ℕ, and −1 is not. One counterexample is enough to destroy a ⊂ claim; no number of examples can prove one.
3
Why the families nest
Every natural number is an integer, and every integer is rational
ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ is not a convention someone chose — it is forced. Every positive whole number is a whole number. Every whole number n is the fraction n1. Every fraction is a point on the number line. Each step is a one-line argument, and each one is worth being able to give.
6 belongs to four families and √7 to two, and both statements are complete answers to different questions. Notice that ℝ appears on every chip: nothing in this course escapes it.
Worked example 3 · Classify −7, 0, 34, √16, 0.416 with 16 recurring and π into every family
1
Simplify anything that can be simplified, before you classify anything
√16 = 4
2
Whole and not negative → all four nesting families
0 and 4: ℕ, ℤ, ℚ, ℝ
3
Whole but negative → drop ℕ
−7: ℤ, ℚ, ℝ
4
A fraction of integers, or a decimal that stops or repeats → ℚ
34 and 0.416 with 16 recurring: ℚ, ℝ
5
Never ends and never repeats → ℚ′, beside ℚ
π: ℚ′, ℝ
Two of these six are traps and both were defused in step 1 or step 4. √16 is not irrational — it is 4. 0.416 with 16 recurring is not irrational — it repeats, so it is 206495. Every number in the list is in ℝ: that column can never be empty.
EXAMPLE“Every family” and “the smallest family” are different questions. The number 6 belongs to ℕ, ℤ, ℚ and ℝ — four correct answers — and its smallest family is ℕ. Read the command term: write down every family wants the list, write down the smallest family wants one symbol, and giving the list when one was asked for is not usually penalised, while giving one when the list was asked for is.
Why each step is forced
- ℕ ⊂ ℤ: a positive whole number is a whole number.
- ℤ ⊂ ℚ: n = n1, and 1 ≠ 0.
- ℚ ⊂ ℝ: every fraction has a place on the number line.
Why the nesting is strict
- −1 ∈ ℤ but −1 ∉ ℕ.
- 12 ∈ ℚ but 12 ∉ ℤ.
- √2 ∈ ℝ but √2 ∉ ℚ.
Practice · Why the families nest · 10 questions
P21Write down the four nesting families in one chain, using ⊂.Fluency
Answerℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ
P22Write down every family that 6 belongs to.Fluency
Answerℕ, ℤ, ℚ, ℝ
P23Write down every family that −6 belongs to.Fluency
Answerℤ, ℚ, ℝ
P24Write down every family that 25 belongs to.Fluency
Answerℚ, ℝ
P25Explain why every integer is also a rational number.Applied
Answerbecause any integer n can be written as the fraction n1The definition of a rational number is “a number that can be written as ab with a and b integers and b ≠ 0”. For an integer n, take a = n and b = 1: 7 = 71 and −6 = −61. Nothing says the fraction has to look like a fraction.
P26Determine every family that √25 belongs to.Applied
Answerℕ, ℤ, ℚ, ℝ√25 = 5, and 5 is a whole number that is not negative. So it is in ℕ, and therefore automatically in ℤ, ℚ and ℝ as well. The root sign changed nothing except the amount of work.
P27Determine every family that −1.75 belongs to.Applied
Answerℚ, ℝ−1.75 stops after two decimal places, so it is a fraction: −1.75 = −175100 = −74. It is not whole, so it is not in ℤ and not in ℕ.
P28Determine every family that √7 belongs to.Applied
Answerℚ′, ℝ7 is not a square number: 2² = 4 and 3² = 9, and there is no whole number in between. So √7 = 2.645751311… never terminates and never repeats, which puts it in ℚ′. Every number in this course is real, so it is in ℝ too — but it is in neither ℕ, ℤ nor ℚ.
P29Explain why ℚ ∩ ℚ′ = ∅.Challenge
Answerno number can be both rational and irrational, so the two families share nothingℚ′ is defined as the real numbers that are not in ℚ. A number that was in both would have to be a fraction of two integers and not a fraction of two integers at the same time. So the overlap is empty — which is why the ℚ′ oval in the diagram sits beside ℚ and not inside it.
