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Mathematics 4024

Quick revision notes ยท Mathematics 4024 ยท Chapter 1 of 35

Arithematics

The number toolkit used in every other chapter: kinds of number, BIDMAS, fractions, decimals and recurring decimals.

6 sectionsWorked examples Common mistakes
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At a glance

The number toolkit used in every other chapter: kinds of number, BIDMAS, fractions, decimals and recurring decimals.

  1. Types of number
  2. Order of operations and negative numbers
  3. Fractions
  4. Decimals and ordering
  5. Halfway values and differences
  6. Recurring decimals to fractions

1Types of number

  • Natural numbers: 1, 2, 3, โ€ฆ Integers: whole numbers including 0 and negatives (โ€ฆ, โˆ’2, โˆ’1, 0, 1, 2, โ€ฆ).
  • Prime: exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, โ€ฆ 1 is not prime; 2 is the only even prime.
  • Factor of 12: divides into it (1, 2, 3, 4, 6, 12). Multiple of 12: in its times table (12, 24, 36, โ€ฆ).
  • Rational: can be written as a fraction a/b (all integers, terminating and recurring decimals). Irrational: cannot โ€” ฯ€, โˆš2, โˆš3, โˆš5 and the square root of any non-square number.
  • Reciprocal of x is 1/x: reciprocal of 4 is 1/4, of 2/5 is 5/2. A number times its reciprocal is 1.
n12345678910
n2149162536496481100
n31827641252163435127291000

Worked example

From โˆ’3, 0, 1/2, โˆš9, โˆš5, 7 and 0.333โ€ฆ, list (a) the irrational numbers, (b) the prime numbers.

  1. โˆš9 = 3 and 0.333โ€ฆ = 1/3, so both are rational.
  2. โˆš5 is not a whole number and does not recur: irrational.
  3. Primes are positive whole numbers with exactly two factors: โˆš9 = 3 and 7.

(a) โˆš5 (b) โˆš9 and 7

Exam tip: Also know 11ยฒ = 121, 12ยฒ = 144, 13ยฒ = 169, 14ยฒ = 196, 15ยฒ = 225 for the non-calculator paper.

2Order of operations and negative numbers

  • BIDMAS: Brackets โ†’ Indices โ†’ Division and Multiplication (left to right) โ†’ Addition and Subtraction (left to right).
  • Two signs next to each other: same signs give +, different signs give โˆ’. So 5 โˆ’ (โˆ’3) = 8 and 4 ร— (โˆ’2) = โˆ’8.
  • (โˆ’3)2 = 9 but โˆ’32 = โˆ’9: without brackets only the 3 is squared.
  • A fraction line acts like brackets: (8 + 4) รท (5 โˆ’ 2) = 4. On a calculator, put brackets round the top and the bottom.

Order to work in

  1. Work out anything in brackets.
  2. Then powers and roots.
  3. Then ร— and รท from left to right.
  4. Finally + and โˆ’ from left to right.

Worked example

Work out (โˆ’3)2 โˆ’ 4 ร— (5 โˆ’ 7).

  1. Brackets: 5 โˆ’ 7 = โˆ’2.
  2. Power: (โˆ’3)2 = 9.
  3. Multiply: 4 ร— (โˆ’2) = โˆ’8.
  4. 9 โˆ’ (โˆ’8) = 9 + 8.

17

3Fractions

  • To add or subtract, use a common denominator (the LCM of the bottoms), then add the tops.
  • To multiply, multiply tops and bottoms (cancel first to keep numbers small).
  • To divide, flip the second fraction and multiply: a/b รท c/d = a/b ร— d/c.
  • Change mixed numbers to improper fractions before multiplying or dividing: 2ยพ = 11/4.
  • Fraction of an amount: divide by the bottom, multiply by the top. 3/8 of 240 = 240 รท 8 ร— 3 = 90.
  • Always give the answer in its simplest form.

Formulas

a/b + c/d = (ad + bc)/bdthen simplify
a/b ร— c/d = ac/bdmultiply across
a/b รท c/d = a/b ร— d/cflip the second, then multiply

Worked example

Work out 2ยพ โˆ’ 1โ…š.

