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IGCSE Mathematics 0580 · Chapter 1 notes · ~20 min read

Number toolkit

The foundations of every IGCSE Maths paper: types of number, primes with HCF and LCM, Venn diagrams, fractions, rounding and bounds, index laws, standard form and surds, with every step shown.

Types of numberPrimes, HCF & LCMSetsFractionsRounding & boundsIndicesStandard formSurds
RSby Riyaz Sir
mentorriyaz.in
No tag = Core (everyone)EXTENDED Extended onlyhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Every number belongs to a family: natural, integer, rational or irrational. Prime factors give the HCF and LCM, and Venn diagrams show how sets overlap.
  • Follow BIDMAS, make denominators equal before adding fractions, and round only at the end. Index laws and standard form make very big and very small numbers easy to handle.
  • Extended: change recurring decimals to fractions, find bounds of calculations, and simplify and rationalise surds.

By the end you can

  • Identify natural numbers, integers, primes, squares, cubes, factors, multiples, rational and irrational numbers and reciprocals.
  • Write a number as a product of prime factors and find the HCF and LCM.
  • Use set notation and Venn diagrams, and solve problems with them.
  • Use BIDMAS, work with negative numbers and calculate with fractions, decimals and percentages.
  • Round to decimal places and significant figures, estimate, and give the limits of accuracy of a rounded value.
  • Use the index laws with positive, zero and negative powers, and write numbers in standard form.
  • Change recurring decimals into fractions. EXT
  • Find upper and lower bounds in calculations. EXT
  • Use fractional indices, calculate in standard form and simplify and rationalise surds. EXT

1Types of number

The number family

REAL NUMBERS: every number on the number lineRATIONAL: can be written as a fractionINTEGERS (whole numbers)NATURAL NUMBERS1, 2, 3, 4, 5, …(counting numbers)… −3, −2, −1, 0negative numbers and zeroare integers too½ 0.75−3.2 2⅓0.333… (0.3̇)terminating orrecurring decimalsIRRATIONALcannot be writtenas a fractionπ = 3.14159…√2 = 1.41421…√3 = 1.73205…√5 = 2.23606…decimals thatnever end andnever repeat
Every number you meet in IGCSE is real. It is either rational (a fraction exists) or irrational (it does not).

Integer — a whole number: positive, negative or zero (…, −2, −1, 0, 1, 2, …).

Prime number — a number with exactly two factors: 1 and itself.

Rational number — a number that can be written as a fraction of two integers. It is a terminating or recurring decimal.

Irrational number — a number that cannot be written as a fraction. Its decimal never ends and never repeats, for example π and √2.

Reciprocal — the number you multiply by to get 1. The reciprocal of x is 1 ÷ x.

  • Factors of 12 are the numbers that divide into it: 1, 2, 3, 4, 6, 12. Multiples of 12 are its times table: 12, 24, 36, …
  • Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Cube numbers: 1, 8, 27, 64, 125.
  • Primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. 1 is not prime, and 2 is the only even prime.
  • The reciprocal of 23 is 32. The reciprocal of 5 is 15. The reciprocal of 0.25 is 4. Zero has no reciprocal.
Common mistake"√9 = 3, so it is irrational." No. A root is only irrational if it is not exact. √9 = 3 is rational, but √2 and √7 are irrational.
Worked example

Which of these are irrational? π, √16, 0.7, √5, 227

  1. π is irrational (its decimal never ends or repeats)
  2. √16 = 4 is rational
  3. 0.7 = 710 is rational
  4. √5 is irrational (5 is not a square number)
  5. 227 is a fraction, so it is rational (it is only an approximation to π)
  6. answer: π and √5

Negative numbers

−2 + 5 = 3−6−5−4−3−2−10123456+53 − 7 = −4−6−5−4−3−2−10123456−73 − (−2) = 3 + 2 = 5−6−5−4−3−2−10123456−(−2) = +2
On the number line, adding moves right and subtracting moves left. Subtracting a negative moves right.
  • Adding a negative = subtracting: 5 + (−2) = 3. Subtracting a negative = adding: 5 − (−2) = 7.
  • Multiplying or dividing: same signs → positive, different signs → negative. (−4) × (−6) = 24, (−12) ÷ 3 = −4.
Worked example

Work out −7 + 12 − (−3) − 5.

  1. −7 + 12 = 5
  2. 5 − (−3) = 5 + 3 = 8
  3. 8 − 5 = 3
Exam tipOn a calculator use the (−) key for a negative number, and put brackets around it: (−3)² = 9 but −3² = −9.

