JCE Mathematics

Types of numbers

Numbers are grouped into different types depending on their properties. Knowing these groups helps you understand which rules apply and how numbers relate to one another.

Natural numbers

A natural number is a counting number, starting from 1: 1, 2, 3, 4, 5... Natural numbers are used for counting whole, separate items and never include zero, fractions, or negative values.

Whole numbers

A whole number is a natural number together with zero: 0, 1, 2, 3, 4... Every natural number is a whole number, but zero is a whole number that is not a natural number.

Integers

An integer is any positive or negative whole number, including zero: ...,-3, -2, -1, 0, 1, 2, 3,... Integers extend whole numbers in both directions on the number line, allowing values below zero.

Prime numbers

A prime number has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13... Two important exceptions to remember: 1 is not a prime number (it has only one factor, not two), and 2 is the only even prime number — every other even number has at least three factors (1, 2, and itself), so it cannot be prime.

Composite numbers

A composite number has more than two factors. Examples include 4 (factors: 1, 2, 4), 6 (factors: 1, 2, 3, 6), 8, and 9. Every whole number greater than 1 is either prime or composite, but never both.

Example: Classify each of the following as prime or composite: 17, 21, 29, 33. 17 → prime (only factors are 1 and 17). 21 → composite (factors include 3 and 7). 29 → prime. 33 → composite (factors include 3 and 11).

Triangular numbers

A triangular number is a number that can be arranged as dots forming an equilateral triangle, built by adding consecutive counting numbers starting from 1: 1, 3, 6, 10, 15, 21... The n-th triangular number is found using T(n) = n(n + 1)2.

Example: Find the 6th triangular number.
T(6) = 6 × 72 = 422 = 21. This matches the pattern: 1, 3, 6, 10, 15, 21.

Rectangular numbers

A rectangular number is a number that can be arranged as dots forming a rectangle (with more than one row and more than one column), meaning it is the product of two whole numbers greater than 1: 6 (2×3), 8 (2×4), 12 (3×4), 10 (2×5)... Every composite number is a rectangular number, since it can be split into two factors greater than 1.

Square numbers

A square number is the result of multiplying a whole number by itself, and can be arranged as dots forming a perfect square: 1, 4, 9, 16, 25, 36... The n-th square number is n² = n × n. Square numbers are a special case of rectangular numbers where both sides are equal.

Example: The first six square numbers are 1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36.

Cube numbers

A cube number is the result of multiplying a whole number by itself three times, and represents the number of unit cubes that fit inside a larger cube of that side length: 1, 8, 27, 64, 125... The n-th cube number is n³ = n × n × n.

Example: The first five cube numbers are 1³=1, 2³=8, 3³=27, 4³=64, 5³=125.

Factors and multiples

A factor of a number divides into it exactly, with no remainder. A multiple of a number is what you get by multiplying it by a whole number. The Highest Common Factor (HCF) is the largest number that divides exactly into two or more numbers. The Lowest Common Multiple (LCM) is the smallest number that two or more numbers divide into exactly.

Example: Find the HCF and LCM of 12 and 18.

Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. HCF = 6.
Multiples of 12: 12, 24, 36, 48... Multiples of 18: 18, 36, 54... LCM = 36.

Integers: operations

When adding or subtracting integers, think of a number line. Adding a negative number is the same as subtracting: 5 + (-3) = 5 - 3 = 2. Subtracting a negative number is the same as adding: 5 - (-3) = 5 + 3 = 8.

When multiplying or dividing integers, use these sign rules:

Signs	Result
Positive × Positive	Positive
Negative × Negative	Positive
Positive × Negative	Negative
Negative × Positive	Negative

The same rules apply to division. Example: -6 × -4 = 24. -6 × 4 = -24. -20-5 = 4.

Order of operations (BODMAS)

Calculations must be done in a set order, remembered by BODMAS: Brackets, Of (powers/indices), Division and Multiplication (left to right), Addition and Subtraction (left to right).

Example: 3 + 4 × (6 - 2) = 3 + 4 × 4 = 3 + 16 = 19.