  1. Improper fractions: 11/4 โˆ’ 11/6.
  2. Common denominator 12: 33/12 โˆ’ 22/12.
  3. = 11/12.

11/12

Exam tip: On the non-calculator paper write every step: the method marks need the common denominator to be seen.

4Decimals and ordering

  • Multiplying decimals: ignore the points, multiply, then give the answer as many decimal places as the question had in total. 0.3 ร— 0.04 = 0.012.
  • Dividing by a decimal: multiply both numbers by 10, 100, โ€ฆ until you divide by a whole number. 4.8 รท 0.06 = 480 รท 6 = 80.
  • Fraction โ†’ decimal: divide the top by the bottom. 3/8 = 0.375.
  • To order a mixture of fractions, decimals and percentages, change them all to decimals (to 3 or 4 places).
  • Symbols: = equal, โ‰  not equal, > greater than, < less than, โฉพ greater than or equal to, โฉฝ less than or equal to.

Worked example

Write in order, smallest first: 0.7, 2/3, 68%, 0.67.

  1. 2/3 = 0.6667, 68% = 0.68.
  2. Compare: 0.6667, 0.67, 0.68, 0.7.

2/3, 0.67, 68%, 0.7

5Halfway values and differences

  • The number halfway between a and b is (a + b) รท 2.
  • The difference between two numbers is the larger minus the smaller; across zero, add the distances: from โˆ’12 to 9 is 12 + 9 = 21.
  • Temperature questions: a rise from โˆ’7 ยฐC to 5 ยฐC is 12 degrees; a fall of 9 degrees from 4 ยฐC gives โˆ’5 ยฐC.

Formulas

halfway = (a + b) รท 2works with negatives and decimals

Worked example

Find the number halfway between โˆ’2.6 and 4.2.

  1. Add: โˆ’2.6 + 4.2 = 1.6.
  2. Halve: 1.6 รท 2.

0.8

6Recurring decimals to fractions

  • A dot over a digit means it repeats: 0.4ฬ‡ = 0.444โ€ฆ, 0.2ฬ‡7ฬ‡ = 0.272727โ€ฆ
  • Every recurring decimal is a fraction. One repeating digit โ†’ over 9; two repeating digits โ†’ over 99.
  • If some digits do not repeat (0.3666โ€ฆ), multiply twice so the repeating parts line up, then subtract.

How to do it

  1. Let x = the decimal.
  2. Multiply by 10 (one repeating digit) or 100 (two) so the repeating part lines up.
  3. Subtract to remove the repeating part.
  4. Divide and simplify.
Fraction1/32/31/91/111/6
Decimal0.3ฬ‡0.6ฬ‡0.1ฬ‡0.0ฬ‡9ฬ‡0.16ฬ‡

Worked example

Write 0.3666โ€ฆ as a fraction in its simplest form.

  1. x = 0.3666โ€ฆ, 10x = 3.666โ€ฆ, 100x = 36.666โ€ฆ
  2. 100x โˆ’ 10x: 90x = 33.
  3. x = 33/90.

11/30

Exam tip: 'Show that 0.2ฬ‡7ฬ‡ = 3/11' needs the algebra (x = โ€ฆ, 100x = โ€ฆ, subtract). Just dividing 3 by 11 gets no marks.

โœ—Common mistakes

Donโ€™t writeWrite instead
Calling 1 a prime number 1 has only one factor, so it is not prime
3 + 4 ร— 2 = 14 Multiply first: 3 + 8 = 11
โˆ’32 = 9 Only the 3 is squared: โˆ’9. (โˆ’3)2 = 9
1/2 + 1/3 = 2/5 Common denominator: 3/6 + 2/6 = 5/6
2ยพ ร— 2 = 4ยพ Improper fraction first: 11/4 ร— 2 = 11/2 = 5ยฝ
Ordering 0.7 before 0.67 because 7 < 67 Compare place by place: 0.70 > 0.67

โ˜…In the exam

  • Non-calculator questions want every step shown โ€” a correct answer with no working may lose marks.
  • Give fractions in their simplest form unless the question asks otherwise.
  • Check an answer is sensible: dividing by a number less than 1 makes it bigger.