Number Toolkit

2. Ordering numbers

Putting numbers in order

• Ascending order runs from smallest to largest; descending order runs from largest to smallest
• To compare whole numbers, compare the highest place value first, then work rightwards until the digits differ
• To compare decimals, give every number the same number of decimal places by padding with zeros
  • 3.6, 3.58 and 3.605 become 3.600, 3.580 and 3.605, so the order is 3.58, 3.6, 3.605
• Length of a decimal is not a guide to its size: 0.9 is larger than 0.4999

Ordering negative numbers

• On a number line, values increase from left to right, so any negative number is smaller than any positive number
• Among negatives, the further a number is from zero, the smaller its value
  • −11 is smaller than −4
• With a mixed list, order the negatives and the positives separately, then place all the negatives first

3. Types of number

Integers and natural numbers

• The integers run in both directions from zero without fractional parts: … −3, −2, −1, 0, 1, 2, 3 …
• Counting from 1 upwards gives the natural numbers 1, 2, 3, 4, …, a set that leaves zero out
• A number written as a decimal or a fraction is only an integer if it simplifies to a whole number
  • 18/3 is an integer because it equals 6

Multiples

• A multiple of an integer is the result of multiplying it by a positive integer
• The multiples of 7 are 7, 14, 21, 28, 35, …
• A common multiple of two integers is a multiple of both
  • 24 is a common multiple of 6 and 8
• Multiples can be written algebraically: the multiples of n are n, 2n, 3n, 4n, …

Factors

• A factor of an integer divides into it exactly, leaving no remainder
• Beyond 1, no integer has fewer than two factors, since 1 and the number itself always divide it
• Factors are found in pairs, which is why testing only up to the square root is enough
  • For 24: 1 × 24, 2 × 12, 3 × 8, 4 × 6, giving factors 1, 2, 3, 4, 6, 8, 12, 24
• A common factor of two integers is a factor of both
  • 5 is a common factor of 20 and 35

Divisibility tests

• Divisibility tests decide whether a number is a factor without dividing, which matters on the non-calculator papers

Primes, squares, cubes and roots

• A prime number has exactly two factors, 1 and itself
  • Among the primes only 2 is even, since every other even number has 2 as a third factor
  • 1 is not prime, because it has only one factor
• A square number is an integer multiplied by itself: 1, 4, 9, 16, 25, 36, …
• A cube number is an integer multiplied by itself twice: 1, 8, 27, 64, 125, …
• The square root of a number is the value that squares to give it, written √
  • √49 = 7, because 7² = 49
• The cube root reverses cubing, written ∛
  • ∛125 = 5, because 5³ = 125

4. Negative numbers

Adding and subtracting

• A subtracted negative reverses into an addition
  • 8 − (−5) = 8 + 5 = 13
• An added negative reverses into a subtraction
  • 6 + (−9) = 6 − 9 = −3
• Two signs written next to each other collapse to one: two like signs give a plus, two unlike signs give a minus

Multiplying and dividing

• Matching signs on the two values produce a positive result
  • (−7) × (−3) = 21 and (−24) ÷ (−6) = 4
• Opposing signs produce a negative result
  • (−7) × 3 = −21 and 24 ÷ (−6) = −4
• With a longer product, count the negative factors: an even count gives a positive, an odd count gives a negative

Negatives in context

• Temperature below zero is the most common context
  • A temperature of 2 °C falling by 9 °C gives 2 − 9 = −7 °C
• Money owed is written as a negative balance
  • A balance of −£150 that receives a £90 repayment becomes −150 + 90 = −£60
• Depth below sea level and floors below ground are also recorded as negatives

5. Reciprocals

Finding a reciprocal

• Inverting a number gives its reciprocal: 1 is divided by the number, and the result is what the original must be multiplied by to reach 1
• Every non-zero value and its reciprocal have a product of exactly 1
• For a fraction, numerator and denominator swap places
  • Inverting 3/8 gives 8/3
• The reciprocal of an integer n is 1/n
  • The reciprocal of 5 is 1/5
• A decimal is easiest to invert by first writing it as a fraction
  • 0.4 = 2/5, so its reciprocal is 5/2 = 2.5
• Zero has no reciprocal, because division by zero is undefined
• In index form the reciprocal of a is a⁻¹

6. Mathematical symbols

Symbols to recognise

• Brackets group part of a calculation so it is carried out first
• Indices, also called powers, show repeated multiplication: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32
• An identity written with ≡ holds for every value of the letter, whereas an equation written with = holds only for particular values

