LEARNING OBJECTIVES
By the end of this chapter, you should be able to:
- classify numbers and calculate HCF, LCM, powers and roots
- use set notation and interpret two- or three-set Venn diagrams
- convert between fractions, decimals and percentages
- order positive and negative quantities and apply operation rules
- use positive, zero, negative and fractional indices
- convert and calculate with numbers in standard form
- round, estimate and calculate upper and lower bounds
- solve ratio, proportion, rate, speed and density problems
- calculate percentage change, interest and reverse percentages
- work accurately with calculators, time, money and currencies
- model exponential growth and decay
- simplify surds and rationalise denominators.
THE BIG IDEA
A calculation is only useful when its meaning is clear
Number work is not a collection of unrelated tricks. The same ideas—place value, equivalence, scale and accuracy—connect almost every topic in mathematics. For example, multiplying by 1.08, finding 108% and applying an 8% increase all describe the same change.
A strong solution has three parts: choose a sensible representation, carry out a correct method, and present the result at an accuracy that fits the context.
SECTION 01
Types of number and prime structure
Classifying a number tells you which methods and properties can be used. One number can belong to several groups: 64 is natural, integer, rational, square and cube.
The counting numbers 1, 2, 3, …
Whole numbers, including negative numbers and zero
Integers greater than 1 with exactly two positive factors
Numbers of the form n², such as 1, 4, 9 and 16
Numbers of the form n³, such as 1, 8, 27 and 64
Numbers that can be written as a/b, where a and b are integers and b ≠ 0
Numbers that cannot be written as a fraction of two integers, such as √2 and π
The number that multiplies a given non-zero number to make 1
Prime factorisation, HCF and LCM
Writing an integer as a product of primes reveals its structure. Divide by the smallest possible prime repeatedly until every factor is prime.
Finding the HCF and LCM of 84 and 126
1Prime factors84 = 2² × 3 × 7 and 126 = 2 × 3² × 7.
2HCF: shared primes with the smaller power2 × 3 × 7 = 42.
3LCM: every prime with the larger power2² × 3² × 7 = 252.
Reciprocal
For a non-zero number x, the reciprocal is 1/x. For example, the reciprocal of 3/5 is 5/3, and their product is 1.
SECTION 02
Sets and Venn diagrams
A set is a collection of clearly defined objects. Venn diagrams make relationships such as overlap and exclusion visible.
n(A)number of elements in Ax ∈ Ax is an element of Ax ∉ Ax is not an element of AA ∪ Bin A or B, including bothA ∩ Bin both A and BA′not in A, but inside the universal setA ⊆ Bevery element of A is in B∅the empty setReading a two-set Venn diagram
In a group of 40 students, 23 study French, 18 study Mandarin and 9 study both.
Important: subtract the overlap before adding regions, or those students are counted twice.
Three-set Venn diagrams (Extended)
With three sets, begin with the region common to all three. Next complete the regions shared by exactly two sets, then the single-set regions, and finally the area outside every circle. Check that all eight regions add to the universal total.
SECTION 03
Powers and roots
A power describes repeated multiplication. In 5³, the base is 5 and the index is 3, so 5³ = 125. A root reverses a power: ∛125 = 5.
SECTION 04
Fractions, decimals and percentages
These are different representations of the same value. Choose the form that makes the next calculation easiest.
Converting 7/16 into other forms
7 ÷ 16 = 0.4375, so 7/16 = 0.4375 = 43.75%.
Recurring decimals (Extended)
A recurring decimal repeats the same digit or block forever. A dot or bar marks the repeating part. Algebra can convert it exactly into a fraction.
Writing 0.272727… as a fraction
Let x = 0.272727…
100x = 27.272727…
100x − x = 27
99x = 27
x = 27/99 = 3/11
SECTION 05
Ordering and the four operations
To compare quantities, first write them in the same form. For negative numbers, the value farther left on a number line is smaller: −7 < −2.
Evaluating 18 − 3(2² + 1) ÷ 5
18 − 3(4 + 1) ÷ 5
= 18 − 3(5) ÷ 5
= 18 − 15 ÷ 5
= 18 − 3 = 15
FRACTION RULES
- Add or subtract using a common denominator.
- Multiply numerators and denominators, cancelling common factors where possible.
- To divide by a fraction, multiply by its reciprocal.
- Convert mixed numbers to improper fractions before multiplying or dividing.
SECTION 06
Index laws
The laws work because indices count repeated factors. They apply when the base is the same.
Evaluating 27²⁄³
The denominator 3 means cube root; the numerator 2 means square.
27²⁄³ = (∛27)² = 3² = 9.
SECTION 07
Standard form
Standard form records very large or small values compactly while keeping place value clear.
The decimal point moves six places left.
The decimal point moves five places right.