More worked examples

Example (powers before multiplication): 5 + 2³ × 3. Work out the power first: 2³ = 8. Then multiply: 8 × 3 = 24. Then add: 5 + 24 = 29.

Example (nested brackets): (12 - (3 + 1)) × 2. Work out the innermost bracket first: 3 + 1 = 4. Then the outer bracket: 12 - 4 = 8. Then multiply: 8 × 2 = 16.

Example (division and multiplication, left to right): 20 ÷ 4 × 2. Division and multiplication have equal priority, so work left to right: 20 ÷ 4 = 5, then 5 × 2 = 10. (Note: this does NOT equal 20 ÷ 8 = 2.5 — multiplication does not automatically come before division.)

Example (addition and subtraction, left to right): 15 - 6 + 2. Work left to right: 15 - 6 = 9, then 9 + 2 = 11. (Note: this does NOT equal 15 - 8 = 7 — addition does not automatically come before subtraction.)

Example (combining several rules): 3² + (8 - 5) × 4 ÷ 2. Bracket first: 8 - 5 = 3. Power next: 3² = 9. Multiplication and division, left to right: 3 × 4 = 12, then 12 ÷ 2 = 6. Finally add: 9 + 6 = 15.

Example (fraction bar acting as a bracket): 7 + 34 - 2. Treat the top and bottom of a fraction as separate brackets: top = 7 + 3 = 10, bottom = 4 - 2 = 2. Then divide: 102 = 5.

Fractions

A fraction has a numerator (top number) and a denominator (bottom number). A proper fraction has a numerator smaller than the denominator (e.g. 3/4). An improper fraction has a numerator equal to or larger than the denominator (e.g. 7/4). A mixed number combines a whole number and a proper fraction (e.g. 13/4).

To add or subtract fractions, first find a common denominator. To multiply fractions, multiply the numerators together and the denominators together, then simplify. To divide fractions, multiply by the reciprocal (flip the second fraction, then multiply).

Example: 2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3 = 22/3.

More worked examples

Example (adding fractions with different denominators): 1/4 + 2/5. Find a common denominator (LCM of 4 and 5 is 20): 1/4 = 5/20 and 2/5 = 8/20. Add: 5/20 + 8/20 = 13/20.

Example (subtracting fractions with different denominators): 3/4 - 1/6. LCM of 4 and 6 is 12: 3/4 = 9/12 and 1/6 = 2/12. Subtract: 9/12 - 2/12 = 7/12.

Example (adding mixed numbers): 21/3 + 13/4. Convert to improper fractions: 21/3 = 7/3, 13/4 = 7/4. Common denominator 12: 7/3 = 28/12, 7/4 = 21/12. Add: 28/12 + 21/12 = 49/12 = 41/12.

Example (subtracting mixed numbers with borrowing): 31/4 - 11/2. Convert to improper fractions: 31/4 = 13/4, 11/2 = 3/2 = 6/4. Subtract: 13/4 - 6/4 = 7/4 = 13/4.

Example (multiplying fractions): 2/5 × 3/7 = (2 × 3)/(5 × 7) = 6/35 (already in simplest form since 6 and 35 share no common factor).

Example (multiplying a fraction by a whole number): 3/8 × 4 = 3/8 × 4/1 = 12/8 = 11/2.

Example (multiplying mixed numbers): 11/2 × 21/3. Convert first: 11/2 = 3/2, 21/3 = 7/3. Multiply: 3/2 × 7/3 = 21/6 = 31/2.

Example (dividing a fraction by a whole number): 4/9 ÷ 2 = 4/9 × 1/2 = 4/18 = 2/9.

Example (simplifying to lowest terms): Simplify 18/24. HCF of 18 and 24 is 6. 18 ÷ 6 = 3, 24 ÷ 6 = 4. So 18/24 = 3/4.

Example (comparing and ordering fractions): Order 2/3, 5/8, 3/4 from smallest to largest. Common denominator of 3, 8, 4 is 24: 2/3 = 16/24, 5/8 = 15/24, 3/4 = 18/24. Ordered smallest to largest: 5/8, 2/3, 3/4.

