GCSE Maths - 1. Number

GCSE Maths - 1. Number

1. TYPES

   • Integer or not
     • Integer = whole number
       • +ve
       • -ve
       • 0
     • WHOLE
   • Rational or not
     • Rational number
       • 3 forms
         • Factions
         • Integers
         • Terminating or recurring decimals
     • NEAT
     • Irrational number
       • messy
       • never ending, non repeating decimals
       • can't be written as fractions
       • Surds
         • contains irrational roots
       • Square roots of +ve
       • Pi

2. Operations

   • BODMAS / BIDMAS
     • when there're >1 thing to do & unclear what to do next
   • Reciprocal
     • of a number is 1 /: that number
     • Find a reciprocal of a number

3. BREAKING DOWN

   • MULTIPLES
     • time tables of a number
     • Least common multiples
       • smallest number divides by all numbers in question
       • Break down numbers in prime factors WITH powers
         • Alternative: use ven diagram
           • the overlap area get rid of reappearance
         • ring all broken down factors once for each number value
           • same value factor reappear -> pick highest power
             • LCM = product of ringed prime factors
           • factors DON"T HAVE to be in every number
         • Find LCM
   • FACTORS
     • numbers that divide into a number
     • Highest common factors
       • biggest number divides into all numbers in question
       • Break down numbers in prime factors WITHOUT powers
         • ring those appear in all numbers
           • factors HAVE to be in every number
           • HCF = product of ringed prime factors
         • Find HCF
   • Prime numbers
     • PRIME FACTORS
       • Break down as a product of prime factors
     • doesn't divide by anything but itself and 1
       • 1 is NOT a prime number

4. FORM

   • Standard
     • A x 10^n
       • A = 1 -> 10
       • n = no. of decimal point moves right -> left to shrink A
       • n +ve if big number; -ve if small
         • A x 10 ^ 3 = A move dp RIGHT 3 = A,000
         • A x 10 ^ -3 = A move dp LEFT 3 = 0.00A
       • 35,600 = 3.56 x 10^4
       • dec point moves right if n grows smaller; moves left if n grows bigger
     • a.k.a shrinking/become smaller to multiply 10 with power
   • Ordinary
     • 1,234,567 etc
   • Convert between standard & original forms
   • Calculation with standard forms
     • x and : As together and powers together
     • + and - only with the SAME powers

5. ESTIMATION

   • Rounding
     • d.p.
       • 0. X Y [last digit] [decider]
         • last digit = how many d.p. wanted
           • + 1 if decider 5 or greater
           • unchanged if decider smaller than 5
         • here 3 d.p. are wanted
       • decimal places
     • s.f.
       • significant figures
       • 1st s.f.
         • 1st non 0 digit
           • 0,0001
           • 5000
       • 2nd, 3rd, 4th etc. s.f.
         • if 0
           • only counts if followed by non-0 s.f.
             • 2.001 (4 s.f.)
             • 30900 (3 s.f.)
             • 2.0307 (5 s.f.)
         • if not 0
           • counts
             • 54.8 (3 s.f.)
   • BOUND
     • X number is rounded to A unit (to the nearest A)
       • upper bound = X + half A
         • without = sign (strict)
       • lower bound = X - half A
         • with = sign
   • Estimating Calculations and Square Roots
     • Min/max value evaluation
       • When answer "appropriate degree of accuracy", select no. of s.f. so that both upper and lower bounds of answer round to the same s.f.

6. FRACTIONS & DECIMALS

   • Fraction
     • NUMERATOR: number of the top
     • DENOMINATOR: number of the bottom
   • Improper fraction
     • Heavy top
   • Mixed number
     • Integer / whole number part
     • fraction part
     • convert first before operations
   • Operating fractions
     • Cancel down / Simplify
       • Divide top and bottom by the same number
     • Divide
       • Flip (turn up side down) 2nd fraction then multiply
     • Common denominators
       • Add/subtract
       • Order
     • Fractions of something
       • Multiply top, divide bottom
     • Expressing as fraction of X
       • Divide by X in a fraction
   • Convert to decimals and percentage
     • Fraction --(divide)--> Decimal --(x100)--> percentage
       • Careful: Fraction--> recurring decimals
         • Convert to bottom with 9s
           • top is recurring bit
           • no. of 9s = no. of d.p.
     • Percentage ----(:100)---> decimal --(!@£$%)--> fraction
       • The hard bit: Decimal -> fraction
         • Terminating decimals
           • top: number after decimal point
           • bottom: power of 10 - no. of 0s = no. of decimal places
         • Recurring decimals
           • name decimal as r
             • multiply by power of 10 (x10, x1000 etc.)
               • end up with a subtraction between two different multiples of r...
                 • so that the same recurring bits in both multiples cancel out
     • Some decimal definitions
       • Terminating decimals: finite
       • Recurring decimals: pattern of numbers which repeats forever
         • DOTS on top of beginning and ending numbers of repeating bit