Types of numbers
Square numbers- the difference between two square numbers are all odd.
Cube numbers- they're like the volumbes of cubes e.g. 2x2x2= 8
Prime numbers- all end in 1,3,7,9 (except 2 and 5).
How to work out prime numbers-
- does it end in 1,3,7,5?
- find its square root
- list all prime numbers less than this square root
- divide the number by all the prime numbers less that its square root
- if none of these divide cleanly (the answer has decimal places), the number is a prime number.
Factors, Prime Factors, LCM and HCF
Factors- all the numbers that divide into the asked number (e.g. factors of 12)
Finding Prime Factors- any number can be broken down into a string of prime numbers all multiplied together (called "expressing it as a product of prime factors").
Factor tree method- Split the number into two factors. If the number is a prime number, circle it and if not split it up into more factors until you end up with all the prime factors. You can arrange them in order, smallest first.
Lowest common multiple- the smallest number that will divide by all the numbers in the question.
Highest commont factor- the biggest number that will divide into all the numbers in the question.
Type 1- "Find x% of y"- e.g. Find 15% of £46 - 0.15 x 46 = £6.90
Type 2- "Express x as a percentage of y" e.g. Give 40p as a percentage of £3.34"- (40 divided by 334) x 100 = 12%
Type 3- e.g. "a house increases in a value by 20% to £72,000. Find out what it was worth before the rise." An increase 0f 20% means that £72,000 represents 120% of the orginal value. If it was a drop of 20%, then it would represent 80%. You divide the value (£72,000) by 120% and then times it by 100.
Percentage change- percentage "change"= Change divided by original x 100.
Rational numbers- are either a whole number, a fraction or a terminating/recurring decimal.
Irrational numbers- are never-ending non-repeating decimals. These are surds.
Manipulating Surds- Surds are expressions with irrational square roots in them. They must be used if they ask for an exact answer. There are a few simple rules to follow-
- √a x √b = √ab e.g. √2 x √3 = √2x3 = √6
- (√b)2 = b
- √a/√b = √a/b e.g. √8/√2 = √8/2 = √4 = 2
- √a + √b = do nothing!
- (a + √b)2 = (a + √b) (a + √b) = a2 + 2a√b + b
Bounds and Reciprocals
Upper and lower bounds of a measurement- the real unit can be as much as half the rounded unit above or below the rounded-off value. e.g.- 2.4m to the nearest 0.1 could have a real value of anything up to 2.4m +/- 0.05m giving answers of 2.45m and 2.35m for the upper and lower bounds.
Maximum and minimum values for calculations- when a calcualtion is done using rounded-off values there will be a discrepancy between the calculated value and the actual value.
- the reciprocal of a number is "one over" the number
- you can find the reciprocal of a fraction by turning it upside down
- a number multiplied by its reciprocal gives 1
- 0 has no reciprocal because you can't divide anything by 0
how to work out conversion factor- find the conversion factor, multiply by it and divide by it, choose the common sense answer.
- 1kg = 2 1/4lbs
- 1m = 1 yard + 10%1
- 1 litre = 1 3/4 pints
- 1 inch = 2.5 cm
- 1 gallon = 4.5 litres
- 1 foot = 30cm
- 1 metric tonne = 1 imperial ton
- 1 mile = 1.6 km (or 5 miles = 8 km)
Formular- dn + (a-d)
"a"- the first term of the sequence
"d"- the common difference between terms
Changing difference type- "a + (n-1)d + 1/2 (n-1) (n-2) c"
"a" = first time
"d" = the first difference (between the first and second number)
"c"- the change between one difference and the next