Cambridge O Level Mathematics: Number (4024)

Natural numbers and integers

Natural numbers are the counting numbers 1, 2, 3 and so on. Integers add zero and the negative whole numbers.

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Factors and multiples

A factor divides a number exactly. A multiple is in its times table. 12 cupcakes fit in 1, 2, 3, 4, 6 or 12 equal rows.

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Prime numbers

A prime number has exactly two factors, 1 and itself. So 1 is not prime, and 2 is the only even prime.

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Writing a number as a product of prime factors

Split a number with a factor tree until every branch ends in a prime. 72 = 2³ × 3², and no other set of primes gives 72.

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Common factors and the highest common factor (HCF)

The HCF is the largest number that divides both. From prime factors, take only the primes the two numbers share.

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Common multiples and the lowest common multiple (LCM)

The LCM is the smallest number both divide into. Two buses leaving together meet again after the LCM of their gaps.

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Square numbers and cube numbers

A square number makes a square of dots, like 36 = 6 × 6. A cube number makes a cube of blocks, like 27 = 3 × 3 × 3.

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Rational and irrational numbers

A rational number can be written as a fraction of two integers. √2 and π cannot, so their decimals never end or repeat.

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Reciprocals

The reciprocal of a number is 1 divided by it, so a number times its reciprocal is 1. Flip a fraction to find it.

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Types of number: timed practice

A timed round: name the type of each number, then mixed questions on primes, prime factors, the HCF and the LCM.

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12.

Types of number: exam practice

Past-paper style questions with mark schemes: prime factors, the HCF and LCM, and choosing the irrational number from a list.

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13.

Types of number: summary

Every type of number on one page, the HCF and LCM method, the common mistakes, and a flashcard deck.

1.3 · Powers and roots

Squares and square roots, cubes and cube roots, other powers and roots, and calculations that mix them, with the squares to 15² and the cubes you must know by heart.

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Squares and square roots

Squaring multiplies a number by itself. The square root undoes it: 13² = 169, so √169 = 13. Learn the squares up to 15².

A cube is a number times itself three times. The cube root undoes it: 4³ = 64, so ∛64 = 4. A cube root of a negative is negative.

Cubes and cube roots

Proper fractions, improper fractions and mixed numbers

Other powers and roots

A power counts how many times a number is multiplied by itself: 2⁵ = 32. The fifth root of 32 goes back to 2.

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Calculating with powers and roots

Work out powers and roots before multiplying or adding. 5² × ∛8 is 25 × 2, which is 50.

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Powers and roots: timed practice

A timed recall round on squares, cubes and their roots, then mixed calculations without a calculator.

Powers and roots: exam practice

Paper 1 style questions with mark schemes: write down a root, work out a mixed calculation, find a missing power.

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Powers and roots: summary

The squares to 15², the cubes you must know, the common mistakes, and a flashcard deck for recall.

1.4 · Fractions, decimals and percentages

Proper and improper fractions and mixed numbers, equivalent fractions in simplest form, a fraction of a quantity, and converting between fractions, decimals, recurring decimals and percentages.

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A proper fraction is less than 1. An improper fraction has a top bigger than its bottom, like 7/4, which is the mixed number 1¾.

Equivalent fractions and simplest form

Multiply or divide the top and bottom by the same number and the fraction keeps its size. Divide by the HCF for simplest form.

Further left on the number line means smaller, so -7 is smaller than -3 even though 7 is bigger than 3.

A fraction of a quantity

Divide by the bottom number to find one part, then multiply by the top. Three quarters of 240 taka is 180 taka.

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Converting between fractions and decimals

A fraction is a division: 3/8 means 3 ÷ 8 = 0.375. A decimal goes back to a fraction over 10, 100 or 1000.

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Terminating and recurring decimals

Some divisions stop and some repeat forever. A dot over a digit marks the repeat: 0.1777... is written 0.17 with a dot on the 7.

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Converting a recurring decimal to a fraction

Call the decimal x, multiply by 10 or 100 to line up the repeats, then subtract so the endless part cancels.

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Converting between fractions, decimals and percentages

Per cent means out of 100. Multiply a decimal by 100 to get a percentage: 3/8 = 0.375 = 37.5%.

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Fractions, decimals and percentages: timed practice

A timed matching round of equal fractions, decimals and percentages, then mixed conversions including recurring decimals.

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Fractions, decimals and percentages: exam practice

Past-paper style questions with mark schemes, including writing a recurring decimal as a fraction in its simplest form.

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10.

Fractions, decimals and percentages: summary

The words, the conversions and the recurring decimal method on one page, with the common mistakes and a flashcard deck.