Example (fraction of a quantity): Find 3/5 of P450. 450 ÷ 5 = 90 (one fifth), then 90 × 3 = P270.

Decimals

A decimal is a way of writing a number that is not whole, using a decimal point to separate the whole number part from the fractional part. Each digit after the decimal point represents a fraction with a denominator that is a power of 10: the first digit after the point is tenths, the second is hundredths, the third is thousandths, and so on.

Converting between fractions and decimals

To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, write the digits after the decimal point over the matching power of 10 (10, 100, 1000...), then simplify.

Example: Convert 3/4 to a decimal: 3 ÷ 4 = 0.75. Convert 0.6 to a fraction: 0.6 = 6/10 = 3/5.

Adding and subtracting decimals

Line up the decimal points directly under one another before adding or subtracting, filling any empty places with zeros so each number has the same number of decimal places. Then add or subtract as with whole numbers, keeping the decimal point in the same column in the answer.

Example: 4.6 + 2.35. Line up: 4.60 + 2.35 = 6.95.

Example: 8.2 - 3.75. Line up: 8.20 - 3.75 = 4.45.

Multiplying decimals

Ignore the decimal points first and multiply the numbers as if they were whole numbers. Then count the total number of decimal places in both original numbers, and place the decimal point that many places from the right in the answer.

Example: 0.4 × 0.3. Multiply as whole numbers: 4 × 3 = 12. Total decimal places in the question = 2 (one from each number). Place the point two digits from the right: 0.12.

Example: 1.2 × 0.05. Multiply as whole numbers: 12 × 5 = 60. Total decimal places = 1 + 2 = 3. Place the point three digits from the right: 0.060 = 0.06.

Dividing decimals

When dividing by a decimal, multiply both the number being divided and the divisor by the same power of 10 so that the divisor becomes a whole number, then divide as normal.

Example: 6.40.8. Multiply both by 10: 648 = 8.

Example: 3.150.05. Multiply both by 100: 3155 = 63.

Ordering decimals

To order decimals, compare digit by digit starting from the left (the whole number part first, then tenths, then hundredths, and so on), giving each number the same number of decimal places by adding trailing zeros if needed so the comparison is fair.

Example: Order 0.45, 0.5, 0.408 from smallest to largest. Written with three decimal places: 0.450, 0.500, 0.408. Comparing tenths first (4, 5, 4), then hundredths where tied (0.450 vs 0.408: 5 > 0): order is 0.408, 0.450, 0.5.

Recurring decimals

A recurring decimal has one or more digits that repeat forever, shown with a dot above the repeating digit(s), e.g. 0.̇3 means 0.333... and 0.1̇2̇ means 0.121212... These arise when a fraction's denominator (in simplest form) has prime factors other than 2 or 5.

Example: 1/3 = 0.̇3 (recurring). 1/4 = 0.25 (terminates, since the denominator 4 = 2²).

Rounding and estimation

To round to a given number of decimal places (d.p.), look at the digit after the cut-off point: if it's 5 or more, round up; if less than 5, round down. To round to a given number of significant figures (s.f.), count from the first non-zero digit.

Example (decimal places): Round 4.5723 to 2 decimal places.
Look at the 3rd decimal digit (2): since 2 < 5, round down. Answer: 4.57.

Example (decimal places, rounding up): Round 8.196 to 1 decimal place.
Look at the 2nd decimal digit (9): since 9 ≥ 5, round up. Answer: 8.2.

Example (significant figures): Round 0.03847 to 2 significant figures.
Counting from the first non-zero digit (3): the first 2 significant figures are 3 and 8; the next digit (4) rounds down. Answer: 0.038.

Example (significant figures, larger number): Round 52,847 to 3 significant figures.
The first 3 significant figures are 5, 2, 8; the next digit (4) rounds down, but trailing places must be filled with zeros to keep the place value correct. Answer: 52,800.

Example (rounding to the nearest whole number, 10, or 100): Round 2,368 to the nearest hundred.
The digit in the tens place (6) means round up: Answer: 2,400.

