MYP 5 Standard Mathematics 1.1 Real Number System and Classifications

MYP 5 Standard Mathematics 1.1 Real Number System and Classifications - Full notes

RevisionPrep

MYP 5 Standard Mathematics | 1.1 Real Number System and Classifications

MYP 5 STANDARD MATHEMATICS · CHAPTER 1: NUMBER AND OPERATIONS

1.1

Real Number System and Classifications

Full notes for MYP Year 5 · IB-style

Contents

1	Sets of numbers
2	Rational numbers and decimals
3	Irrational numbers
4	Operations and closure
5	Surds
6	Intervals and absolute value
7	Real numbers in the real world
8	Data-based questions and further practice
9	Common misconceptions and links
10	Exam-style question with markscheme
11	Check your understanding
12	Summary and topic map

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and do every Quick check before moving to the next section.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
classify numbers into ℕ, ℤ, ℚ and ℝ
use set notation and interval notation
convert between fractions, terminating and recurring decimals
explain and prove irrationality; simplify and rationalise surds
use closure, density and absolute value

Guiding questions

How are numbers organised into sets? How can we tell whether a number is rational? Why do we keep some answers in exact form?

How this subtopic is assessed

Criterion A questions test classification, conversions and surds. Criterion B questions investigate patterns such as terminating decimals. Criterion C questions require clear notation and proof. Criterion D questions use exact and approximate values in real contexts.

Always show the method for recurring decimals and surds.

Criterion	Name	What it rewards
A	Knowing and understanding	selecting and applying mathematics to solve problems
B	Investigating patterns	finding, describing and justifying general rules
C	Communicating	notation, representations and clear lines of reasoning
D	Applying mathematics in real-life contexts	modelling real situations and judging whether answers make sense

Command term	What the examiner wants
State / Write down	a brief answer with no working or explanation
Outline	a brief account or summary
Describe	a detailed account without reasons
Explain	a detailed account including causes, reasons or mechanisms
Calculate / Find	a numerical answer with working shown
Show that	obtain the given result, showing every step
Sketch	a graph or diagram showing the general shape and key features
Justify	give valid reasons or evidence for an answer
Evaluate	weigh strengths and limitations and reach a judgement
Suggest / Predict	propose a reasonable answer from the information given

How to use these notes

Each section teaches one topic in full. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Key point	the one idea you must not forget
Common mistake	an error teachers see every year, and how to avoid it
In the real world	where the idea shows up outside the classroom
Try this	something quick you can do yourself to test the idea
Worked example	full solutions with marks (M1 method, A1 answer, R1 reasoning)
Quick check	short questions to test yourself at the end of each section

1 Sets of numbers

From counting to the continuum.

Think about it

Is 0.999… (recurring) less than 1, or equal to it?

Numbers are grouped into sets that fit inside each other. The natural numbers ℕ = {1, 2, 3, …} are used for counting. Including zero and the negative whole numbers gives the integers ℤ. A rational number ℚ is any number that can be written as p/q where p and q are integers and q ≠ 0. Numbers that cannot be written this way are irrational. Rational and irrational numbers together form the real numbers ℝ, which fill the whole number line.

1.1 The number sets

Each set is contained in the next: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. Every integer n is rational because n = n/1. Every real number is either rational or irrational, never both. Some textbooks include 0 in ℕ; in IB-style MYP work it is safest to state your convention.

Figure 1. The real number system as nested sets.

Figure 2. Classifying real numbers.

Key point

ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. Rationals are p/q with q ≠ 0 and have terminating or recurring decimals; irrationals do not. Surds are simplified using √(ab) = √a × √b.

WORKED EXAMPLE 1 Classify (basic)

State the smallest set (ℕ, ℤ, ℚ or ℝ) that contains each number: 7, −4, 0.6, √5.

Solution

7 ∈ ℕ; −4 ∈ ℤA1

0.6 ∈ ℚ; √5 ∈ ℝ (irrational)A1

WORKED EXAMPLE 2 Classify (standard)

Classify √49, √50, −3/1, π/π as rational or irrational.

Solution

√49 = 7, −3/1 = −3, π/π = 1: rationalA1

√50 = 5√2: irrationalA1

WORKED EXAMPLE 3 Explain (standard)

Explain why every integer is rational.

Solution

any integer n can be written as n/1R1

which has the form p/q with q ≠ 0R1

WORKED EXAMPLE 4 Explain (exam level)

Explain why 0.999… = 1.