P30Determine whether there is a real number belonging to neither ℚ nor ℚ′.Challenge
Answerno — ℚ ∪ ℚ′ = ℝTake any real number and ask one question: can it be written as a fraction of two integers? If yes it is in ℚ; if no it is in ℚ′. There is no third answer, so nothing can escape both. Together with P29 this says the two families partition ℝ: no gaps, no overlap.
4
Decimals that stop, and decimals that repeat
Both are rational — and N05 already showed you how to prove it
A decimal that stops is a fraction with a power of ten underneath: 0.25 = 25100 = 14. A decimal that repeats a block for ever is also a fraction, and N05 gave you the method that produces it. Both kinds are therefore rational — and producing the fraction is not just an answer, it is the proof.
0.25 stops. 0.333… and 0.416161… never stop, but each repeats a block for ever, so the ×10ⁿ subtraction method from N05 turns them into 13 and 206495. √2 does neither, and that is the whole difference.
Worked example 4 · Show that 0.416 with 16 recurring ∈ ℚ
1
Give the number a name and write the tail out in full
x = 0.416 with 16 recurring = 0.4161616…
2
One digit sits outside the block, so step past it first
10x = 4.161616…
3
The block is two digits long, so go 100 times further again
1000x = 416.161616…
4
Subtract the pair whose tails match — the tails cancel exactly
1000x − 10x = 412, so 990x = 412
5
Divide, then cancel the factor of 2
x = 412990 = 206495, so x ∈ ℚ
This is the N05 method, unchanged, doing a new job. In N05 it turned a recurring decimal into a fraction; here the fraction is the proof of membership, because “rational” means exactly “can be written as a fraction of two integers”. Producing the fraction ends the argument.
| Decimal | What the method does | Fraction | Family |
|---|---|---|---|
| 0.25 | it stops, so put it over a power of ten | 14 | ℚ |
| 0.3 with 3 recurring | 10x − x = 3 | 13 | ℚ |
| 0.27 with 27 recurring | 100x − x = 27 | 311 | ℚ |
| 0.416 with 16 recurring | 1000x − 10x = 412 | 206495 | ℚ |
| 0.9 with 9 recurring | 10x − x = 9 | 1 | ℕ |
EXAMPLE0.9 with 9 recurring = 1 is the line that stops most people. Run the method: x = 0.999…, so 10x = 9.999…, so 9x = 9 and x = 1. There is no rounding anywhere in that argument. 0.9 with 9 recurring and 1 are two names for one number, in the same way that 24 and 12 are.
Practice · Decimals that stop, and decimals that repeat · 10 questions
P31Write down whether 0.25 is rational.Fluency
Answeryes
P32Write down 0.4 as a fraction in its lowest terms.Fluency
Answer25
P33Write down 0.3 with 3 recurring as a fraction in its lowest terms.Fluency
Answer13
P34Write down whether a decimal that stops is always rational.Fluency
Answeryes
P35Show that 0.7 with 7 recurring ∈ ℚ.Applied
Answer0.7 with 7 recurring = 79Let x = 0.7 with 7 recurring = 0.777… The block is one digit long, so multiply by 10: 10x = 7.777… Subtract: 10x − x = 7, so 9x = 7 and x = 79. That is a fraction of two integers, so 0.7 with 7 recurring ∈ ℚ.
P36Show that 0.27 with 27 recurring ∈ ℚ.Applied
Answer0.27 with 27 recurring = 311The block is two digits long, so multiply by 100, not 10. x = 0.272727… and 100x = 27.272727… Subtract: 99x = 27, so x = 2799 = 311.
P37Determine the fraction equal to 0.416 with 16 recurring.Applied
Answer206495One digit sits outside the block, so step past it first: x = 0.4161616… and 10x = 4.161616… The block is two digits long, so also form 1000x = 416.161616… Subtract the pair whose tails match: 1000x − 10x = 412, so 990x = 412 and x = 412990 = 206495.
