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 months, there are 120 investment periods, so, n = 120. S = P(1 + r/k)n S = 4000(1 + 0.06/12)120 S = 4000(1.005)120 S = 4000(1.819396734) S = $7277.59 Coordinate Geometry Formulas From: http://www.dummies.com/how-to/content/coordinate-geometry-formulas.html

5 | P a g e Algebraic Manipulation x = y+z y = x-z x = y-z y = x+z x = yz y = x/z ; z = x/y x = y/z y = xz ; z = y/x wx = yz w = yz/x ; x=yz/w ; y = wx/z ; z = wx/y x = y2 y = +/-sqrt.x x = sqrt.y y = x2 x = y3 y = cuberoot.x x = cuberoot.y y = x3 ax + bx = x(a+b) ax + bx + kay + kby = x(a+b) + ky(a+b) = (a+b)(x+ky) (a+b)2 = a2 + 2ab + b2 (a-b)2 = a2 - 2ab + b2 - a2 - b2 = (a + b)(a - b) Solving algebraic fractional equations Avoid these common mistakes! Solution of Quadratic Equations

6 | P a g e Completing the Square Step 1: Take the number or coefficient before x and square it Step 2: Divide the square of the number by 4 Eg. y = x2 + 6x - 11 y = x2 + 2x(6/2) + (6/2)2 - 11 - (6/2)2 y = (x + 3)2 - 20 Sketching Graphs of Quadratic Equations A. eg. y= +/-(x - h)2 + k Steps: 1. Identify shape of curve  look at sign in front of(x - h) to determine if it is "smiley face" or "sad face". 2. Find turning point  (h, -k) 3. Find y-intercept  sub x = 0 into the equation --> (0, y) 4. Line of symmetry reflect  x = h, reflect to get (2x, y) B. eg. y = +/-(x - a)(x - b) Steps: 1. Identify shape of curve  look at the formula ax2 + bx + c.  if a>1, it is positive; otherwise, it is negative 2. Find turning point  (a + b)/2, sub answer into equation --> (a,b)

7 | P a g e 3. Find y-intercept  sub x = 0 into the equation --> (0, y) 4. Line of symmetry reflect  x = a, reflect to get (2a, y) Inequalities Ways to solve equalities: 1. Add or subtract numbers from each side of the inequality eg 10 - 3 < x - 3 2. Multiply or divide numbers from each side of the inequality by a constant eg 10/3 < x/3 3. Multiply or divide by a negative number AND REVERSE THE INEQUALITY SIGNS eg. 10 < x becomes 10/-3 > x/-3 Example

8 | P a g e Geometrical terms and relationships Parallel Lines Perpendicular Lines Right Angle Acute Angles: angles less than 90o Obtuse Angles: angles between 90o and 190o

9 | P a g e Obtuse Angles: angles between 180o and 360o Polygons Polygon: a closed figure made by joining line segments, where each line segment intersects exactly 2 others Irregular polygon: all its sides and all its angles are not the same Regular Polygon: all its sides and all its angles are the same The sum of angles in a polygon with n sides, where n is 3 or more, is Name of Polygons Number of sides Polygon 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon

10 | P a g e Triangles Triangle Property Equilateral All sides of equal length All angles are equal Each angle is 60o Isoceles 2 sides are equal 2 corresponding angles are equal Scalene All sides are of unequal length Acute All 3 angles in the triangle are acute angles Obtuse 1 of the 3 angles is obtuse Right-angled 1 of the 3 angles is 90o Quadrilaterals Quadrilateral Property Rectangle All sides meet at 90o Square All sides meet at 90o All sides are of equal length Parallelogram 2 pairs of parallel lines Rhombus All sides are of equal length 2 pairs of parallel lines Trapezium Exactly 1 pair of parallel sides Similar Plane Figures Figures are similar only if  their corresponding sides are proportional  their corresponding angles are equal