Solution

let x = 0.999…, then 10x = 9.999…M1

10x − x = 9, so 9x = 9 and x = 1A1

there is no real number between them, so they are equalR1

1.2 Set notation

Mathematicians use symbols to describe sets: ∈ means 'is an element of', ∉ 'is not an element of', and ⊂ 'is a subset of'. Set-builder notation describes a set by a rule, for example {x ∈ ℤ | −2 ≤ x < 3} = {−2, −1, 0, 1, 2}.

Figure 3. Set notation.

WORKED EXAMPLE 5 List (basic)

List the elements of {x ∈ ℤ | −3 < x ≤ 2}.

Solution

−2, −1, 0A1

1, 2A1

WORKED EXAMPLE 6 Decide (standard)

State whether true or false: (i) √16 ∈ ℕ, (ii) −5 ∈ ℕ, (iii) π ∉ ℚ.

Solution

(i) true (= 4); (ii) falseA1

(iii) trueA1

WORKED EXAMPLE 7 Write (standard)

Write in set-builder notation: the real numbers greater than −1 and at most 4.

Solution

{x ∈ ℝ | −1 < x ≤ 4}A1

2 Rational numbers and decimals

Terminating and recurring.

2.1 Decimal forms

Every rational number has a decimal that either terminates or recurs. A fraction in lowest terms terminates exactly when its denominator has no prime factors other than 2 and 5. Otherwise the decimal recurs. A dot over the digits shows the repeating block, for example 0.333… (written with a dot over the 3).

Figure 4. Types of decimal.

Figure 5. Which unit fractions 1/q terminate?

Figure 6. Proportion of unit fractions that terminate.

WORKED EXAMPLE 8 Decide (standard)

Without dividing, decide whether 7/40 and 5/12 terminate.

Solution

40 = 2³ × 5: 7/40 terminates (= 0.175)A1

12 = 2² × 3: 5/12 recurs (= 0.41666…)A1

WORKED EXAMPLE 9 Convert (basic)

Write 0.375 as a fraction in lowest terms.

Solution

375/1000M1

= 3/8A1

WORKED EXAMPLE 10 Explain (exam level)

Explain why the decimal of 1/7 must recur, with a period of at most 6.

Solution

in long division by 7 the only possible remainders are 1 to 6R1

a remainder must repeat within 6 steps, so the digits repeatR1

2.2 Converting recurring decimals to fractions

Let x be the recurring decimal. Multiply by 10ⁿ, where n is the number of repeating digits, so the repeating parts line up, then subtract to remove them.

Figure 7. Converting 0.2727… to a fraction.

WORKED EXAMPLE 11 Convert (standard)

Write 0.444… as a fraction.

Solution

x = 0.444…; 10x = 4.444…; 9x = 4M1

x = 4/9A1

WORKED EXAMPLE 12 Convert (standard)

Write 0.1515… as a fraction in lowest terms.

Solution

100x − x = 15, so 99x = 15M1

x = 15/99 = 5/33A1

WORKED EXAMPLE 13 Convert (exam level)

Write 0.3161616… (16 recurring) as a fraction in lowest terms.

Solution

10x = 3.1616… and 1000x = 316.1616…M1

990x = 313M1

x = 313/990A1

WORKED EXAMPLE 14 Show (exam level)

Show that 0.1222… = 11/90.

Solution

10x = 1.222…, 100x = 12.222…M1

90x = 11, so x = 11/90AG

3 Irrational numbers

Numbers that never repeat.

Think about it

How can we be sure that √2 is not some very complicated fraction?

3.1 Examples and approximations

√n is irrational whenever n is a positive integer that is not a perfect square. π and e are irrational. Irrational numbers have decimals that neither terminate nor recur, so in calculations we use exact forms (such as √2 or π) or approximations. The more decimal places, the smaller the error.

Figure 8. The spiral of Theodorus constructs √2, √3, … as lengths.

Figure 9. Error of decimal approximations of √2.

Figure 10. Errors of common approximations to π (log scale).

WORKED EXAMPLE 15 Calculate (standard)

Calculate the percentage error when 22/7 is used for π.

Solution

(22/7 − π) ÷ π × 100M1

≈ 0.040%A1

WORKED EXAMPLE 16 Explain (standard)

Explain how the spiral of Theodorus shows that √2 is a real length.

Solution

a right triangle with legs 1 and 1 has hypotenuse √(1² + 1²) = √2R1

so √2 is a point on the number lineR1

WORKED EXAMPLE 17 Compare (standard)

Which is the better approximation to π: 3.14 or 22/7?

Solution

errors: 0.00159 and 0.00126M1

22/7 is slightly betterA1

WORKED EXAMPLE 18 Estimate (standard)

Without a calculator, find two consecutive integers between which √90 lies.