P38Determine the fraction equal to 0.12 with 12 recurring.Applied
Answer433x = 0.121212…, and the block is two digits, so 100x = 12.121212… Subtract: 99x = 12, so x = 1299 = 433.
P39Show that 0.9 with 9 recurring = 1, and write down the smallest family that 0.9 with 9 recurring belongs to.Challenge
Answer0.9 with 9 recurring = 1, so the smallest family is ℕx = 0.999…, so 10x = 9.999… Subtract: 9x = 9, so x = 1 exactly. This is not a rounding error and it is not “nearly 1” — the two expressions name the same number. Since that number is 1, the smallest family containing it is ℕ.
P40Determine the fraction equal to 0.16 with 6 recurring, where only the 6 recurs.Challenge
Answer16One digit is outside the block and the block is one digit long. x = 0.1666…, 10x = 1.666… and 100x = 16.666… Subtract the matching tails: 100x − 10x = 15, so 90x = 15 and x = 1590 = 16. Moving the dot changes the number: 0.16 with 16 recurring is 1699, not 16.
5
What irrational really means
Never ends AND never repeats — and why √16 is not irrational
A number is irrational when its decimal never stops AND never repeats. Both halves matter. Failing only the first half — a decimal that never stops but does repeat — is what a recurring decimal does, and recurring decimals are rational. The root sign tells you nothing on its own: carry it out first.
Two of these four roots are natural numbers and two are irrational, and you cannot tell which is which without carrying out the instruction. Simplify first, then classify.
Worked example 5 · Determine whether √16 and √20 are irrational
1
Never classify a number by the way it is written. Simplify first
√16 = 4
2
4 is whole and not negative
√16 ∈ ℕ, so it is rational
3
Now √20. Look for a whole number whose square is 20
4² = 16, 5² = 25 — nothing in between
4
So its decimal never stops and never repeats
√20 = 4.472135955…
5
The two roots land in different families
√16 ∈ ℕ · √20 ∈ ℚ′
The root sign is an instruction, not a label. √n is rational exactly when n is a square number, and irrational otherwise — and the first thing you do with any root is find out which. Of √1 to √10, only three are rational.
Worked example 6 · Determine whether π ∈ ℚ
1
Write down what would have to be true
π = ab for some integers a and b, b ≠ 0
2
Look at the decimal
π = 3.14159265358979…
3
First test: does it stop?
no — it has no last digit
4
Second test: does a block of digits repeat for ever?
no — and it never will
5
Both tests fail, so no such fraction exists
π ∉ ℚ; π ∈ ℚ′
Both tests have to fail. A decimal that stops is rational; a decimal that repeats is rational; only “never stops AND never repeats” is irrational. 227 = 3.142857 with 142857 recurring… repeats, so it is rational — it is a good approximation to π, not π.
EXAMPLEOf √1, √2, √3, …, √10, exactly three are rational: √1 = 1, √4 = 2 and √9 = 3. The other seven are irrational. So a root picked at random is far more likely to be irrational than not — which is precisely why assuming it always is feels safe, and is wrong.
Practice · What irrational really means · 10 questions
P41Write down whether √16 is irrational.Fluency
Answerno — √16 = 4
P42Write down the value of √49 and the smallest family it belongs to.Fluency
Answer7, and the smallest family is ℕ
P43Write down whether π is rational.Fluency
Answerno
P44Write down the two things the decimal of an irrational number does.Fluency
Answerit never ends and it never repeats
P45Determine which of √36 and √37 is irrational.Applied
Answer√37 is the irrational one36 is a square number, so √36 = 6 ∈ ℕ. 37 is not a square number: 6² = 36 and 7² = 49. So √37 = 6.082762530… never ends and never repeats, and it is the irrational one. Two roots that look alike can land in completely different families.
P46Explain why 227 is rational even though it is often used as an approximation for π.Applied
Answer227 is a fraction of two integers, so it is rationalBeing close to an irrational number does not make a number irrational. 227 = 3.142857142857… — the block 142857 repeats for ever, which is the signature of a rational number. π = 3.14159265… never repeats. The two numbers agree to two decimal places and then part company for good.