Calculating (3.6 × 10⁷)(5 × 10⁻³)
Multiply ordinary numbers and add powers:
(3.6 × 5) × 10⁷⁻³
= 18 × 10⁴
= 1.8 × 10⁵
18 × 10⁴ is not in standard form because 18 is not less than 10.
SECTION 08
Rounding, estimation and limits of accuracy
Decimal places and significant figures
Decimal places count digits after the decimal point. Significant figures begin at the first non-zero digit. Inspect the next digit: 5 or more rounds up; 4 or less stays.
Estimation
For a quick estimate, round each value to one significant figure before calculating. The estimate checks whether a calculator answer has a sensible size.
Estimating (59.7 × 0.418) ÷ 8.12
(60 × 0.4) ÷ 8 = 24 ÷ 8 = 3. A calculator answer near 3 is reasonable.
Bounds
Lower and upper bounds
The smallest possible value is the lower bound. The greatest limiting value is the upper bound, which is normally not included.
Area bounds for a measured rectangle (Extended)
A length is 8.4 cm and a width is 3.2 cm, each correct to the nearest 0.1 cm.
8.35 ≤ length < 8.45
3.15 ≤ width < 3.25
Lower area = 8.35 × 3.15 = 26.3025 cm²
Upper area = 8.45 × 3.25 = 27.4625 cm²
SECTION 09
Ratio and proportion
A ratio compares quantities using the same units. Proportion describes how quantities change together.
Dividing $840 in the ratio 3 : 4
1Total parts3 + 4 = 7.
2One part$840 ÷ 7 = $120.
3Shares3 × $120 = $360 and 4 × $120 = $480.
Direct proportion and unitary method
If five notebooks cost $17.50 at the same rate, one costs $3.50, so eight cost 8 × $3.50 = $28. Finding one unit first is the unitary method.
SECTION 10
Rates and compound measures
A rate compares quantities with different units, such as kilometres per hour or grams per cubic centimetre.
Average speed with mixed time units
A train travels 132 km in 1 hour 36 minutes. Since 36 minutes = 36/60 = 0.6 hours, the total time is 1.6 hours.
Average speed = 132 ÷ 1.6 = 82.5 km/h.
Other common rates include pay per hour, currency exchange, water flow and fuel use. Always write units beside the result; they show which quantity was divided by which.
SECTION 11
Percentage calculations
A percentage multiplier turns a verbal change into one multiplication. This is especially useful for repeated changes.
Reverse percentage after a discount
A jacket costs $68 after a 15% discount. The new price is 85% of the original, so its multiplier is 0.85.
Original price = 68 ÷ 0.85 = $80.
Compound interest
$2500 is invested for four years at 3.2% compound interest each year.
Value = 2500(1.032)⁴ = $2835.69, to the nearest cent.
SECTION 12
Calculator use, time and money
Efficient calculator use
- Enter the whole calculation using brackets so the intended structure is preserved.
- Keep the full display during intermediate steps and round only the final answer.
- Use an estimate to detect an input or mode error.
- Interpret context: 4.8 dollars means $4.80, but 4.8 hours means 4 hours 48 minutes.
Time
Finding a finishing time
A flight leaves at 22:45 and lasts 3 hours 38 minutes. Adding 3 hours gives 01:45 the next day; adding 38 minutes gives 02:23.
For timetables and time zones, decide which location is ahead before adding or subtracting the difference. Include any date change when crossing midnight.
Money and exchange rates
Read the direction of the exchange rate. If 1 SGD = 3.48 MYR, convert SGD to MYR by multiplying by 3.48. Convert MYR to SGD by dividing by 3.48.
SECTION 13
Exponential change and surds (Extended)
Exponential models repeat the same multiplier, while surds keep irrational roots exact.
Exponential growth and decay
Three years of depreciation
A machine worth $2400 loses 12% of its value each year. The decay multiplier is 0.88.
Value after 3 years = 2400(0.88)³ = $1635.53, to the nearest cent.
Surds
Surd
An irrational root written in exact form, such as √3. It has not been replaced by a rounded decimal.
Simplifying and rationalising
√72 = √(36 × 2) = 6√2
5/√3 = (5/√3)(√3/√3) = 5√3/3
1/(3 − √2) = (3 + √2)/(9 − 2) = (3 + √2)/7
QUICK CHAPTER SUMMARY
The ideas to carry forward
- Prime factorisation gives an efficient route to HCF and LCM.
- Sets use precise symbols for overlap, union, complement and membership.
- Fractions, decimals and percentages are equivalent representations.
- Index laws require a common base; standard form requires 1 ≤ A < 10.
- Round only when required and use estimates to check magnitude.
- A rounded measurement represents an interval between two bounds.
- Ratios compare like units; rates compare different units.
- Percentage multipliers handle increases, decreases and repeated change.
- Time decimals are parts of an hour, not minutes.
- Surds preserve exact values and denominators can be rationalised.