Solution

81 < 90 < 100M1

so 9 < √90 < 10A1

3.2 Proof that √2 is irrational

A proof by contradiction assumes the opposite of what we want to prove and shows it leads to something impossible.

Figure 11. Proof by contradiction that √2 is irrational.

WORKED EXAMPLE 19 Prove (exam level)

Prove that √2 is irrational.

Solution

assume √2 = p/q, integers with no common factorM1

p² = 2q², so p² even, so p even; p = 2k gives q² = 2k², so q evenR1

p and q share factor 2: contradiction, so √2 is irrationalR1

WORKED EXAMPLE 20 Explain (exam level)

Explain why the same proof fails for √4.

Solution

4q² = p² gives p = 2qR1

no contradiction arises because √4 = 2 is rationalR1

4 Operations and closure

Combining numbers.

4.1 Closure

A set is closed under an operation if combining any two members always gives a member of the same set. The rationals are closed under addition, subtraction, multiplication and division (except by zero). Irrational numbers are not closed: √2 × √2 = 2 and √2 + (−√2) = 0. The sum of a rational and an irrational number is always irrational.

Figure 12. Closure of number sets.

WORKED EXAMPLE 21 Give (standard)

Give a counterexample to show ℕ is not closed under subtraction.

Solution

3 − 5 = −2A1

−2 ∉ ℕA1

WORKED EXAMPLE 22 Decide (standard)

State whether each result is rational: (i) √3 × √12, (ii) 2 + √5, (iii) π − π.

Solution

(i) √36 = 6: rational; (ii) irrationalA1

(iii) 0: rationalA1

WORKED EXAMPLE 23 Prove (exam level)

Prove that the sum of a rational number r and an irrational number x is irrational.

Solution

suppose r + x = s is rationalM1

then x = s − r is rational, since ℚ is closed under subtractionR1

contradiction, so r + x is irrationalR1

4.2 Density of rationals

Between any two rational numbers there is another rational number, for example their mean. So there are infinitely many rationals between any two of them. Irrational numbers are also found between any two rationals.

Figure 13. Density of the rational numbers.

WORKED EXAMPLE 24 Find (standard)

Find a rational number between 3/8 and 2/5.

Solution

mean = (3/8 + 2/5) ÷ 2 = (15/40 + 16/40) ÷ 2M1

= 31/80A1

WORKED EXAMPLE 25 Find (exam level)

Find an irrational number between 1.4 and 1.5.

Solution

√2 ≈ 1.414 is between themA1

or e.g. 1.41010010001… (non-repeating)A1

5 Surds

Exact forms of roots.

5.1 Simplifying and combining surds

A surd is an irrational root such as √2. Use √(ab) = √a × √b to simplify by taking out square factors, and √(a/b) = √a ÷ √b. Like surds can be added: 3√2 + 5√2 = 8√2, but √2 + √3 cannot be simplified. Writing numbers as products of primes helps find square factors.

Figure 14. Working with surds.

Figure 15. Prime factorisation of 360.

WORKED EXAMPLE 26 Simplify (basic)

Simplify √50.

Solution

√(25 × 2)M1

= 5√2A1

WORKED EXAMPLE 27 Simplify (standard)

Simplify √12 + √75.

Solution

2√3 + 5√3M1

= 7√3A1

WORKED EXAMPLE 28 Simplify (standard)

Simplify √360 using its prime factorisation.

Solution

360 = 2³ × 3² × 5 = 36 × 10M1

√360 = 6√10A1

WORKED EXAMPLE 29 Expand (standard)

Expand and simplify (3 + √2)(3 − √2).

Solution

9 − 3√2 + 3√2 − 2M1

= 7A1

5.2 Rationalising the denominator

To remove a surd from a denominator, multiply top and bottom by the surd, or by the conjugate when the denominator is a sum or difference: (a + √b)(a − √b) = a² − b.

WORKED EXAMPLE 30 Rationalise (standard)

Rationalise 10/√5.

Solution

10√5/5M1

= 2√5A1

WORKED EXAMPLE 31 Rationalise (exam level)

Rationalise 4/(3 − √5).

Solution

multiply by (3 + √5)/(3 + √5)M1

denominator 9 − 5 = 4M1

= 3 + √5A1

WORKED EXAMPLE 32 Show (exam level)

Show that 1/(√3 − 1) = (√3 + 1)/2.

Solution

multiply by (√3 + 1)M1

denominator 3 − 1 = 2, giving (√3 + 1)/2AG

6 Intervals and absolute value

Describing parts of the number line.

