Reference Sheet: Programme F.1 — Arithmetic

Rules, procedures and drill, in the order of the book's programme. Every rule links to the unit in the lesson that derives it — if a rule here looks arbitrary, the link is where it stops being arbitrary.

1. Natural numbers, place value and order

From Unit 01 and Unit 02.

The natural numbers are 0,1,2,3,\dots. Position gives value: 246=2(10^2)+4(10)+6, the digits being the coefficients. On a number line, further right is greater: 8>5, 3<6.

2. Integers and brackets

From Unit 02.

The integers are the whole numbers, zero and their negatives; order still holds: -5<3, -2>-4. Never write two operation signs together — bracket the negative number: 5-(-3), 7\times(-2).

3. Addition and subtraction

From Unit 02.

Adding moves right, subtracting moves left. a+(-b)=a-b and a-(-b)=a+b.

Examples. 8+(-3)=5; 9-(-6)=15; (-4)+(-8)=-12.

4. Multiplication and division

From Unit 02.

Like signs give +, unlike signs give -. Forced by the distributive law, not chosen.

5. Brackets and precedence rules

From Unit 03. A convention for reading expressions.

1. Brackets — the innermost first when nested.
2. Powers.
3. Multiplication and division, working from the left, as met.
4. Addition and subtraction, working from the left, as met.

With brackets the \times may be dropped, 5(6-4), and \div may be written as a fraction line.

Examples. 14-3\times4=2. 34+10\div(2-3)\times5=-16. 4(10-2[7-4])=16.

6. Basic laws of arithmetic

From Unit 03. Facts about numbers, not conventions.

• Commutative: a+b=b+a, ab=ba. Not - or \div.
• Associative: (a+b)+c=a+(b+c), (ab)c=a(bc). Not - or \div.
• Distributive: a(b\pm c)=ab\pm ac and (b\pm c)a=ba\pm ca. Division distributes from the right only: (b\pm c)\div a=b\div a\pm c\div a, but a\div(b+c)\ne a\div b+a\div c.

7. Estimating and rounding

From Unit 06 and Unit 08.

Round to the nearest 10, 100, 1000: below halfway round down, above round up, exactly halfway round up. For a negative number round the size and keep the sign: -4550\to-4600 (nearest 100).

Estimate every calculator result with rounded values first: 18\times21-19\div11\approx20\times20-20\div10=398.

8. Factors, primes and prime factorization

From Unit 04.

A factor divides a number exactly. A prime has exactly two factors, 1 and itself: 2,3,5,7,11,13,\dots; 1 is not prime. Factorize by dividing by increasing primes:

Fundamental Theorem of Arithmetic: the prime factorization is unique, apart from order. (That is why 1 is excluded — extra 1s would break uniqueness.)

9. HCF and LCM

From Unit 04.

Line up the prime powers. HCF: common primes, each to its smaller power. LCM: every prime present, each to its larger power. \operatorname{HCF}\times\operatorname{LCM}=ab.

Example. 144=2^4\cdot3^2, 66=2\cdot3\cdot11: HCF 6, LCM 1584.

10. Fractions

From Unit 05.

Numerator over denominator; a rational number. Proper \frac47, improper \frac{12}5, mixed 2\frac25; fractions may be negative.

11. Multiplying fractions, and "of"

From Unit 05.

12. Equivalent fractions and lowest terms

From Unit 05.

\frac ab=\frac{ak}{bk} (k\neq0). Cancel common factors to reduce to lowest terms: \frac{84}{108}=\frac79.

13. Dividing fractions; the reciprocal

From Unit 05.

The reciprocal of \frac cd is \frac dc; of -5 it is -\frac15.

Example. \frac23\div\frac57=\frac{14}{15}.

14. Adding and subtracting fractions

From Unit 05.

Rewrite over a common denominator — the LCM of the denominators — then add or subtract the numerators.

Example. \frac59+\frac16=\frac{10}{18}+\frac3{18}=\frac{13}{18}.

15. Ratios

From Unit 05.

From proportions to a ratio: put the fractions over a common denominator; the numerators are the ratio and add up to the denominator. A component given as "the rest" is found by subtraction.

Example. \frac12 gravel, \frac13 sand, the rest cement: \frac36,\frac26,\frac16, so 3:2:1.

Sharing in a ratio: divide the total by the number of parts. 120 kg in 3:2:1: one part is 20 kg, giving 60, 40, 20 kg.

16. Percentages

From Unit 05.

