Mathematics-notes

1 | P a g e Mathematics - Important Notes and Formulas Numbers Type Definition Natural numbers All whole numbers except 0 eg: 1, 2, 3, 4, 5... Even numbers 0, 2, 4, 6, 8, 10... Odd numbers 1, 3, 5, 7, 9... Integers whole numbers that can be positive, negative, or zero eg: -1, -2, -3, 1, 2, 3... Prime number a natural number which has only 2 different factors eg: 2, 3, 5, 7, 11, 13... Composite number a natural number that has more than 2 different factors eg: 4, 6, 8, 9... Real number Include rational and irrational numbers, fractions, and integers Rational number a number that can be expressed as a fraction or as a ratio Irrational number a number that cannot be expressed as a fraction or a ratio of 2 integers. eg: pi and roots Test of Divisibility Divisible by Test 2 if the number is even 3 if the sum of the digits is divisible by 3 4 if the number formed by the last 2 digits is divisible by 4 5 if the last digit is 0 or 5 9 if the sum of its digits is divisible by 9 10 if the last digit is 0 11 if the difference between the sum of the digits in the odd places and the sum of the digits in the even places is equal to 0 or is a multiple of 11 Standard form This is a convenient way to write very large or very small numbers, using the from a x 10n, where n is a positive or negative integer, and a s between 1 to 10 inclusive. More examples: 123 400 written as standard form is 1.234 x 105 0.0000987 written as standard form is 9.87 x 10-5

2 | P a g e Multiplying numbers in standard form Dividing numbers in standard form Adding and Subtracting numbers in standard form - Make the index between the 2 numbers the same so that it is easier to factorise the numbers before adding eg Scales and Maps Given that a map has a scale of 1:10 000, this means that 1cm on the map represents 10,000cm on the actual ground. 1cm : 200m = 1cm : 0.2km = 1cm2 : 0.04km2 Proportion A. Direct Proportion This means that when y increases, x increases, and vice versa. Use this equation: y = kx B. Indirect Proportion This means that when y increases, x decreases, and vice versa. Use this equation: y=k/x

3 | P a g e Percentage Change Percentage Profit and Loss Simple Interest and Compound Interest A. Simple Interest Formula B. Compound Interest Formula C. Compound interest compounded MONTHLY Formula: S = P(1 + r/k)n S = final value P = principal r = interest rate (expressed as decimal eg 4% = 0.04) k = number of compounding periods

4 | P a g e Note:  if compounded monthly, number of periods = 12  if compounded quarterly, number of periods = 4 Example: If $4000 is invested at an annual rate of 6.0% compounded monthly, what will be the final value of the investment after 10 years? Since the interest is compounded monthly, there are 12 periods per year, so, k = 12. Since the investment is for 10 years, or 120 m