6.1 Interval notation

Intervals describe sets of real numbers. Square brackets include an endpoint, round brackets exclude it: [−2, 3) means −2 ≤ x < 3. On a number line a closed dot shows an included endpoint and an open dot an excluded one. ∞ is not a number, so it always has a round bracket.

Figure 16. The interval [−2, 3) on a number line.

Figure 17. Real numbers on a number line.

WORKED EXAMPLE 33 Write (basic)

Write −1 < x ≤ 5 in interval notation.

Solution

(−1, 5]A1

WORKED EXAMPLE 34 Write (standard)

Write the set of real numbers greater than or equal to 2 in interval and set-builder notation.

Solution

[2, ∞)A1

{x ∈ ℝ | x ≥ 2}A1

WORKED EXAMPLE 35 Order (standard)

Order from smallest: √10, 3.16, 19/6, π.

Solution

π ≈ 3.1416, 3.16, 19/6 ≈ 3.1667, √10 ≈ 3.1623M1

π < 3.16 < √10 < 19/6A1

6.2 Absolute value

The absolute value |x| is the distance of x from 0 on the number line, so it is never negative: |−5| = 5. The distance between a and b is |a − b|. The inequality |x| < 3 means −3 < x < 3.

Figure 18. The graph of y = |x|.

Figure 19. Sketch graphs for this topic.

WORKED EXAMPLE 36 Evaluate (basic)

Evaluate |−7| − |3 − 8|.

Solution

7 − 5M1

= 2A1

WORKED EXAMPLE 37 Solve (standard)

Solve |x − 2| = 5.

Solution

x − 2 = 5 or x − 2 = −5M1

x = 7 or x = −3A1

WORKED EXAMPLE 38 Solve (exam level)

Write |x − 1| ≤ 4 as an interval.

Solution

−4 ≤ x − 1 ≤ 4M1

−3 ≤ x ≤ 5, so [−3, 5]A1

7 Real numbers in the real world

Exact values and approximations in use.

7.1 Paper sizes and the golden ratio

A-series paper is designed so that folding a sheet in half gives the next size with the same shape. This requires the ratio of long side to short side to be √2, an irrational number, so real sheets use rounded millimetre values. The golden ratio φ = (1 + √5)/2 ≈ 1.618 is also irrational; ratios of consecutive Fibonacci numbers are rational numbers that get closer and closer to it.

Figure 20. A-series paper sizes.

Figure 21. Ratios of consecutive Fibonacci numbers approach the golden ratio.

WORKED EXAMPLE 39 Calculate (standard)

Calculate the ratio of long to short side for A4 (297 mm by 210 mm) and compare with √2.

Solution

297 ÷ 210 ≈ 1.4143M1

√2 ≈ 1.4142: agree to 3 decimal placesA1

WORKED EXAMPLE 40 Show (exam level)

Show that if a rectangle with sides L and S keeps its shape when halved, then L/S = √2.

Solution

halved sheet has sides S and L/2, so L/S = S/(L/2)M1

L² = 2S², so L/S = √2AG

WORKED EXAMPLE 41 Calculate (standard)

Calculate 13/8 and 21/13 and state which is closer to φ ≈ 1.6180.

Solution

13/8 = 1.625; 21/13 ≈ 1.6154A1

21/13 is closer (error 0.0026 vs 0.0070)A1

WORKED EXAMPLE 42 Explain (standard)

Explain why A4 paper cannot have exactly the ratio √2 when cut to whole millimetres.

Solution

whole-millimetre sides give a rational ratioR1

√2 is irrational, so it can only be approximatedR1

WORKED EXAMPLE 43 Justify (exam level)

A calculator shows √2 = 1.414213562. Justify whether this proves √2 is rational.

Solution

the display is rounded to a fixed number of digitsR1

the true decimal continues without repeating, so it does not show √2 is rationalR1

8 Data-based questions and further practice

Applying the ideas.

8.1 Investigation: which fractions terminate?

A student wrote 1/q as a decimal for q = 2 to 16 and recorded the results.

q	2	3	4	5	6	8	10	12	16
1/q	0.5	0.333…	0.25	0.2	0.1666…	0.125	0.1	0.08333…	0.0625

WORKED EXAMPLE 44 Pattern (exam level)

Describe the pattern and state a general rule.

Solution

2, 4, 5, 8, 10, 16 terminate; 3, 6, 12 recurA1

terminating denominators have only prime factors 2 and 5R1

rule: 1/q terminates if and only if q = 2ᵃ × 5ᵇA1

WORKED EXAMPLE 45 Verify (standard)

Use the rule to predict whether 1/20 and 1/24 terminate, and check.