Estimation

Estimation uses rounded values to quickly approximate the answer to a calculation, useful for checking whether an exact answer is reasonable. Round each number in the calculation to 1 significant figure first, then calculate.

Example: Estimate the value of 38.7 × 5.2.
Round each number to 1 s.f.: 38.7 ≈ 40, 5.2 ≈ 5.
Estimate = 40 × 5 = 200 (the exact answer, 201.24, is close to this).

Example (estimation with division): Estimate the value of 28931.
Round each number to 1 s.f.: 289 ≈ 300, 31 ≈ 30.
Estimate = 30030 = 10.

Example (estimation with a mixed calculation): Estimate the value of 19.6 × 5.14.9.
Round each number to 1 s.f.: 19.6 ≈ 20, 5.1 ≈ 5, 4.9 ≈ 5.
Estimate = 20 × 55 = 1005 = 20.

Standard form

Standard form (also called scientific notation) writes numbers as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. It is used to write very large or very small numbers in a compact, manageable way, and makes it easier to compare the size of numbers at a glance.

Large numbers (10 or more) use a positive power of 10, matching how many places the decimal point moves to the left to get A. Small numbers (less than 1) use a negative power of 10, matching how many places the decimal point moves to the right to get A.

Example: 45,000 = 4.5 × 10⁴ (decimal point moved 4 places left). 0.00072 = 7.2 × 10^-4 (decimal point moved 4 places right).

Converting numbers into standard form

Example: Write 3,200,000 in standard form. Move the decimal point until only one non-zero digit remains before it: 3.2. Count the places moved: 6. So 3,200,000 = 3.2 × 10⁶.

Example: Write 0.000091 in standard form. Move the decimal point right until one non-zero digit remains before it: 9.1. Count the places moved: 5. So 0.000091 = 9.1 × 10^-5.

Converting standard form back to an ordinary number

Example: Write 6.03 × 10⁵ as an ordinary number. A positive power means the number is large, so move the decimal point 5 places to the right, filling gaps with zeros: 603,000.

Example: Write 8.4 × 10^-3 as an ordinary number. A negative power means the number is small, so move the decimal point 3 places to the left: 0.0084.

Multiplying and dividing numbers in standard form

To multiply, multiply the A-values together and add the powers of 10. To divide, divide the A-values and subtract the powers of 10. If the resulting A-value is not between 1 and 10, adjust it and change the power of 10 to compensate.

Example: (2 × 10³) × (3 × 10⁴) = (2 × 3) × 10³⁺⁴ = 6 × 10⁷.

Example: 8 × 10⁶4 × 10² = 84 × 10^6-2 = 2 × 10⁴.

Example (adjusting the result): (5 × 10³) × (4 × 10²) = 20 × 10⁵. Since 20 is not between 1 and 10, rewrite 20 as 2 × 10¹: 2 × 10¹ × 10⁵ = 2 × 10⁶.

Adding and subtracting numbers in standard form

Numbers in standard form cannot be added or subtracted directly unless the powers of 10 match. Convert them to the same power of 10 first (or convert to ordinary numbers), then add or subtract the A-values.

Example: (3.2 × 10⁴) + (5 × 10³). Rewrite 5 × 10³ as 0.5 × 10⁴. Add: (3.2 + 0.5) × 10⁴ = 3.7 × 10⁴.

Standard form with negative powers in calculations

Example: (6 × 10^-3) × (3 × 10⁵) = (6 × 3) × 10^-3+5 = 18 × 10² = 1.8 × 10³.

Real-world use of standard form

Standard form is commonly used for very large quantities (distances in space, populations, national budgets) and very small quantities (sizes of cells, wavelengths of light, masses of atoms), since writing out all the zeros would be impractical.

Example: The distance from Earth to the Sun is approximately 150,000,000 km, written in standard form as 1.5 × 10⁸ km.

Squares, cubes, and roots

The square of a number is the number multiplied by itself: n² = n × n. The square root (√) is the inverse operation — it asks "what number, multiplied by itself, gives this value?" The cube of a number is the number multiplied by itself three times: n³ = n × n × n. The cube root (∛) reverses this, asking "what number, multiplied by itself three times, gives this value?"