1.5 · Ordering

Order integers, decimals, fractions and percentages by size, and read and write the symbols =, ≠, <, >, ≤ and ≥.

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Ordering integers

The symbols =, ≠, <, >, ≤ and ≥

The open end of < and > faces the bigger number. ≤ and ≥ also allow equal, which shows as a filled circle on a number line.

Numbers below zero: temperature changes

Ordering decimals

Compare place by place from the left. 0.5 is bigger than 0.45, because 5 tenths beats 4 tenths. More digits is not bigger.

Ordering fractions

Write the fractions over a common denominator, then compare the tops. 2/3 is bigger than 5/8, because 16/24 beats 15/24.

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Ordering fractions, decimals and percentages

Change every value to a decimal first, then order the decimals and write the answer in the original forms.

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Ordering: timed practice

A timed round on the right symbol between two values, then mixed lists to put in order of size.

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Ordering: exam practice

Past-paper style questions with mark schemes: order a mixed list, and insert the correct symbol between two values.

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Ordering: summary

The six symbols, the rule for each kind of number, the common mistakes, and a flashcard deck.

1.6 · The four operations

Add, subtract, multiply and divide with negative numbers, decimals, fractions and mixed numbers, and apply the order of operations with brackets, without a calculator.

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A rise moves right on the number line and a fall moves left. A freezer at -18 °C warming by 5 degrees reaches -13 °C.

Adding and subtracting with negative numbers

Subtracting a negative turns the move around, so 5 - (-3) is 5 + 3, which is 8.

The index says how many times the base is multiplied by itself: 3⁴ = 3 × 3 × 3 × 3 = 81.

Multiplying and dividing with negative numbers

Same signs give a positive answer and different signs give a negative one. The pattern 3 × -2, 2 × -2, 1 × -2 shows why.

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Order of operations and brackets

Brackets first, then powers, then multiply and divide from left to right, then add and subtract from left to right.

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Multiplying decimals

Multiply as whole numbers, then count the decimal places in the question and put the same number in the answer.

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Dividing integers and decimals

Make the divisor a whole number by multiplying both numbers by 10 or 100, then use short division.

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Adding and subtracting fractions and mixed numbers

Pieces of different sizes cannot be counted together, so change both fractions to the same denominator first.

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Multiplying fractions and mixed numbers

Multiply the tops and multiply the bottoms. Change mixed numbers to improper fractions before you start.

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Dividing fractions and mixed numbers

Dividing by a fraction asks how many of it fit. That is the same as multiplying by its reciprocal: 3 ÷ ¾ = 3 × 4/3 = 4.

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10.

The four operations: timed practice

A timed round of non-calculator sums with negatives, decimals and fractions, each with a worked answer.

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11.

The four operations: exam practice

Paper 1 style questions with mark schemes: a temperature change, a mixed-number division, and brackets in the right place.

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12.

The four operations: summary

The sign rules, the order of operations and the fraction methods on one page, with the common mistakes and a flashcard deck.

1.7 · Indices

Index notation, the rules for multiplying, dividing and raising powers, and what zero, negative and fractional indices mean, all worked without a calculator.

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Index notation

Multiplying numbers in index form

Write the factors out and count them: 2³ × 2⁴ has seven 2s, so it is 2⁷. Add the indices when the bases match.

Standard form for large numbers

Dividing numbers in index form

Common factors cancel, so 5⁶ ÷ 5² leaves four 5s: 5⁴. Subtract the indices when the bases match.

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Raising a power to a power

(2³)² is two lots of 2³ multiplied together, which is 2⁶. Multiply the indices.

The zero index

Go down the powers of 2 and each step halves: 8, 4, 2, 1. So 2⁰ = 1, and any non-zero number to the power 0 is 1.

Negative indices

Keep halving past 2⁰ and you reach ½, ¼, ⅛. A negative index means the reciprocal: 7⁻² = 1/49.

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Fractional indices: roots

Two halves make a whole, so 9^½ × 9^½ = 9¹, which makes 9^½ the square root, 3. The index 1/n means the nth root.

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Fractional indices: powers of roots

In 8^(2/3), the bottom 3 is a cube root and the top 2 is a square. Root first to keep numbers small: 2² = 4.

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Using the rules of indices together

Pick one rule per step. 2⁻³ × 2⁴ = 2¹, and (1/16)^(-¼) turns over to 16^¼, which is 2.

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10.

Indices: timed practice

A timed round naming which index rule applies, then mixed evaluations with zero, negative and fractional indices.