Words used for the four operations

7. Addition and subtraction

Column addition

• Write the numbers one above the other with the place-value columns lined up
• Add each column working right to left
• When a column totals ten or more, write the ones digit below the line and carry the tens digit into the next column

Column subtraction

• Write the number being subtracted underneath the number it is taken from
• Work right to left, and where the upper digit is too small, exchange one unit down from the neighbouring column
• Exchanging turns one unit of the left-hand column into ten of the current column

8. Multiplication and division

Column multiplication

• Multiply the top number by each digit of the bottom number in turn, working right to left
• Write a zero as a place holder for each column moved leftwards through the bottom number
• Add the partial products to reach the final answer

The grid method

• The grid method splits each number into its place-value parts, so no carrying is needed until the final addition
• For 246 × 34, split 246 into 200, 40 and 6, and 34 into 30 and 4

• Adding every cell gives 6 000 + 1 200 + 180 + 800 + 160 + 24 = 8 364

The lattice method

• Every product stays a single-digit pair in this layout, which is what makes it reliable for longer numbers
• Each cell is split by a diagonal, with the tens digit above the diagonal and the ones digit below
• The digits are then summed along each diagonal, carrying leftwards where a diagonal totals ten or more

Short division

• Short division divides by a single digit, carrying any remainder into the next digit of the number
• The remainder at each step is written as a small digit in front of the next digit

• Only one written method for each operation needs to be known, provided it is used accurately

Shortcuts worth spotting

• Multiplying or dividing by a power of ten shifts every digit along the place-value columns, one column for each zero
  • 380 ÷ 10 = 38, 45 ÷ 100 = 0.45 and 28 ÷ 1000 = 0.028
  • 3.7 × 100 = 370
• Writing the leading zeros in first makes the shift easier to see, so 0028 ÷ 1000 becomes 0.028
• A division by a larger number can be simplified first, exactly as a fraction is cancelled
  • 1008 ÷ 28 is 1008/28, which halves to 504/14 and again to 252/7, and short division finishes it as 36
• Halving repeatedly divides by a power of 2, so 568 ÷ 8 is 284, then 142, then 71

9. Operations with decimals

Adding and subtracting decimals

• Line the decimal points up underneath one another before adding or subtracting
• Fill any short columns with zeros so every number has the same number of decimal places
• The decimal point in the answer sits directly below the points in the question

Multiplying decimals

• Multiplying by a power of ten turns a decimal into an integer, and the change is undone at the end
  • For 1.4 × 2.05, work out 14 × 205 = 2 870, then divide by 10 and by 100 to give 2.87
• Counting decimal places gives the same result: the answer has as many decimal places as the two numbers together
  • 1.4 has one and 2.05 has two, so the answer has three before trailing zeros are removed
• An estimate settles where the point belongs: 1.4 × 2.05 is roughly 1.5 × 2 = 3, so 2.87 is sensible and 28.7 is not

Dividing decimals

• Multiply both numbers by the same power of ten until the divisor is a whole number
  • 3.15 ÷ 0.05 becomes 315 ÷ 5 = 63
• The answer is unchanged because both numbers were scaled equally

10. Order of operations

BIDMAS

• Operations are carried out in a fixed order, remembered as BIDMAS or BODMAS

• Division and multiplication share a rank, so 24 ÷ 6 × 2 is 8, not 2
• Addition and subtraction likewise share a rank, so 10 − 4 + 3 is 9, not 3

Hidden brackets in fractions and roots

• A fraction line acts as a bracket around the whole numerator and the whole denominator
  • (7 + 5)/(9 − 3) is worked as 12 ÷ 6 = 2
• A root sign acts as a bracket around everything underneath it
  • √(9 + 16) is √25 = 5, which is not the same as √9 + √16 = 7
• These brackets have to be typed in explicitly on a calculator, because the layout that makes them obvious on paper is lost on the keypad

11. Rational and irrational numbers

Rational numbers

• Any value expressible as one integer over another, with a non-zero denominator, is a rational number
• Every integer is rational, because it can be written over a denominator of 1
• Every terminating decimal is rational
  • 0.35 = 7/20
• Every recurring decimal is rational
  • 0.333… = 1/3

Irrational numbers

• An irrational number admits no such integer-over-integer form
• Its decimal form never terminates and never settles into a repeating block
• The square root of a positive integer that is not a square number is irrational, and is called a surd
  • √7 is irrational, but √36 is not, because it equals 6
• π is irrational
• A non-zero rational number multiplied by an irrational number is still irrational
  • 5√2 and π/4 are both irrational