Example: 5² = 25, so √25 = 5. 2³ = 8, so ∛8 = 2.

Squares and square roots of larger numbers

Example: Find 12². 12 × 12 = 144.

Example: Find √144. Since 12² = 144, √144 = 12.

Example: Find √169. Testing nearby squares: 13² = 169, so √169 = 13.

Squares and roots of fractions and decimals

To square a fraction, square the numerator and denominator separately. To find the square root of a fraction, find the square root of the numerator and denominator separately.

Example: 3/4² = 9/16. √4/25 = 2/5.

Example: 0.3² = 0.09. √0.81 = 0.9 (since 0.9 × 0.9 = 0.81).

Cube roots of larger numbers

Example: Find ∛64. Since 4³ = 4 × 4 × 4 = 64, ∛64 = 4.

Example: Find ∛1000. Since 10³ = 1000, ∛1000 = 10.

Negative numbers and roots

A negative number cannot have a real square root, since a number multiplied by itself (whether positive or negative) always gives a positive result. However, a negative number does have a real cube root, since a negative number multiplied by itself three times stays negative.

Example: (-3)² = 9, and (-3)² is also 9 — so √9 = 3 only (the positive root is used unless the question asks for both). ∛-27 = -3, since -3 × -3 × -3 = -27.

Estimating roots that are not exact

When a square root is not a whole number, estimate it by finding the two whole numbers it falls between.

Example: Estimate √50. Since 7² = 49 and 8² = 64, √50 lies between 7 and 8, and is close to 7.1 (nearer to 49 than to 64).

Using squares, cubes, and roots together

Example: Evaluate √(5² + 12²). Work inside the root first: 5² + 12² = 25 + 144 = 169. Then √169 = 13. (This is the same calculation used in Pythagoras' theorem.)

Working with negative fractions and decimals

The sign rules for multiplying and dividing integers apply equally to fractions and decimals: two like signs (positive × positive, or negative × negative) give a positive result; two unlike signs give a negative result. When more than two values are combined, keep track of the sign separately from the size of the calculation — count the number of negative values involved: an even count gives a positive answer, an odd count gives a negative answer.

Example: -0.4 × -0.5 × 2 = 0.4 (two negatives, so positive).

Adding and subtracting negative decimals

Treat a negative decimal the same way as a negative integer: adding a negative value is the same as subtracting, and subtracting a negative value is the same as adding.

Example: 3.6 + (-1.2) = 3.6 - 1.2 = 2.4.

Example: -2.5 - (-4.1) = -2.5 + 4.1 = 1.6.

Multiplying and dividing negative decimals

Example: -0.6 × 0.5 = -0.3 (one negative, so the answer is negative).

Example: -4.8-0.6 = 8 (two negatives, so the answer is positive).

Negative fractions

A negative fraction can be written with the negative sign in front, on the numerator, or on the whole fraction — all are equivalent: -3/4 = -3/4 = -(3/4).

Example: -2/3 + 1/6. Common denominator 6: -4/6 + 1/6 = -3/6 = -1/2.

Example: -3/5 × -1/4 = 3/20 (two negatives, so positive).

Example: 2/3 ÷ -4/5. 2/3 × -5/4 = -10/12 = -5/6 (one negative, so the answer is negative).

Combined operations with negative fractions and decimals

Example: -1.5 + 1/2 × -0.4. Work out the multiplication first (BODMAS): 1/2 × -0.4 = -0.2. Then add: -1.5 + (-0.2) = -1.7.

HCF and LCM using prime factors

Every whole number greater than 1 can be broken down into a unique product of prime numbers, called its prime factorisation. This is usually found using a factor tree, repeatedly splitting a number into two factors until every branch ends in a prime number.

For larger numbers, HCF and LCM are found more reliably using prime factorisation than by listing factors or multiples. Write each number as a product of prime factors, then: HCF = product of the lowest power of each common prime; LCM = product of the highest power of every prime that appears (common or not).

Example: 36 = 2² × 3², 60 = 2² × 3 × 5. HCF = 2² × 3 = 12. LCM = 2² × 3² × 5 = 180.