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Indices: exam practice

Paper 1 style questions with mark schemes: find the value of a negative or fractional power, and find a missing index.

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12.

Indices: summary

Every rule of indices on one card, the meaning of zero, negative and fractional indices, the common mistakes and a flashcard deck.

1.8 · Standard form

Write very large and very small numbers as A × 10ⁿ, convert them back, and multiply, divide, add and subtract numbers in standard form with and without a calculator.

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Put the decimal point after the first non-zero digit, then count how many places it moved. 4 500 000 = 4.5 × 10⁶.

Standard form for small numbers

Numbers less than 1 get a negative power of 10. 0.00032 = 3.2 × 10⁻⁴, because the point moves 4 places right.

Rounding to the nearest 10, 100 or 1000

Converting numbers into and out of standard form

Change an answer like 45 × 10³ into proper standard form, 4.5 × 10⁴, and write any standard form number as an ordinary number.

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Multiplying and dividing in standard form

Multiply the numbers, add the powers of 10, then tidy the answer so the first part is between 1 and 10.

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Adding and subtracting in standard form

Powers of 10 cannot be added. Write both numbers with the same power, or as ordinary numbers, then add.

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Standard form: timed practice

A timed round spotting which numbers are in standard form, then mixed conversions and calculations.

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Standard form: exam practice

Past-paper style questions with mark schemes: write in standard form, then calculate with it and give the answer in standard form.

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Standard form: summary

The A × 10ⁿ rule, the method for each calculation, the common mistakes and a flashcard deck.

1.9 · Estimation

Round to the nearest 10, 100 or 1000, to decimal places and to significant figures, estimate a calculation by rounding to 1 significant figure, and give answers to a sensible accuracy.

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Find the two values either side and see which is closer. 5764 to the nearest thousand is 6000. A 5 rounds up.

Rounding to a given number of decimal places

Count places after the decimal point, then look at the next digit. 3.476 to 1 decimal place is 3.5.

A length of 8 m to the nearest metre could be anything from 7.5 m up to, but not including, 8.5 m. Go half a unit each way.

Rounding to a given number of significant figures

Start counting at the first non-zero digit. 0.004 072 to 2 significant figures is 0.0041, and 38 250 is 38 000.

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Estimating answers to calculations

Round every number to 1 significant figure, then work it out in your head. The estimate tells you if a calculator answer is sensible.

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Rounding answers to a reasonable degree of accuracy

Let the problem decide. 250 students need 5.56 buses, so book 6. Money goes to 2 decimal places, and 3 significant figures is the default.

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Estimation: timed practice

A timed rounding round in decimal places and significant figures, then estimates to compare with exact answers.

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Estimation: exam practice

Past-paper style questions with mark schemes: round to a stated accuracy, and estimate a fraction calculation, showing each rounded value.

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Estimation: summary

The rounding rules, the 1 significant figure estimate, when to round up in context, the common mistakes and a flashcard deck.

1.10 · Limits of accuracy

Give the upper and lower bounds of a rounded value, then find the bounds of a sum, a difference, a product and a quotient, such as a perimeter, an area and a speed.

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Upper and lower bounds

Bounds for values rounded to decimal places or significant figures

First find the unit it was rounded to. 2.4 kg to 1 decimal place was rounded to 0.1, so the bounds are 2.35 and 2.45.

Writing a ratio in its simplest form

Bounds of sums and differences

The biggest sum adds the two upper bounds. The biggest difference takes the lower bound away from the upper bound.

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Bounds of products: area

The largest possible area of a rectangle multiplies the two upper bounds, and the smallest multiplies the two lower bounds.

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Bounds of quotients: speed

Dividing works the other way round. The lowest speed takes the shortest distance over the longest time.

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Limits of accuracy: timed practice

A timed round choosing upper or lower bound for each part of a calculation, then mixed bounds questions.

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Limits of accuracy: exam practice

Past-paper style questions with mark schemes: bounds of a length, the upper bound of a perimeter, and the lower bound of a speed.

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Limits of accuracy: summary

Half a unit each way, the table of which bounds give the biggest and smallest result, the common mistakes and a flashcard deck.

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1.11 · Ratio and proportion

Simplify a ratio, use a ratio to find an unknown amount, divide a quantity in a given ratio, and use direct proportion, inverse proportion and map scales.

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A ratio compares parts. Divide every part by the same number: 20 : 30 : 40 in its simplest form is 2 : 3 : 4.

Simplifying ratios with units, fractions and decimals

Put both parts in the same unit first: 50 cm : 2 m is 50 : 200, which is 1 : 4. Multiply out decimals and fractions.