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1a · E1.1–E1.8

Types of number, sets, powers and standard form

Topic 1 Number · Core and Extended · Papers 1–4

🎯What you need to be able to do

  • Identify natural numbers, integers, primes, squares, cubes, factors, multiples, rational and irrational numbers and reciprocals.
  • Write a number as a product of its prime factors, and use that to find the HCF and LCM.
  • Use set notation and Venn diagrams: two sets for Core, up to three sets and the full notation for Extended EXTENDED.
  • Recall squares up to 15² and cubes of 1, 2, 3, 4, 5 and 10, with their roots.
  • Convert between fractions, decimals and percentages; order quantities; calculate with fractions and negative numbers.
  • Convert recurring decimals to fractions EXTENDED.
  • Use the rules of indices — integer indices for Core, fractional indices too for Extended EXTENDED.
  • Convert to and from standard form and calculate with it.

📚The mathematics

Types of number

  • Natural numbers: 1, 2, 3, …  Integers: …, −2, −1, 0, 1, 2, …
  • Prime: exactly two factors, 1 and itself. 1 is not prime; 2 is the only even prime.
  • Rational: can be written as \( \tfrac{a}{b} \) with integers \(a, b\) — includes every terminating and recurring decimal. Irrational: cannot, e.g. \( \sqrt{2} \), \( \pi \), \( \sqrt{7} \).
  • The reciprocal of \(x\) is \( \tfrac{1}{x} \): the reciprocal of \( \tfrac{3}{5} \) is \( \tfrac{5}{3} \), and of 0.4 is 2.5.

Prime factors, HCF and LCM

Split a number with a factor tree until every branch ends in a prime, then write the primes in index form. From two such products:

HCF: each prime to the lowest power that appears in both
LCM: every prime to the highest power that appears in either
A factor tree for 360: 360 splits into 36 and 10, 36 into 4 and 9, 4 into 2 and 2, 9 into 3 and 3, and 10 into 2 and 5. The prime leaves are circled, giving 360 = 2 cubed times 3 squared times 5.
Any split works; the prime leaves are always the same: \( 360 = 2^{3} \times 3^{2} \times 5 \).

Sets and Venn diagrams

\( n(A) \): number of elements
\( A' \): complement (not in \(A\))
\( A \cup B \): union (in \(A\) or \(B\) or both)
\( A \cap B \): intersection (in both)

Extended adds \( \in \) (“is an element of”), \( \notin \), \( \varnothing \) (the empty set), \( A \subseteq B \) (subset) and \( A \not\subseteq B \), and Venn diagrams with three sets EXTENDED. When filling a Venn diagram from totals, start in the middle (the intersection) and work outwards, subtracting as you go.

A Venn diagram with universal set of 30 students and two overlapping circles, F (football) and B (basketball). The regions contain 13 in F only, 5 in both, 7 in B only and 5 outside both circles.
Practice question 4: fill the intersection first, then the rest of each circle, then the outside.

Fractions, decimals and recurring decimals

To divide by a fraction, multiply by its reciprocal; turn mixed numbers into improper fractions first. To order a mix of fractions, decimals and percentages, convert everything to decimals.

EXTENDED A recurring decimal becomes a fraction by multiplying so that the repeating part lines up, then subtracting. For \( x = 0.\dot{4}\dot{7} = 0.4747\ldots \): \( 100x = 47.4747\ldots \), so \( 99x = 47 \) and \( x = \tfrac{47}{99} \). If the repeat starts later, as in \( 0.1\dot{3} = 0.1333\ldots \), use two multipliers: \( 100x - 10x = 13.3\ldots - 1.3\ldots = 12 \), so \( x = \tfrac{12}{90} = \tfrac{2}{15} \).

Indices

\( a^{m} \times a^{n} = a^{m + n} \)
\( a^{m} \div a^{n} = a^{m - n} \)
\( (a^{m})^{n} = a^{mn} \)
\( a^{0} = 1 \), \( a^{-n} = \dfrac{1}{a^{n}} \)
\( a^{\frac{1}{n}} = \sqrt[n]{a} \), \( a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m} \) EXTENDED

For a fractional power, take the root first — the numbers stay small: \( 27^{\frac{2}{3}} = (\sqrt[3]{27})^{2} = 3^{2} = 9 \). A negative power means “reciprocal”, not “negative”: \( 5^{-2} = \tfrac{1}{25} \).

Standard form

\( A \times 10^{n} \) with \( 1 \le A < 10 \) and \(n\) an integer. To multiply, multiply the \(A\)s and add the powers; to divide, divide and subtract. Then re-adjust so that \(A\) is between 1 and 10: \( 14.4 \times 10^{3} = 1.44 \times 10^{4} \). (Core candidates calculate with standard form only on the calculator paper.)