2Factors, multiples and primes

Prime factors

728924332260610232572 = 23 × 3260 = 22 × 3 × 5green = prime: stop splitting
Keep splitting until every branch ends in a prime. Write the primes with powers.
  • Prime factor decomposition: write the number as a product of primes. Use a factor tree: split into any two factors, then keep splitting the composite ones.
  • 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32. 60 = 22 × 3 × 5. Write the smallest prime first.
Worked example

Write 360 as a product of prime factors.

  1. 360 = 36 × 10
  2. 36 = 6 × 6 = 2 × 3 × 2 × 3 and 10 = 2 × 5
  3. collect: 2 × 2 × 2 × 3 × 3 × 5
  4. = 23 × 32 × 5

HCF and LCM

prime factors of 72prime factors of 60232235only in 72only in 60in bothHCF = multiply the overlap = 2 × 2 × 3 = 12(the biggest number that divides both)LCM = multiply every number, overlap once: 2×3×2×2×3×5 = 360(the smallest number both go into)
Put the prime factors in a Venn diagram. The overlap gives the HCF, and everything (overlap once) gives the LCM.

HCF — the highest common factor: the biggest number that divides into both numbers.

LCM — the lowest common multiple: the smallest number that both numbers divide into.

  • Prime factor method: HCF = lowest power of each shared prime. LCM = highest power of every prime that appears.
  • 72 = 23 × 32 and 60 = 22 × 3 × 5. HCF = 22 × 3 = 12. LCM = 23 × 32 × 5 = 360.
Exam tipCheck: HCF × LCM = the product of the two numbers. 12 × 360 = 4 320 = 72 × 60. ✓
Worked example

A bus leaves a station every 12 minutes and a tram every 18 minutes. Both leave at 08:00. When do they next leave together?

  1. you need the LCM of 12 and 18
  2. 12 = 22 × 3 and 18 = 2 × 32
  3. LCM = 22 × 32 = 36
  4. they leave together again at 08:36
Worked example

Find the HCF and LCM of 24 and 36.

  1. 24 = 23 × 3 and 36 = 22 × 32
  2. HCF = lowest powers = 22 × 3 = 12
  3. LCM = highest powers = 23 × 32 = 72
  4. check: 12 × 72 = 864 = 24 × 36 ✓

3Sets and Venn diagrams

Set language

SymbolMeaningExample
ξthe universal set: everything we are consideringξ = {1, 2, …, 10}
∈is an element of3 ∈ {1, 2, 3}
∉is not an element of5 ∉ {1, 2, 3}
A ∪ Bunion: in A or B (or both){1, 2} ∪ {2, 3} = {1, 2, 3}
A ∩ Bintersection: in A and B{1, 2} ∩ {2, 3} = {2}
A′the complement: not in Aeverything in ξ except A
n(A)the number of elements in An({4, 6, 8}) = 3
∅ or { }the empty setno elements
A ⊂ BA is a subset of B (all of A is inside B){2} ⊂ {1, 2, 3}

Shading regions

A ∪ BξABA or B (or both)A ∩ BξABA and BA′ξABnot A(A ∪ B)′ξABneither A nor BA′ ∩ BξABB onlyA ∩ B′ξABA only
Shade the region that matches the words: or = union, and = intersection, not = complement.
  • Read the symbols as words: ∪ = "or", ∩ = "and", ′ = "not". Then find which parts of the diagram match.
  • For a complement, shade everything outside the set, including the area outside both circles.
Common mistakeLeaving out the area outside the circles. In A′ the shading covers B-only and the region inside the rectangle but outside both circles.

Problems with Venn diagrams

ξ = 30Football (F)Cricket (C)12693F onlybothC onlyneithern(F) = 18, n(C) = 15n(F ∩ C) = 6, none = 31 Both: write 6 first2 F only = 18 − 6 = 123 C only = 15 − 6 = 94 Neither = 30 − 27 = 3n(F ∪ C) = 18 + 15 − 6 = 27
Fill the overlap first, then work outwards, then find the number in neither set.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
subtract the overlap so it is not counted twice
  • Method: (1) write the number in both sets first; (2) subtract it from each total to find "only" numbers; (3) add everything and take it from the total to find "neither".
Worked example

In a class of 40, 24 speak French (F), 20 speak Spanish (S) and 8 speak both. How many speak neither?