P47Determine whether √0.25 is irrational.Applied
Answerno — √0.25 = 0.5, which is rational0.25 = 14, and the square root of 14 is 12. So √0.25 = 0.5 ∈ ℚ. A root does not have to be a root of a whole number to come out exact.
P48Determine whether 3 + √2 is rational or irrational.Applied
AnswerirrationalSuppose 3 + √2 were rational, say equal to r. Then √2 = r − 3, and a rational number minus a rational number is rational — which would make √2 rational. It is not, so 3 + √2 cannot be rational either. Adding a rational number to an irrational one always leaves it irrational.
P49Determine how many of √1, √2, √3, …, √10 are rational.Challenge
Answer3 — √1, √4 and √9√n is rational exactly when n is a square number. Between 1 and 10 the square numbers are 1, 4 and 9, giving √1 = 1, √4 = 2 and √9 = 3. The other seven roots are irrational. Notice how few of them are rational — which is why the trap in this unit is worth so many marks.
P50Explain why 3.14159 is rational but π is not.Challenge
Answer3.14159 stops, so it equals 314159100000; π never stops and never repeats3.14159 is a terminating decimal, and every terminating decimal is a fraction with a power of ten underneath. π has no last digit and no repeating block, so no such fraction exists. The two numbers agree to five decimal places, and agreeing to any number of decimal places is not the same as being equal.
6
Classifying any number
One routine, and how to answer a “find a number such that” question
Every classification question in this unit is the same four questions asked in the same order. Work down them and the first “Yes” names the smallest family; everything larger in the chain comes free. Reaching the bottom without a Yes is what irrational means.
Work down the left-hand column. The first “Yes” you reach names the smallest family the number belongs to, and everything to the right of it in the chain comes free. Reaching the bottom without a “Yes” is what irrational means.
Worked example 7 · Find x with x ∈ ℚ and x ∉ ℤ, and x with x ∈ ℤ and x ∉ ℕ
1
Read the demand as two tests, not one
inside the bigger family, outside the smaller
2
For ℚ but not ℤ, take a fraction that will not cancel
x = 12
3
Run both tests before you write it down
12 ∈ ℚ ✓ · 12 ∉ ℤ ✓
4
For ℤ but not ℕ, go negative — that is all ℕ leaves out
x = −1
5
Run both tests again
−1 ∈ ℤ ✓ · −1 ∉ ℕ ✓
One example is a complete answer to find a number such that; one counterexample is a complete answer to is it always true?. Neither needs a list. What both need is the two tests written down — the marks are for showing that your number passes them.
EXAMPLE“Find a number x with x ∈ ℚ and x ∉ ℤ” is asking for one number that passes two tests. Write the tests down beside your answer: 12 ∈ ℚ because it is a fraction of two integers, and 12 ∉ ℤ because it is not whole. The marks are for the tests, not for the cleverness of the number.
Practice · Classifying any number · 10 questions
P51Write down the smallest family containing 12.Fluency
Answerℕ
P52Write down the smallest family containing −12.Fluency
Answerℤ
P53Write down the smallest family containing 0.6.Fluency
Answerℚ
P54Write down the smallest family containing √11.Fluency
Answerℚ′
P55Determine a value of x for which x ∈ ℚ and x ∉ ℤ.Applied
Answerany non-whole fraction, for example x = 12The demand is two tests, not one: in the bigger family, out of the smaller. 12 is a fraction of two integers, so 12 ∈ ℚ ✓; and it is not a whole number, so 12 ∉ ℤ ✓. Any answer that passes both tests earns full marks — 0.7, −34 and 2.5 all work.
P56Determine a value of x for which x ∈ ℤ and x ∉ ℕ.Applied
Answerany negative integer, for example x = −1ℕ holds the positive integers and zero, so the integers it leaves out are exactly the negative ones. −1 ∈ ℤ ✓ and −1 ∉ ℕ ✓. Answering 0 fails the second test, because 0 is in ℕ.
