- Integers are whole numbers; a prime has exactly two factors, 1 and itself, so 1 is not prime.
- The first primes are 2, 3, 5, 7, 11, 13, and 2 is the only even prime.
- Square numbers come from n squared (1, 4, 9, 16); cube numbers from n cubed (1, 8, 27, 64).
- A rational number can be written as a fraction; irrational numbers like pi and root 2 cannot.
Watch out: Thinking 1 is a prime number. It only has one factor, so it does not meet the definition of prime.
- Each place is ten times the one to its right: thousands, hundreds, tens, units, tenths, hundredths.
- To order decimals, compare digit by digit from the left, not by how many digits there are.
- Multiplying by 10, 100 or 1000 moves every digit left; dividing moves them right.
- Negative numbers get smaller as you move left, so -7 is less than -3.
Watch out: Judging a decimal as bigger just because it has more digits, for example thinking 0.096 is bigger than 0.6.
- A factor divides into a number exactly; a multiple is in that number times table.
- Every whole number above 1 breaks into a unique product of primes.
- Use a prime factor tree, then write the answer with powers, e.g. 60 = 2 squared x 3 x 5.
- Factors come in pairs, so only square numbers have an odd number of factors.
Watch out: Confusing factors with multiples, for example listing the times table of a number when asked for its factors.
- The HCF is the largest number that divides into both; the LCM is the smallest number both divide into.
- From prime factors: HCF multiplies the shared primes, LCM multiplies the highest power of every prime.
- A Venn diagram of prime factors gives the HCF in the overlap and the LCM from everything.
- HCF x LCM equals the product of the two original numbers.
Watch out: Confusing HCF and LCM, and finding the highest common factor when the question actually asks for the lowest common multiple, or the other way round.
- Simplify by dividing top and bottom by the same number until they share no common factor.
- Equivalent fractions have the same value: 2/3 = 4/6 = 6/9.
- To compare fractions, rewrite them over a common denominator, then compare numerators.
- A fraction equals 1 when the numerator and denominator match.
Watch out: Simplifying a fraction partway and stopping, for example simplifying 8/12 to 4/6 and not noticing it can still be simplified further to 2/3.
- You can only add or subtract fractions that have the same denominator.
- Rewrite each fraction over the common denominator, then add or subtract the numerators only.
- For mixed numbers, convert to improper fractions first, or handle the whole and fraction parts separately.
- Simplify the answer and turn improper fractions back into mixed numbers.
Watch out: Adding or subtracting the denominators as well as the numerators, for example writing 1/3 + 1/4 as 2/7 instead of finding a common denominator.
- To multiply, multiply the numerators and multiply the denominators; no common denominator needed.
- To divide, flip the second fraction and multiply (keep, change, flip).
- Turn mixed numbers into improper fractions before multiplying or dividing.
- Cancel common factors before multiplying to keep the numbers small.
Watch out: Trying to find a common denominator before multiplying fractions, which is only needed for adding and subtracting, not multiplying.
- Line up the decimal points when adding or subtracting, filling gaps with zeros.
- To multiply decimals, ignore the points, multiply, then put back as many decimal places as both numbers had together.
- To divide by a decimal, multiply both numbers by 10 or 100 until the divisor is whole.
- A terminating decimal stops; a recurring decimal repeats a block forever.
Watch out: Not lining up the decimal points when adding or subtracting, especially when the numbers have a different number of decimal places.
- Percent means out of 100, so 30% is 30/100 or 0.3.
- Non-calculator: 10% is divide by 10, 1% is divide by 100, 5% is half of 10%.
- Calculator: multiply the amount by the percentage as a decimal, so x 0.3 for 30%.
- To find 17.5%, add 10% then 5% then 2.5%.
Watch out: Dividing by 10 instead of 100 when converting a percentage to a decimal, for example writing 35% as 3.5 instead of 0.35.
- Percentage change is the change divided by the original amount, times 100.