  1. both = 8
  2. French only = 24 − 8 = 16, Spanish only = 20 − 8 = 12
  3. in at least one language = 16 + 8 + 12 = 36
  4. neither = 40 − 36 = 4

4Fractions, decimals and percentages

Order of operations

BBrackets( )IIndicespowers and rootsDMDivide / Multiplyleft to rightASAdd / Subtractleft to rightWork out 3 + 4 × 223 + 4 × 223 + 4 × 4power first3 + 16multiply19add lastNOT 7 × 4 = 28 (adding first is wrong)
Brackets, then Indices, then Divide and Multiply (left to right), then Add and Subtract (left to right).
  • BIDMAS (or BODMAS). Operations of the same level are done left to right: 12 ÷ 3 × 2 = 4 × 2 = 8 (not 2).
  • A fraction line acts as a bracket: 5 + 72 × 3 means (5 + 7) ÷ (2 × 3) = 2.

Fractions

two thirds = 8 twelfthsone quarter = 3 twelfths2/3 + 1/4 = 8/12 + 3/12 = 11/12Same denominator first (12), then add the numerators.
To add or subtract, make the denominators the same. The numerators then add as whole numbers.
  • Add / subtract: common denominator, then add or subtract the numerators: 23 + 14 = 812 + 312 = 1112.
  • Multiply: top × top, bottom × bottom (cancel first): 23 × 910 = 35.
  • Divide: keep, change, flip: 56 ÷ 23 = 56 × 32 = 1512 = 54.
  • Mixed numbers: change to improper fractions first: 312 = 72.
  • Fraction of an amount: divide by the denominator, multiply by the numerator. 38 of 64 = 64 ÷ 8 × 3 = 24.
Worked example

Work out 413 − 156.

  1. change to improper fractions: 133 − 116
  2. common denominator 6: 266 − 116
  3. = 156
  4. simplify: 52 = 212
Worked example

Write in order of size (smallest first): 25, 0.38, 37%.

  1. change all to decimals
  2. 25 = 0.4, 0.38, 37% = 0.37
  3. order: 37%, 0.38, 25

Fraction, decimal and percentage equivalents

FractionDecimalPercentage
1/20.550%
1/40.2525%
3/40.7575%
1/50.220%
1/80.12512.5%
1/30.333…33.3…%
2/30.666…66.6…%
  • Learn these: they save time in non-calculator papers. To change a fraction to a percentage, multiply by 100: 38 = 37.5%.

Recurring decimals as fractionsEXTENDED

  • EXT Algebra method: let x = the recurring decimal. Multiply by 10, 100… so the recurring parts line up, then subtract.
Worked example

EXT Write 0.454545… (the digits 45 repeat) as a fraction in its simplest form.

  1. let x = 0.454545…
  2. 100x = 45.454545…
  3. subtract: 99x = 45
  4. x = 4599 = 511
Worked example

EXT Write 0.16666… (only the 6 repeats) as a fraction.

  1. let x = 0.1666…
  2. 10x = 1.666… and 100x = 16.666…
  3. subtract: 90x = 15
  4. x = 1590 = 16

5Accuracy

Rounding

Round 3.847 to 1 decimal place3.83.853.9rounds DOWN to 3.8rounds UP to 3.93.847halfway = 3.85Round 0.004739 to 3 significant figures0.004739leading zeros do not count3 significant figuresnext digit 9 ≥ 5: round up0.004745 or more: round UP. 4 or less: stay the same. Count s.f. from the first non-zero digit.
Look at the digit after the one you keep. 5 or more rounds up. Zeros at the start do not count as significant figures.
  • Decimal places (d.p.) count after the decimal point. Significant figures (s.f.) count from the first non-zero digit.
  • 45 678 → 46 000 (2 s.f.). 0.004739 → 0.00474 (3 s.f.). 5.9996 → 6.00 (2 d.p.).
Exam tipGive answers to 3 significant figures unless the question says otherwise (angles to 1 decimal place). Keep the full calculator value until the last step, then round.
Common mistakeRounding in the middle of a calculation. This builds up errors and loses the accuracy mark.

Estimation

  • Round every number to 1 significant figure first, then calculate. It checks that a calculator answer is sensible.
Worked example

Estimate 48.7 × 3.10.52.