Solution

20 = 2² × 5: terminates (0.05)A1

24 = 2³ × 3: recurs (0.041666…)A1

8.2 Graded practice

WORKED EXAMPLE 46 Recall (basic)

Give an example of a number that is rational but not an integer, and one that is real but not rational.

Solution

e.g. 3/4A1

e.g. √3 or πA1

WORKED EXAMPLE 47 Convert (standard)

Write 0.666… + 0.333… as a single number.

Solution

2/3 + 1/3M1

= 1A1

WORKED EXAMPLE 48 Simplify (standard)

Simplify (√5 + √3)(√5 − √3).

Solution

5 − 3M1

= 2A1

WORKED EXAMPLE 49 Analyse (exam level)

A student says √2 + √8 = √10. Analyse the error.

Solution

√a + √b ≠ √(a + b)R1

√8 = 2√2, so the sum is 3√2 ≈ 4.24A1

√10 ≈ 3.16, so the statement is falseA1

8.3 Graded practice 2

WORKED EXAMPLE 50 Recall (basic)

State whether −8.5 is an integer, rational or irrational.

Solution

rational (= −17/2), not an integerA1

WORKED EXAMPLE 51 Calculate (standard)

Find the exact perimeter of a square with area 72 cm².

Solution

side √72 = 6√2 cmM1

perimeter 24√2 cmA1

WORKED EXAMPLE 52 Rationalise (standard)

Rationalise 3/(2√3).

Solution

3√3/(2 × 3)M1

= √3/2A1

WORKED EXAMPLE 53 Justify (exam level)

Justify whether the product of two irrational numbers can be rational.

Solution

yes: √2 × √8 = 4R1

but not always: √2 × √3 = √6 is irrationalR1

so the irrationals are not closed under multiplicationA1

8.4 Multi-step problem: a garden design

WORKED EXAMPLE 54 Garden (exam level)

A square garden has area 50 m². (a) Find the exact side length. (b) Find the exact diagonal. (c) Fencing costs $12 per metre: find the cost of fencing the perimeter to the nearest dollar.

Solution

(a) √50 = 5√2 mA1

(b) 5√2 × √2 = 10 mM1 A1

8.5 Exam technique

•Give the smallest set a number belongs to when classifying.
•Simplify roots before deciding if a number is rational (√49 = 7).
•For recurring decimals multiply by 10ⁿ to line up the repeats.
•Keep surds exact unless asked for a decimal.
•Use square brackets for included endpoints; ∞ always gets a round bracket.
•In proofs state the assumption and the contradiction clearly.

8.6 Key terms

Term	Meaning
natural numbers ℕ	1, 2, 3, …
integers ℤ	…, −2, −1, 0, 1, 2, …
rational ℚ	can be written p/q, q ≠ 0
irrational	real but not rational
real numbers ℝ	all rational and irrational numbers
surd	irrational root, e.g. √3
closure	operation stays inside the set
absolute value	distance from zero

8.7 Self-test

Try these without notes, then check the answers.

Quick check

1. Is 0 a natural number?

2. 0.555… as a fraction?

3. √98 simplified?

4. Does 1/25 terminate?

5. Interval for 1 ≤ x < 4?

6. |−3.5|?

Answers: 1. depends on convention; state it 2. 5/9 3. 7√2 4. yes (0.04) 5. [1, 4) 6. 3.5

8.8 Self-test 2

Students often think…	Actually…
recurring decimals are irrational	they are rational
√(a + b) = √a + √b	only √(ab) = √a√b
22/7 equals π	it is an approximation
the sum of irrationals is irrational	√2 + (−√2) = 0
[2, ∞]	use [2, ∞)

Topic	Connection
1.5 Exponents and roots	surds and powers
1.4 Rounding	approximations
1.7 Error bounds	accuracy
3.2 Inequalities	intervals

Idea	Remember
Sets	ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ
Decimals	terminate or recur if rational
Recurring	multiply by 10ⁿ, subtract
Irrational	√2, π; proof by contradiction
Surds	simplify, rationalise
Intervals	[ ] and ()
\|x\|	distance from 0

Part	Answer	Marks
(a)	√81 = 9 and −2/5 rational	A1
	√8 = 2√2 irrational	A1
(b)	100x − x = 36	M1
	x = 36/99	A1
	= 4/11	A1
(c)	4√3 − 2√3	M1
	2√3	A1
(d)	multiply by (√5 − √2)	M1
	denominator 5 − 2 = 3	M1
	2(√5 − √2) = 2√5 − 2√2	A1