Finding a prime factorisation from scratch

Example: Find the prime factorisation of 84. 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7. So 84 = 2² × 3 × 7.

HCF and LCM of three numbers

The same method extends to three or more numbers: write each as a product of primes, then take the lowest power of each prime common to all numbers for the HCF, and the highest power of every prime appearing in any of the numbers for the LCM.

Example: Find the HCF and LCM of 12, 18, and 30. 12 = 2² × 3. 18 = 2 × 3². 30 = 2 × 3 × 5. HCF = 2 × 3 = 6 (lowest power of primes common to all three). LCM = 2² × 3² × 5 = 180 (highest power of every prime appearing).

Using HCF and LCM in word problems

Example: Two bells ring every 12 minutes and every 18 minutes. If they ring together at 8:00, when will they next ring together? This requires the LCM of 12 and 18. 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 36. They will next ring together after 36 minutes, at 8:36.

Example: A shopkeeper wants to pack 48 oranges and 60 apples into identical boxes, with no fruit left over and the same number of each fruit per box. What is the largest possible number of boxes? This requires the HCF of 48 and 60. 48 = 2⁴ × 3, 60 = 2² × 3 × 5. HCF = 2² × 3 = 12. The largest possible number of boxes is 12.

Number line placement

Being able to place integers, fractions, and decimals accurately on a number line is a common exam skill, especially for ordering values from smallest to largest. A number line has equally spaced marks representing values in order, with numbers increasing to the right and decreasing to the left. To place a value correctly, first identify the two labelled marks it falls between, then judge its position based on how close it is to each.

Placing integers

Integers are placed at their exact marked position on the number line. Remember that negative numbers get smaller (further from zero, further left) the larger their digit becomes: -5 is smaller than -2.

Example: Place -4, 3, -1, and 0 on a number line in order from left to right (smallest to largest): -4, -1, 0, 3.

Placing fractions

To place a fraction on a number line marked in whole numbers, first work out which two whole numbers it lies between, then judge its position within that gap based on the size of the fraction.

Example: Place 7/4 on a number line. Since 7/4 = 13/4 (one and three quarters), it lies between 1 and 2, three-quarters of the way along.

Example: Place -2/3 on a number line. Since 2/3 is between 0 and 1, -2/3 lies between -1 and 0, two-thirds of the way from 0 towards -1.

Placing decimals

Decimals are placed by first identifying the whole number they are closest to, then using the tenths (and hundredths, if needed) to judge the exact position within that interval.

Example: Place 2.6 on a number line. It lies between 2 and 3, six-tenths of the way along (closer to 3 than to 2).

Example: Place -1.25 on a number line. It lies between -2 and -1, one quarter of the way from -1 towards -2.

Placing and comparing mixed types together

Exam questions often mix integers, fractions, and decimals on the same number line, requiring you to convert them to a common form (usually decimals) before comparing.

Example: Place -1, 1/2, -0.75, and 3/2 in order from smallest to largest. Convert to decimals: -1, 0.5, -0.75, 1.5. Ordered smallest to largest: -1, -0.75, 0.5, 1.5. So the order is -1, -0.75, 1/2, 3/2.

Using a number line for inequalities

A number line can also represent a range of values rather than a single point. An open (unfilled) circle marks a value that is not included in the range (used for < or >); a closed (filled) circle marks a value that is included (used for ≤ or ≥). An arrow extending from the circle shows the direction the range continues.

Example: Represent x > -2 on a number line. Draw an open circle at -2, with an arrow extending to the right (towards larger values).

Example: Represent -3 ≤ x < 1 on a number line. Draw a closed circle at -3 (included) and an open circle at 1 (not included), with a solid line joining them to show all the values in between.

Finding a value's approximate position without a scale

Some exam questions give a number line without every value marked, and ask you to estimate where a given value should go relative to a few labelled points, using proportional reasoning rather than an exact scale.

Example: On a number line where only 0 and 10 are marked, estimate where 7.5 should be placed. Since 7.5 is three-quarters of the way from 0 to 10, mark it three-quarters of the distance along the line from 0 towards 10.