✏️Worked example (non-calculator)

(a) Write 360 and 84 as products of prime factors, and find their HCF and LCM. [4] (b) Work out \( (4.8 \times 10^{5}) \times (3 \times 10^{-2}) \), giving your answer in standard form. [2] (c) EXTENDED Write \( 0.1\dot{3} \) as a fraction in its simplest form. [2]

(a) \( 360 = 2^{3} \times 3^{2} \times 5 \) and \( 84 = 2^{2} \times 3 \times 7 \). HCF: lowest powers of the shared primes, \( 2^{2} \times 3 = 12 \). LCM: highest powers of all the primes, \( 2^{3} \times 3^{2} \times 5 \times 7 = 2520 \).

(b) \( 4.8 \times 3 = 14.4 \) and \( 10^{5} \times 10^{-2} = 10^{3} \), so \( 14.4 \times 10^{3} = 1.44 \times 10^{4} \).

(c) \( x = 0.1333\ldots \). \( 10x = 1.333\ldots \) and \( 100x = 13.333\ldots \), so \( 90x = 12 \) and \( x = \tfrac{12}{90} = \tfrac{2}{15} \).

Check it. HCF × LCM should equal the product of the two numbers: \( 12 \times 2520 = 30240 = 360 \times 84 \) ✓. And \( 2 \div 15 = 0.1333\ldots \) ✓.
Leaving \(A\) outside 1–10. \( 14.4 \times 10^{3} \) is the right number but is not in standard form, and loses the final mark. Moving the decimal point one place left means adding 1 to the power.

📝Practise

Questions in the style of the current papers. EXTENDED marks Extended-only content.

1. (Non-calculator.) Write 252 as a product of its prime factors. Hence find the HCF and the LCM of 252 and 90. [4]
\( 252 = 2^{2} \times 3^{2} \times 7 \) and \( 90 = 2 \times 3^{2} \times 5 \). HCF \( = 2 \times 3^{2} = 18 \). LCM \( = 2^{2} \times 3^{2} \times 5 \times 7 = 1260 \).
2. (Non-calculator.) Work out \( 2\tfrac{1}{3} \div 1\tfrac{3}{4} \), giving your answer as a mixed number in its simplest form. [3]
\( \tfrac{7}{3} \div \tfrac{7}{4} = \tfrac{7}{3} \times \tfrac{4}{7} = \tfrac{4}{3} = 1\tfrac{1}{3} \).
3. (Non-calculator.) Write these in order of size, smallest first: \( \tfrac{5}{8} \), 0.6, 62%, \( \tfrac{2}{3} \). [2]
As decimals: 0.625, 0.6, 0.62, 0.666… So 0.6, 62%, \( \tfrac{5}{8} \), \( \tfrac{2}{3} \).
4. (Non-calculator.) In a class of 30 students, 18 play football (F), 12 play basketball (B) and 5 play neither. Find \( n(F \cap B) \) and \( n(F \cap B') \). [3]
\( 30 - 5 = 25 \) play at least one. \( 18 + 12 = 30 \) counts the students who play both twice, so \( n(F \cap B) = 30 - 25 = 5 \). Then \( n(F \cap B') = 18 - 5 = 13 \) (football but not basketball). The Venn diagram above shows the complete picture.
5. (Non-calculator.) Find the value of (a) \( 5^{-2} \), (b) \( 27^{\frac{2}{3}} \) EXTENDED, (c) \( 16^{-\frac{3}{4}} \) EXTENDED. [3]
(a) \( \tfrac{1}{25} \). (b) \( (\sqrt[3]{27})^{2} = 9 \). (c) \( \dfrac{1}{(\sqrt[4]{16})^{3}} = \dfrac{1}{8} \).
6. (Non-calculator.) Work out \( (2.4 \times 10^{7}) \div (8 \times 10^{-3}) \), giving your answer in standard form. [2]
\( 2.4 \div 8 = 0.3 \) and \( 10^{7} \div 10^{-3} = 10^{10} \): \( 0.3 \times 10^{10} = 3 \times 10^{9} \).
7. (Non-calculator.) EXTENDED Write (a) \( 0.\dot{2}\dot{7} \) and (b) \( 0.5\dot{8} \) as fractions in their simplest form. [4]
(a) \( 100x - x = 27 \), \( x = \tfrac{27}{99} = \tfrac{3}{11} \). (b) \( 100x - 10x = 58.8\ldots - 5.8\ldots = 53 \), so \( x = \tfrac{53}{90} \).
8. (Non-calculator.) EXTENDED \( A = \{x : x \text{ is a factor of } 12\} \) and \( B = \{x : x \text{ is a prime number less than } 12\} \). List \( A \cap B \) and find \( n(A \cup B) \). [3]
\( A = \{1, 2, 3, 4, 6, 12\} \), \( B = \{2, 3, 5, 7, 11\} \). \( A \cap B = \{2, 3\} \). \( n(A \cup B) = 6 + 5 - 2 = 9 \).

🔗Go deeper — other people’s work

These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.

  • Corbettmaths — short videos and worksheets on HCF/LCM, recurring decimals and standard form
  • Khan Academy — exponent rules and fractional exponents