Backnotes

Number

1.1 – Number Facts and Sequences

  • Natural Numbers:
  • Used for counting purposes
  • Made up of all irrational and rational numbers
  • Integer: A whole number
  • Prime Numbers:
  • Only divisible by itself and 1
  • 1 is NOT a prime number
  • Rational Numbers: Can be written as a fraction
  • Irrational Numbers: Cannot be written as a fraction

                                      E.g. π

  • Common Factors: Common divisors of a number
  • Common Multiples: Multiples of two or more numbers that are the same
  • Sequences:
  • Finding the nth term:
  • Linear Sequence / Arithmetic Sequence:
  • Formula to find nth term : a+(n – 1)d
  • a = first number in the sequence
  • d = difference

E.g.  1,  4 , 7 , 10 , 13

a = 1

d = 3 (4 – 1 = 3, 7 – 4 = 3 etc.)

a+(n – 1)d

= 1 + (n – 1) 3

= 3n – 3 + 1

= 3n – 2              Expression for the nth term

1.2 – Fractions and Decimals

  • Converting Fractions to Decimals:
  • Divide the numerator by the denominator
  • E.g.  7/8  to a decimal

                     7÷8 = 0.875

  • Converting Decimals to a Fraction:
  • Write down the decimal divided by 1
  • E.g.  0.35/1
  • Multiply both the numerator and the denominator by 10 for every number after the decimal point
  • E.g. 0.35/1           35/100
  • Simplify the Fraction
  • E.g. 35/100           7/20

            1.3 – Approximations and Estimations

  • Rounding Decimals to the Nearest Whole Number:
  • If the number after the decimal is less than 5 then round the number down by removing the decimal part of the number
  • E.g.  7.3176           7

              This figure is ‘less than 5’

  • If the number after the decimal is 5 or more then round the number up by adding 1 on to the ones digit and removing the decimal part of the number.
  • E.g.  7.8176           8

                            This figure is ‘5 or more’

  • Rounding Decimals to Two/Three Significant Figures:
  • Count from the first non–zero digit for two/three digits (0 is not counted as a digit)
  • Then round the last digit
  • E.g.
  • 7.8176            7.82 (3 s.f)

                                        This figure is ‘5 or more’

  • 0.078176           0.0782 (3 s.f)

                                              7 is the first significant figure

  • Rounding to Two/Three Decimal Places:
  • Count from the first digit for two/three digits (0 is counted as a digit)
  • Then round the last digit
  • E.g. 7.8176            7.818 ( 3 decimal places)

                   This figure is ‘5 or more’

  • Measurements and Bounds:
  • The ‘unit’ is 1 so ‘half a unit’ is 0.5  (+ 0.5 or – 0.5)           upper bound , lower bound
  • The ‘unit’ is 0.1 so ‘half a unit’ is 0.05  (+ 0.05 or – 0.05)           upper bound , lower bound
  • The ‘unit’ is 0.001 so ‘half a unit’ is 0.0005  (+ 0.0005 or – 0.0005)           upper bound , lower bound
  • E.g.  A length is measured 135 cm to the nearest cm

        The actual length could be anything from 134.5 cm to 135.49999…cm using the normal convention

        which is to round up a figure of 5 or more. Clearly 135.49999… is effectively 135.5 and we say the

        upper bound is 135.5. The lower bound is 134.5

        As an inequality we can write 134.5 ≤ length < 135.5

1.4 – Standard Form

  • The number a×10^n is in standard form when 1 ≤ a < 10 and n is a positive or negative integer
10⁴ = 1000010^ -1 = 0.1
10³ = 100010^ -2 = 0.01
10² = 10010^ -3 = 0.001
10¹ = 1010^ -4 = 0.0001
10⁰ = 110^ -5 = 0.00001
  • E.g.
  • 2000           2 × 10³
  • 150           1.5 × 10²
  • 0.0004           4 × 10^ -4

1.5 – Ratio and Proportion

  • Ratio:
  • Used to describe a fraction
  • E.g. 4:7
  • Changing to the Form 1:n:
  • E.g.  2:5

        2:5 = 1: 5/2

              = 1: 2.5

  • Changing to the Form n:1:
  • E.g.  2:5

       2:5 = 2/5 :1

              = 0.4:1

  • Foreign Exchange:
  • Money changed from one currency to another using proportion
  • E.g.
  • Covert $22.50 to dinars

   $1 = 0.30 dinars (KWD)

   $22.50 = 0.30 × 22.50

               = 6.75 KWD

  • Map Scales: Using proportion to work out map scales
  • Metric Equivalents:
  • 1 km = 1000 m
  • 1 m = 100 cm
  • 1 cm = 10 mm

1.6 – Percentages

  • Converting Percentages to a Fraction:
  • Write down the percent divided by 100
  • E.g.  35%           35/100
  • If the percent is not a whole number, then multiply both top and bottom by 10 for every number after the decimal point
  • E.g.  0.35%          0.35/100          35/10000
  • Simplify the fraction
  • E.g.  35/100          7/20
  • Converting Fractions to a Percentage:
  • Convert the fraction to a decimal
  • Divide the numerator by the denominator
  • E.g. ¼  to a decimal

                     1 ÷ 4 = 0.25

  •  Multiply by 100 to get percent value
  • E.g. 0.25 × 100 = 25

       25%

  • Converting Percentages to a Decimal:
  • Divide the number (in percentage format) by 100
  • E.g. 25% to a decimal

       25/100 = 0.25

  • Percentage Increase or Decrease:
  • Simple Interest:
  • A sum of money $P is invested for T years at R% interest per year the interest gained is given by:
  • Compound Interest:
  • Formula:

1.7 – Speed, Distance and Time

  • Distance = Speed × Time
  • Speed = Distance/Time
  • Time = Distance/Speed
  • Units of Speed:
  • km/hr  –  Kilometers per hour
  • m/s  –  Meters per second
  • Units of Distance:
  • km  –  Kilometers
  • m  –  Meters
  • Units of Time:
  • hr  –  Hours
  • sec  –  Seconds