P57Determine a value of x for which x ∈ ℝ and x ∉ ℚ.Applied
Answerany irrational number, for example x = √2A real number that is not rational is by definition irrational, so the question is asking for a member of ℚ′. √2, √3, π and 1 + √5 all do the job. √4 does not: it is 2.
P58Determine whether it is possible to find x with x ∈ ℕ and x ∉ ℝ.Applied
Answerno — ℕ ⊂ ℝℕ sits inside ℤ, which sits inside ℚ, which sits inside ℝ. So every natural number is real, and there is nothing to find. A question can ask for a counterexample that does not exist, and saying so — with the reason — is the full answer.
P59List every family that each of −3, 0, 58, √64, 0.416 with 16 recurring and π belongs to.Challenge
Answer−3: ℤ, ℚ, ℝ · 0: ℕ, ℤ, ℚ, ℝ · 58: ℚ, ℝ · √64: ℕ, ℤ, ℚ, ℝ · 0.416 with 16 recurring: ℚ, ℝ · π: ℚ′, ℝSimplify first: √64 = 8, so it behaves exactly like any other natural number. 0.416 with 16 recurring recurs, so it is the fraction 206495 and therefore rational. −3 is whole but negative. 58 is a fraction that will not cancel to a whole number. π is the only member of ℚ′ in the list. Every one of the six is in ℝ — that column is never empty.
P60Determine whether “a number written with a decimal point is never an integer” is true, and justify your answer.Challenge
Answerfalse — 4.0 = 4, and 4 ∈ ℤThe claim is about how a number is written, not about what it is. 4.0 has a decimal point and is the integer 4; so does 12.00. Even 0.9 with 9 recurring is the integer 1. One counterexample settles it, and the counterexample has to be a number, not an argument.
The classifying routine
Four moves, whatever the number.
1
Simplify before you classify√16 is 4. 0.416 with 16 recurring is 206495. 63 is 2. Classify what the number is, never what it looks like on the page — this single habit defuses the two most expensive traps in the unit.
2
Ask the whole-number questions firstIs it a whole number and not negative? Then ℕ, and ℤ, ℚ, ℝ come free. Is it whole but negative? Then ℤ, and ℚ, ℝ come free. The first Yes you reach names the smallest family.
3
Then ask the fraction question — and remember what countsA fraction of two integers. A decimal that stops. A decimal that repeats a block for ever. All three are the same thing, and all three are ℚ. Only “never stops AND never repeats” reaches ℚ′.
✓
Answer the question that was asked“The smallest family” wants one symbol. “Every family” wants a list, and that list always ends with ℝ. “Find a number such that” wants one number that passes both tests, written down with the tests.
The mistakes that cost the most marks
1Assuming that anything with a root sign is irrational
Wrong√16 has a root sign, so √16 ∈ ℚ′
Right√16 = 4, and 4 ∈ ℕ, so √16 is rational — it is not in ℚ′ at all
The root sign is an instruction to find a number, not a label saying what kind of number will appear. Carry it out first. √n is rational exactly when n is a square number: √1, √4, √9, √16, √25 are 1, 2, 3, 4, 5, and every one of them is natural. This is the single most expensive slip in the topic, because it turns a one-mark classification into a wrong answer that looks confident.
2Assuming that a long or recurring decimal must be irrational
Wrong0.4161616… goes on for ever, so it must be irrational
Rightthe block 16 repeats for ever, so 0.416 with 16 recurring = 206495 ∈ ℚ
Going on for ever is only half the test. An irrational number never ends and never repeats; a recurring decimal fails the second half, and the ×10ⁿ subtraction method from N05 turns it into a fraction every time. If you can find the repeating block, you can find the fraction, and finding the fraction is the proof of rationality. Length proves nothing at all.