  1. round to 1 s.f.: 48.7 → 50, 3.1 → 3, 0.52 → 0.5
  2. 50 × 30.5 = 1500.5
  3. = 300

Upper and lower bounds

7.4 cm measured to 1 decimal place7.307.357.407.457.50lower boundupper bound7.47.35 ≤ L < 7.45 (half a unit either side; the upper bound is not reached)Rectangle 5 cm × 3 cm, each side to the nearest cm5 × 3smallest area: 4.5 × 2.5 = 11.25largest area: 5.5 × 3.5 = 19.25lower × lower = lowestupper × upper = highestinner = lower bounds, outer = upper bounds
A value of 7.4 cm to 1 d.p. lies from 7.35 up to, but not including, 7.45.
  • A rounded value is somewhere in an interval. Go half a unit either side: 7.4 cm (1 d.p.) → 7.35 ≤ L < 7.45. The upper bound is not reached, but we use it in calculations.
  • EXT Smallest sum / difference / product / quotient:
CalculationLargest answerSmallest answer
a + bupper + upperlower + lower
a − bupper − lowerlower − upper
a × bupper × upperlower × lower
a ÷ bupper ÷ lowerlower ÷ upper
Worked example

EXT A runner covers 120 m (nearest 10 m) in 15 s (nearest second). Find the greatest possible speed.

  1. distance: 115 ≤ d < 125, time: 14.5 ≤ t < 15.5
  2. speed = distance ÷ time
  3. greatest speed = upper distance ÷ lower time
  4. = 125 ÷ 14.5 = 8.62 m/s (3 s.f.)

6Powers, roots and standard form

Index laws

24 =16÷ 223 =8÷ 222 =4÷ 221 =2÷ 220 =1÷ 22−1 =½÷ 22−2 =¼keep dividing by 2multiply (same base): add the powersam × an = am+ndivide (same base): subtract the powersam ÷ an = am−npower of a power: multiply the powers(am)n = amnzero and negative powersa0 = 1, a−n = 1 ÷ anfractional powers (root first)a1/n = ⁿ√a, am/n = (ⁿ√a)m
Add the powers when multiplying, subtract when dividing, multiply when raising a power to a power.
  • Same base only. a3 × a4 = a7; a8 ÷ a3 = a5; (a2)3 = a6. (Algebra uses the same laws with letters.)
  • Zero: a0 = 1. Negative: a−n = 1an, so 2−3 = 18. A negative power means reciprocal, not a negative number.
  • EXT Fractional: a1/n = n-th root of a; am/n = (ⁿ√a)m. Root first, then the power.
Worked example

EXT Work out 8−2/3.

  1. the denominator 3 means cube root: ∛8 = 2
  2. the numerator 2 means square: 2² = 4
  3. the negative means reciprocal: 14
Worked example

Simplify (2x3)2 × 3x−1.

  1. (2x3)2 = 2² × x6 = 4x6
  2. multiply: 4x6 × 3x−1 = 12x5 (multiply numbers, add powers of x)
Worked example

EXT Solve 2x+1 = 32.

  1. write 32 as a power of 2: 32 = 25
  2. the bases match, so the powers match: x + 1 = 5
  3. x = 4

Standard form

45 000 → 4.5 × 104450004 places left4.5 × 1040.00062 → 6.2 × 10−40.000624 places right6.2 × 10−4big number: power is +small number: power is −
Standard form is A × 10ⁿ with A between 1 and 10. Count how many places the decimal point moves.
  • A × 10n with 1 ≤ A < 10 and n an integer. Big numbers: n positive. Small numbers: n negative.
  • 45 000 = 4.5 × 104. 0.00062 = 6.2 × 10−4. And 7.5 × 10−3 = 0.0075.
  • EXT Multiply / divide: multiply or divide the A parts, and use the index laws on the powers. Adjust the answer if A is not between 1 and 10.
  • EXT Add / subtract: write both with the same power of 10 first.
Common mistake45 × 103 or 0.45 × 105 is not standard form. A must be at least 1 and less than 10.
Worked example

EXT Work out (4 × 106) × (5 × 103). Give the answer in standard form.

  1. multiply the numbers: 4 × 5 = 20
  2. add the powers: 106 × 103 = 109
  3. 20 × 109 is not standard form
  4. 2 × 10 × 109 = 2 × 1010
Worked example

EXT Work out 2.5 × 104 + 3.5 × 103.

  1. same power: 3.5 × 103 = 0.35 × 104
  2. add: 2.5 + 0.35 = 2.85
  3. answer: 2.85 × 104

SurdsEXTENDED

11√2Pythagoras: 1² + 1² = 2so the diagonal is √2 = 1.41421…The decimal never ends and never repeats,so √2 is irrational.21√51² + 2² = 5
A surd is a root that is not exact. √2 is the diagonal of a unit square, and its decimal never ends.

Surd — EXT an irrational root that cannot be written exactly as a decimal or fraction, for example √2, √3 and 5√7.

  • EXT Rules: √a × √b = √(ab), and √a ÷ √b = √(a ÷ b). Simplify by taking out the biggest square factor: √50 = √25 × √2 = 5√2.
  • EXT Add / subtract only like surds: 2√3 + 3√3 = 5√3.
  • EXT Rationalise the denominator: multiply top and bottom by the surd (or by the conjugate if the denominator is a sum): 6√3 = 6√33 = 2√3.
Worked example

EXT Simplify √12 + √27.