A percentage is a fraction over 100. Fraction to %: multiply by 100 (\frac{12}{25}=48\%). % of a quantity: 15\% of 240=36. Change by a factor: +r gives \times(1+r), and repeated changes multiply: 500(1.08)^3=629.86.

17. Decimal numbers, significant figures and decimal places

From Unit 06.

3.125=3+\frac1{10}+\frac2{100}+\frac5{1000}. Significant figures count from the first non-zero digit; decimal places count after the point. If the first digit dropped is 5 or more, round the last kept digit up.

Examples. 7.846\to7.85 (2 dp). 0.004736\to0.00474 (3 sf). 2.345\to2.35 (3 sf). Trailing zeros meet the requirement: 12\,645\to13\,000 (2 sf), 13.1\to13.100 (3 dp).

18. Fractions as decimals, and decimals as fractions

From Unit 06.

Fraction to decimal: divide. \frac38=0.375. Terminating decimal to fraction: write over a power of ten and reduce. 0.52=\frac{52}{100}=\frac{13}{25}. A reduced fraction terminates exactly when its denominator's only primes are 2 and 5.

19. Unending decimals

From Unit 06.

Dot notation: a dot over a repeating digit, or over the first and last digits of a repeating block: \frac13=0.\dot3, \frac2{11}=0.\dot1\dot8, \frac17=0.\dot14285\dot7.

To a fraction: multiply by 10^k (k = length of the block), subtract, divide. If the repeat starts later, shift the non-repeating part out first.

Examples. 0.\dot2\dot1: 99x=21, x=\frac7{33}. 0.4\dot6: 10x-x=4.2, x=\frac{42}{90}=\frac7{15}.

20. Rational, irrational and real numbers

From Unit 06.

Rational: expressible as a fraction; its decimal ends or repeats. Irrational: not expressible so; decimal never repeats (\sqrt2, \pi, e). Together: the real numbers.

21. Powers and the laws of powers

From Unit 07.

In a^n, a is the base and n the index or power.

(a^m)^n\ne a^{(m^n)}: (2^3)^2=64 but 2^9=512. Different bases combine only under a common power: 2^3\times5^3=10^3; 2^3\times3^2 does not simplify.

22. Fractional powers, roots and surds

From Unit 07.

a^{1/n} is a number whose nth power is a; a^{m/n}=\left(a^{1/n}\right)^m.

• Odd roots are unique; of a negative, negative: (-27)^{1/3}=-3.
• Even roots of positives: two values, \pm. \sqrt{\ } and, in these notes, a^{1/n} mean the positive one: 16^{1/4}=2. The book writes \pm for even fractional powers (16^{1/4}=\pm2).
• Even roots of negatives: no real value ((-16)^{1/2}).
• A surd is an irrational root kept exact: \frac1{\sqrt2}=\frac{\sqrt2}2.

23. Powers of ten, and precedence with powers

From Unit 08.

Multiplying by 10^n moves the point n places right (n>0) or left (n<0): 1.2345\times10^3=1234.5, 144.032\div10^5=0.00144032. Powers are evaluated before multiplication and division.

24. Standard form

From Unit 08.

Mantissa 1\le A<10 times 10 to an exponent: 52\,674=5.2674\times10^4, 0.000047=4.7\times10^{-5}.

• ×, ÷: multiply or divide the mantissas, add or subtract the exponents, then renormalize. (3.2\times10^4)(5\times10^{-7})=1.6\times10^{-2}.
• +, −: first make the exponents equal. 4.1\times10^3+2.5\times10^4=2.91\times10^4.

25. Preferred standard form

From Unit 08.

Exponent a multiple of 3 (SI), so up to three digits before the point. Write in standard form, then adjust: 3.472\times10^8=347.2\times10^6, 4.7\times10^{-5}=47\times10^{-6}.

26. Checking calculations

From Unit 08.

Write each factor in standard form, round the mantissas to one figure, and track the powers of ten separately: \frac{41.8\times0.0213}{7.92}\approx\frac{4\times2}{8}\times10^{1-2}=0.1 (exact 0.1124).

27. Accuracy

From Unit 08.

Rule: a result from measured data is given to no more significant figures than the least in any measurement. Exact numbers do not count. 12.6\ \text{m}\times4.1\ \text{m}=51.66\to52 m².

Bounds (worst case): a value to the nearest d lies within \pm\frac d2; push the interval ends through the formula. Relative uncertainties approximately add under multiplication and division.

28. Number systems

From Unit 01.

