Mathematics 4024 ยท Syllabus tracker
Arithematics
Types of number, the order of operations, fractions and decimals, halfway values and recurring decimals.
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Quick revision
Arithematics on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
The number toolkit used in every other chapter: kinds of number, BIDMAS, fractions, decimals and recurring decimals.
1Types of number
- Natural numbers: 1, 2, 3, โฆ Integers: whole numbers including 0 and negatives (โฆ, โ2, โ1, 0, 1, 2, โฆ).
- Prime: exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, โฆ 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, โฆ).
- Rational: can be written as a fraction a/b (all integers, terminating and recurring decimals). Irrational: cannot โ ฯ, โ2, โ3, โ5 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.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| n2 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
| n3 | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |
Worked example
From โ3, 0, 1/2, โ9, โ5, 7 and 0.333โฆ, list (a) the irrational numbers, (b) the prime numbers.
- โ9 = 3 and 0.333โฆ = 1/3, so both are rational.
- โ5 is not a whole number and does not recur: irrational.
- Primes are positive whole numbers with exactly two factors: โ9 = 3 and 7.
(a) โ5 (b) โ9 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 โ. So 5 โ (โ3) = 8 and 4 ร (โ2) = โ8.
- (โ3)2 = 9 but โ32 = โ9: without brackets only the 3 is squared.
- A fraction line acts like brackets: (8 + 4) รท (5 โ 2) = 4. On a calculator, put brackets round the top and the bottom.
Order to work in
- Work out anything in brackets.
- Then powers and roots.
- Then ร and รท from left to right.
- Finally + and โ from left to right.
Worked example
Work out (โ3)2 โ 4 ร (5 โ 7).
- Brackets: 5 โ 7 = โ2.
- Power: (โ3)2 = 9.
- Multiply: 4 ร (โ2) = โ8.
- 9 โ (โ8) = 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
Worked example
Work out 2ยพ โ 1โ .
- Improper fractions: 11/4 โ 11/6.
- Common denominator 12: 33/12 โ 22/12.
- = 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, โฆ 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.
- 2/3 = 0.6667, 68% = 0.68.
- 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 โ12 to 9 is 12 + 9 = 21.
- Temperature questions: a rise from โ7 ยฐC to 5 ยฐC is 12 degrees; a fall of 9 degrees from 4 ยฐC gives โ5 ยฐC.
Formulas
Worked example
Find the number halfway between โ2.6 and 4.2.
- Add: โ2.6 + 4.2 = 1.6.
- Halve: 1.6 รท 2.
0.8
6Recurring decimals to fractions
- A dot over a digit means it repeats: 0.4ฬ = 0.444โฆ, 0.2ฬ7ฬ = 0.272727โฆ
- Every recurring decimal is a fraction. One repeating digit โ over 9; two repeating digits โ over 99.
- If some digits do not repeat (0.3666โฆ), multiply twice so the repeating parts line up, then subtract.
How to do it
- Let x = the decimal.
- Multiply by 10 (one repeating digit) or 100 (two) so the repeating part lines up.
- Subtract to remove the repeating part.
- Divide and simplify.
| Fraction | 1/3 | 2/3 | 1/9 | 1/11 | 1/6 |
|---|---|---|---|---|---|
| Decimal | 0.3ฬ | 0.6ฬ | 0.1ฬ | 0.0ฬ9ฬ | 0.16ฬ |
Worked example
Write 0.3666โฆ as a fraction in its simplest form.
- x = 0.3666โฆ, 10x = 3.666โฆ, 100x = 36.666โฆ
- 100x โ 10x: 90x = 33.
- x = 33/90.
11/30
Exam tip: 'Show that 0.2ฬ7ฬ = 3/11' needs the algebra (x = โฆ, 100x = โฆ, subtract). Just dividing 3 by 11 gets no marks.
โCommon mistakes
| Donโt write | Write 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 |
| โ32 = 9 | Only the 3 is squared: โ9. (โ3)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.