  1. √12 = √4 × √3 = 2√3
  2. √27 = √9 × √3 = 3√3
  3. add like surds: 2√3 + 3√3 = 5√3
Worked example

EXT Rationalise the denominator of 12 + √3.

  1. multiply top and bottom by the conjugate (2 − √3)
  2. top = 1 × (2 − √3) = 2 − √3
  3. bottom = (2 + √3)(2 − √3) = 4 − 3 = 1
  4. answer: 2 − √3
Memory trickThe conjugate of (a + √b) is (a − √b). Multiplying them gives a² − b: the square root vanishes.

7Test yourself

Test yourself — cover the answers first

Quick check

  1. Is 1 a prime number?
  2. List the factors of 18.
  3. Write 84 as a product of prime factors.
  4. Find the HCF of 36 and 48.
  5. Find the LCM of 6 and 8.
  6. What is the reciprocal of 34?
  7. Work out (−5) × (−3) − 4.
  8. Work out 3 + 4 × 2².
  9. Work out 34 ÷ 38.
  10. Write 0.375 as a fraction.
  11. Write 0.00473 to 2 significant figures.
  12. Write 3.456 to 2 decimal places.
  13. Write 7.4 cm (1 d.p.) as an inequality.
  14. Simplify a5 × a−2.
  15. Evaluate 50 + 2−2.
  16. Write 360 000 in standard form.
  17. Write 0.0000805 in standard form.
  18. EXT Evaluate 163/4.
  19. EXT Write 0.777… (7 repeating) as a fraction.
  20. EXT Simplify √72.
  21. EXT Rationalise 10√5.
  22. EXT Work out (3 × 105) ÷ (6 × 102) in standard form.
  23. EXT A length is 12 cm to the nearest cm. Write its upper bound.

Exam-style

  1. Write 180 as a product of its prime factors. Hence find the HCF of 180 and 150. [4]
  2. In a class of 30 students, 18 play tennis (T), 14 play hockey (H) and 6 play neither. Find how many play both, and draw the Venn diagram numbers. [4]
  3. Work out 23 + 34 and give the answer as a mixed number. [3]
  4. Estimate the value of 19.8 × 4.10.48, showing the rounding. [2]
  5. EXT Write 0.272727… (27 repeating) as a fraction in its simplest form. [3]
  6. EXT A square has sides of length 8.5 cm correct to 1 decimal place. Find the lower and upper bounds of its perimeter. [3]
  7. EXT Simplify fully (2 + √3)² and show that it is not a rational number. [3]
  8. EXT Work out (6.4 × 105) ÷ (2 × 102) and give your answer in standard form. [2]
Show answers
  1. no (it has only one factor)
  2. 1, 2, 3, 6, 9, 18
  3. 2² × 3 × 7
  4. 12
  5. 24
  6. 43
  7. 11
  8. 19
  9. 2
  10. 38
  11. 0.0047
  12. 3.46
  13. 7.35 ≤ L < 7.45
  14. a³
  15. 54
  16. 3.6 × 105
  17. 8.05 × 10−5
  18. 8
  19. 79
  20. 6√2
  21. 2√5
  22. 5 × 102
  23. 12.5 cm
  24. 180 = 2² × 3² × 5
    150 = 2 × 3 × 5²
    HCF = lowest powers of shared primes = 2 × 3 × 5
    = 30
  25. in at least one sport = 30 − 6 = 24
    n(T) + n(H) = 18 + 14 = 32
    both = 32 − 24 = 8
    tennis only 10, both 8, hockey only 6, neither 6
  26. common denominator 12: 812 + 912
    = 1712
    = 1512
  27. 20 × 4 and 0.5
    800.5 = 160
  28. x = 0.2727…, 100x = 27.2727…
    99x = 27
    x = 2799 = 311
  29. side bounds: 8.45 ≤ s < 8.55
    lower perimeter = 4 × 8.45 = 33.8
    upper perimeter = 4 × 8.55 = 34.2
  30. (2 + √3)(2 + √3) = 4 + 4√3 + 3
    = 7 + 4√3
    √3 is irrational, so 7 + 4√3 is irrational
  31. 6.4 ÷ 2 = 3.2
    105 ÷ 102 = 103, so 3.2 × 103
Practise this chapter
Types of numberSetsPowers & rootsFractions, decimals & percentagesThe four operationsIndices IStandard formEstimationLimits of accuracySurds
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