3Reading ℚ′ as “not a number”, or thinking ℚ ∪ ℚ′ leaves something out
Wrongℚ′ is where things go when they are not really numbers, and some real numbers are in neither ℚ nor ℚ′
Rightℚ′ is the set of real numbers that are not rational; ℚ ∪ ℚ′ = ℝ and ℚ ∩ ℚ′ = ∅
√2 and π are numbers: you can locate them on a number line, measure them and calculate with them. The prime on ℚ′ means “the rest of ℝ”, so the two families divide the real numbers between them exactly — nothing is left out and nothing is in both. That is why the diagram draws ℚ′ beside ℚ inside ℝ, and never inside ℚ.
Tasks to submit
Criterion B · Investigating patterns
Which families survive their own arithmetic?
Add two natural numbers and you get a natural number. Subtract two natural numbers and you might not: 3 − 5 = −2, which has left ℕ altogether. A family is called closed under an operation when the answer always lands back inside the family, and not closed when a single pair of members can throw you out of it.
That word is the whole investigation. Five families, four operations, twenty cells — and one row of the table does something that surprises almost everybody the first time they see it.
Part 1 · structured
| Family | + | − | × | ÷ |
|---|---|---|---|---|
| ℕ the positive integers and zero | closed | not closed 3 − 5 = −2 ∉ ℕ | | |
| ℤ the integers | | | | |
| ℚ the rational numbers | | | | |
| ℚ′ the irrational numbers | | | | |
| ℝ the real numbers | | | | |
- Predict, before you calculate anything. Write down which of the five families you think are closed under subtraction, and which under division. Keep the prediction — you will be asked to compare it with what you find.
- Complete the table. In each cell write closed, or write not closed together with one counterexample: two members of the family whose answer escapes it. The ℕ row’s first two cells are done for you. Ignore division by zero throughout — it is not an answer that escapes the family, it is not an answer at all.
- Identify what happens as you read down the ℕ, ℤ, ℚ rows. Count the closed cells in each row and say in one sentence what growing the family buys you.
- Identify what is unusual about the ℚ′ row, and write down the four counterexamples in full.
- Determine a general rule. Complete this sentence so that it works for any set S and any operation ∗: “S is closed under ∗ when …” Then determine the rule for disproving closure — how many counterexamples does it take, and why?
- Test and verify your rule on a family that is not in the table. Try the even integers under + and under ×, and the odd integers under + and under ×. Predict first, then test, then say whether your rule survived.
- Justify why one counterexample settles “not closed” for ever, but no number of successful examples can settle “closed”. Use your ℚ′ row to explain why a family can look closed for a long time and not be.
Part 2 · open
Now you choose. Here are three openings — take one, or propose your own. You decide which cases to try, how many you need, how to record them, and how you will test whatever you find.
- Investigate whether the product of two irrational numbers is always irrational. Choose your own cases and decide for yourself how many you need. If you find the answer is no, determine a rule that describes exactly when the product comes out rational.
- Investigate whether a rational number plus an irrational number is always irrational. This one is true — so justify it rather than testing it. Start by supposing the sum were rational and see what that would force √2 to be.
- Choose your own family and investigate it: the multiples of 3, the integers from 0 to 10, the numbers of the form a + b√2, or one you invent. Determine which of the four operations it is closed under, verify each answer, and justify the pattern you find.
Whichever you choose, take it through the same ladder: predict, identify the pattern, determine a general rule, test and verify it on cases you have not already used, and justify why it must hold and where it stops holding.
Submit: your completed table with every counterexample written out, your general rule for closure, your test of it on a family outside the table, and your Part 2 enquiry — as a scan or a typed document. Marked out of 8 against Criterion B. Filling in the table earns the lower bands; the general rule, the verification and the justification are what reach levels 5–8. Part 2 must show your own choice of method.
Remember, you should communicate in an organized and coherent manner.
Criterion D · Applying maths in real-life contexts
Fencing the round kraal at Tshimologo Farm
Rre Kgosietsile is re-fencing the round cattle kraal at Tshimologo Farm, outside Serowe. He has measured across the middle of the kraal with a 30 m tape and written the measurement down to the nearest 0.1 m. He wants to know how much fencing wire to buy, whether to buy it in rolls or loose, and what the whole job will cost. The site notes contain more information than you need.