- To increase by 12%, multiply by 1.12; to decrease by 12%, multiply by 0.88.
- The multiplier is 1 plus rate/100 for a rise, 1 minus rate/100 for a fall.
- Profit and loss percentages are worked out on the cost price, not the selling price.
Watch out: Working out the percentage of the amount and giving that as the final answer, instead of adding it to or subtracting it from the original amount.
- If a price already includes a percentage change, that price is not 100%.
- After a 20% rise the amount is 120%, so divide by 1.2 to get the original.
- After a 15% sale the price is 85%, so divide by 0.85 to find the original.
- Never take the percentage straight off the new amount; that gives the wrong original.
Watch out: Finding a percentage of the given (final) amount and adding or subtracting it, instead of dividing by the correct decimal multiplier.
- Compound interest is added to the total each year, so you earn interest on interest.
- Use amount equals principal x multiplier to the power of the number of years.
- The multiplier is 1 plus rate/100 for growth, 1 minus rate/100 for depreciation.
- Simple interest adds the same amount every year; compound interest grows faster.
Watch out: Using simple interest instead of compound interest, by multiplying the percentage by the number of years and adding it all in one step, instead of applying the multiplier repeatedly.
- Standard form is A x 10 to the power n, where A is at least 1 and less than 10.
- A positive power means a large number; a negative power means a small number below 1.
- To multiply, multiply the A parts and add the powers; to divide, divide the A parts and subtract the powers.
- Always check A is between 1 and 10 and adjust the power if it is not.
Watch out: Miscounting the number of decimal places moved, giving a power of 10 that is one too high or too low.
- Multiplying powers of the same base adds the indices; dividing subtracts them.
- A power raised to a power multiplies the indices.
- Anything to the power 0 is 1; a negative power means one over the positive power.
- A fractional power is a root: x to the power one half is root x, x to the power two thirds is the cube root of x squared.
Watch out: Adding or subtracting indices when the bases are different, which only works when the base is exactly the same on both sides.
- A surd is a root that cannot be simplified to a whole number, like root 2 or root 5.
- root(a b) equals root a x root b, so root 50 is root 25 x root 2, which is 5 root 2.
- You can only add or subtract like surds: 3 root 2 plus 4 root 2 is 7 root 2.
- Rationalise a denominator by multiplying top and bottom by the surd, or by its conjugate.
Watch out: Not using the largest square factor when simplifying a surd, leaving the answer only partially simplified.
- Round up if the next digit is 5 or more, down if it is 4 or less.
- The first significant figure is the first non-zero digit from the left.
- Keep place value: 3847 to 2 significant figures is 3800, not 38.
- To estimate a calculation, round every number to 1 significant figure first.
Watch out: Rounding a whole number to the nearest 100 or 1000 but forgetting to replace the remaining digits with zeros, for example writing 38 instead of 3800.
- Round every number to 1 significant figure, then do the easier calculation.
- Use the approximately equal sign for an estimate, not the equals sign.
- Dividing by a number less than 1 makes the answer bigger.
- Say whether your estimate is an overestimate or an underestimate, and why.
Watch out: Rounding to a different number of significant figures than asked for, for example rounding to 2 significant figures instead of 1.
- A value rounded to the nearest 10 could be up to 5 either side of the stated value.
- The lower bound is included; the upper bound is the value you never quite reach.
- For the largest possible result, use the largest values, but take care with subtraction and division.
- Maximum of a difference uses upper minus lower; maximum of a quotient uses upper divided by lower.
Watch out: Using the whole rounding unit instead of half of it when finding the upper and lower bounds.
- Every recurring decimal is a fraction, so it is a rational number.
- Let x be the decimal, then multiply by 10, 100 or 1000 to line up the repeating block.
- Subtract the two equations to remove the recurring part, then solve for x.
- One repeating digit means multiply by 10; two repeating digits means multiply by 100.
Watch out: Multiplying by the wrong power of 10, for example multiplying by 10 when two digits repeat instead of 100.