To denary: multiply each digit by its place value and add. 1011.101_2=11.625; \text{X}5_{12}=125.

29. The nested multiplication method

From Unit 01.

Whole part: from the left, multiply the running total by the base and add the next digit. 2753_8: 2\to23\to189\to1515. Fractional part: same process across the digits after the point, then multiply the final total by b^{-k} for k digits.

30. Changing from denary to a new base

From Unit 01.

Whole part: divide repeatedly by the base; read remainders bottom to top. 245_{10}=11110101_2.

Fractional part: multiply the fractional part repeatedly by the base; read the whole-number parts top to bottom. 0.8125_{10}=0.1101_2.

31. Octal as a stepping stone; the reverse method

From Unit 01.

Denary → octal → each octal digit as three bits → regroup in fours outward from the point (pad with zeros) → hex. 245\to365_8\to011\,110\,101_2\to1111\,0101_2=\text{F5}_{16}. Reverse: hex → four-bit groups → regroup in threes → octal → denary.

Traps

• Adding percentages. +8\% three times is \times1.08^3, not +24\%.
• Adding fractions across denominators. \frac12+\frac13\ne\frac25.
• Losing a sign through a bracket. -(a-b)=-a+b.
• Dividing from the left. 24\div(4+2)\ne24\div4+24\div2.
• (a^m)^n vs a^{(m^n)}. (2^3)^2=64, 2^9=512.
• Even roots. The book expects \pm for even fractional powers; \sqrt{\ } alone is positive.
• Rounding early. Round once, at the end.
• Reading remainders downward in a whole-number conversion. The first remainder is the last digit.
• Claiming precision. A result cannot have more significant figures than the least precise measurement.

Self-check

1. Evaluate 20-4\times(3-5)^2\div8.
2. Round -2650 to the nearest 100, and 3.4050 to 3 sf.
3. Find the HCF and LCM of 90 and 84.
4. A mortar is \frac14 cement and \frac23 sand, the rest water. Find the ratio cement : sand : water.
5. Evaluate \frac23\div\frac49+\frac12.
6. Write 0.3\dot7 as a fraction in lowest terms.
7. Evaluate (-32)^{1/5}, 27^{-2/3} and (2^2)^3\div2^4.
8. Evaluate (6.4\times10^5)(2.5\times10^{-8}) in standard form and in preferred standard form.
9. A panel is measured as 3.62 m by 1.8 m. State its area to an appropriate accuracy.
10. Convert 99.625_{10} to binary and to hexadecimal.
11. Convert 2\text{A}_{16} and 1\Lambda_{12} to denary.

Fully worked solutions

1. Brackets: (3-5)^2=(-2)^2=4. Then 4\times4\div8=2 (from the left). 20-2=18.

2. -2650 is exactly halfway between -2600 and -2700; round the size up: -2700. 3.4050: the first dropped digit is 5, so 3.41.

3. 90=2\cdot3^2\cdot5, 84=2^2\cdot3\cdot7. HCF =2\cdot3=6; LCM =2^2\cdot3^2\cdot5\cdot7=1260. Check: 6\times1260=7560=90\times84.

4. Over 12: cement \frac3{12}, sand \frac8{12}, water 1-\frac{11}{12}=\frac1{12}. Ratio 3:8:1.

5. \frac23\times\frac94=\frac{18}{12}=\frac32; \frac32+\frac12=2.

6. x=0.3777\ldots, 10x=3.777\ldots, so 9x=3.4 and x=\frac{3.4}9=\frac{34}{90}=\frac{17}{45}.

7. (-32)^{1/5}=-2 (odd root of a negative). 27^{-2/3}=\frac1{(27^{1/3})^2}=\frac1{9}. (2^2)^3\div2^4=2^6\div2^4=2^2=4.

8. 6.4\times2.5=16 and 10^{5-8}=10^{-3}: 16\times10^{-3}=1.6\times10^{-2} (standard form) =16\times10^{-3} (preferred).

9. 3.62\times1.8=6.516; 1.8 has 2 sf, so 6.5 m².

10. 99: remainders on dividing by 2 give 1100011_2 (64+32+2+1). 0.625: 1.25\to1, 0.5\to0, 1.0\to1, so 0.101_2. Together 1100011.101_2. Regroup in fours from the point: 0110\,0011.1010_2=63.\text{A}_{16}. Check: 6\times16+3=99 and \frac{10}{16}=0.625.

11. 2\text{A}_{16}=2\times16+10=42. 1\Lambda_{12}=1\times12+11=23.