The site notes
| Item | Figure |
|---|---|
| Kraal, diameter across the middle | 24.0 m (30 m tape, nearest 0.1 m) |
| Gateway already built, width | 3.0 m |
| Strands of wire to run round the kraal | 4 |
| Fence posts, one every | 3.0 m |
| Fencing wire, 50 m roll | P489.95 |
| Fencing wire, cut loose, per metre | P11.50 |
| Fence post, each | P78.00 |
| Water reservoir, internal diameter | 5.4 m |
| Water reservoir, depth | 2.1 m |
| Borehole yield | 1 800 litres per hour |
| Cattle on the farm | 46 |
| Distance to the hardware shop in Serowe | 38 km |
The measurements, prices and quantities in these notes were written for this task. They are realistic for a farm outside Serowe, but they are not a real price survey and should not be quoted as one. All prices are in Pula, to 2 decimal places.
What Rre Kgosietsile tells you
- The kraal is a circle, and the measurement across the middle is its diameter.
- The gateway is already built. No wire runs across it.
- Wire is sold either in whole 50 m rolls or cut loose by the metre — he will not buy part of a roll.
- Posts go all the way round, including past the gateway.
- He collects the materials himself in the farm bakkie.
- Identify which rows of the site notes you actually need in order to answer the questions below, and explain why each of the others is not relevant. There is more here than you need. (Strand i — identifying the relevant elements)
- Select your strategy and explain it. Say which formula gives the distance round the kraal, and explain the decision you are going to make about π: at what point in the working will you turn it into a decimal, and why not earlier? (Strand ii — selecting appropriate strategies)
- Apply it. (a) Calculate the distance round the kraal, first as an exact value in terms of π and then to 3 significant figures. (b) The gateway is already built and needs no wire. Calculate the total length of wire for 4 strands round the rest. (c) Determine how many whole 50 m rolls that needs and what they cost, and compare it with buying the same wire loose by the metre. (d) Determine the number of posts and the total bill for wire and posts. (Strand iii — applying the strategy)
- Explain how accurate your answers are. Three things to deal with. First, explain which of your figures is exact and which is not, and why the distance round the kraal can never be written exactly as a decimal. Second, the diameter was measured to the nearest 0.1 m: calculate the largest and smallest wire lengths that measurement allows, and explain whether the difference changes what you would buy. Third, explain why the number of rolls is rounded up rather than to the nearest whole number, and why that is not a breach of the round-don’t-truncate rule. (Strand iv — accuracy of the solution)
- Discuss whether your answers make sense on the farm. Is an answer left in terms of π more useful to a builder buying materials than a decimal, or less? Discuss what the spare wire on the last roll is worth, whether 4 strands and a post every 3 m is a sensible specification for cattle, and one cost the site notes do not mention at all. (Strand v — does it make sense in context)
Submit: your exact and rounded lengths, your roll-versus-loose comparison with both costs, your total bill, and written answers to strands i, iv and v. Marked out of 8 against Criterion D. This is the Standard band, so strands iv and v ask you to explain and discuss, not merely to describe — a correct total with no account of what π did to the accuracy will not reach the upper bands.
Where you go next
N13 How Far From Zero is next. It takes the number line you have just populated and asks a single question about it — how far is this number from 0? — which is what the absolute value |x| measures.
Working with surds lives in Algebra Unit 19. This unit needed only to know that √2 is irrational. Simplifying √45, adding like surds and rationalising a denominator are taught once, there, and the Standard-band bridge page links to them. Do not go looking for them here.
When you are ready
The mastery quiz is 10 questions drawn at random from a bank of 25. You need 70% to pass and you have three attempts. Five of the 25 are one-mark naming questions — learn the five words in Concept 1 and the three symbols in Concept 2 and they are free marks. Work through Concept 5 before you attempt it: the √16 trap appears in the bank three times.
Printable version
Download Unit 12 as a PDF — the same lesson, laid out for printing and for working offline.
Download Unit 12 as a PDF — the same lesson, laid out for printing and for working offline.