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.An example:
More examples:
123 400 written as standard form is 1.234 x 1050.0000987 written as standard form is 9.87 x 10-5
Multiplying 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 ProportionThis 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
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
Note:
- if compounded monthly, number of periods = 12
- if compounded quarterly, number of periods = 4
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)120S = 4000(1.005)120S = 4000(1.819396734)
S = $7277.59
Coordinate Geometry Formulas
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
Completing the Square
Step 1: Take the number or coefficient before x and square itStep 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 + kSteps
1. Identify shape of curve
- look at sign in front of(x - h) to determine if it is "smiley face" or "sad face".
- (h, -k)
- sub x = 0 into the equation --> (0, y)
- x = h, reflect to get (2x, y)
Steps
1. Identify shape of curve
- look at the formula ax2 + bx + c.
- if a>1, it is positive; otherwise, it is negative
- (a + b)/2, sub answer into equation --> (a,b)
- sub x = 0 into the equation --> (0, y)
- x = a, reflect to get (2a, y)
Inequalities
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
Geometrical terms and relationships
Parallel Lines
Right Angle
Acute Angles: angles less than 90o
Obtuse Angles: angles between 90o and 190o
Obtuse Angles: angles between 180o and 360o
Polygons
Polygon: a closed figure made by joining line segments, where each line segment intersects exactly 2 othersIrregular polygon: all its sides and all its angles are not the same
Regular Polygon: all its sides and all its angles are the same
Name of Polygons
| Number of sides | Polygon |
| 5 | Pentagon |
| 6 | Hexagon |
| 7 | Heptagon |
| 8 | Octagon |
| 9 | Nonagon |
| 10 | Decagon |
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
Similar Solid Figures
Solids are similar if their corresponding linear dimensions are proportional.
Congruent Figures
Congruent figures are exactly the same size and shape.2 triangles are congruent if they satisfy any of the following:
a. SSS property: All 3 sides of one triangle are equal to the corresponding sides of the other triangle.
c. AAS property: 2 given angles and a given side of one triangle are equal to the corresponding angles and side of the other triangle.
Bearings
A bearing is an angle, measured clockwise from the north direction.
Symmetry
| Shape | Number of lines of symmetry | Order of rotational symmetry | Centre of point symmetry |
| Equilateral triangle | 3 | 3 | Yes |
| Isosceles triangle | 1 | 1 | None |
| Square | 4 | 4 | Yes |
| Rectangle | 2 | 2 | Yes |
| Kite | 1 | 1 | None |
| Isosceles trapezium | 1 | 1 | None |
| Parallelogram | 0 | 2 | Yes |
| Rhombus | 2 | 2 | Yes |
| Regular pentagon | 5 | 5 | Yes |
| Regular hexagon | 6 | 6 | Yes |
Angle properties
| No. | Property | Explanation | Example |
| 1 | Angles on a straight line |
|  |
| 2 | Angles at a point | Angles at a point add up to 360o |  |
| 3 | Vertically opposite angles | Vertically opposite angles are equal |  |
| 4 | Angles formed by parallel lines | Alternate interior angles are equal |  |
| 5 | Angles formed by parallel lines | Alternate exterior angles are equal |  |
| 6 | Angles formed by parallel lines | Corresponding angles are equal |  |
| 7 | Angle properties of triangles | The sum of angles in a triangle adds up to 180o |  |
| 8 | Angle properties of triangles | The sum of 2 interior opposite angles is equal to the exterior angle |  |
| 9 | Angle properties of polygons |
|  |
| 10 | Angle properties of polygons |
|  |
Angle Properties of Circles
Mensuration
All the mensuration formulas you'll ever need can by found here...http://oscience.info/math-formulas/mensuration-formulas/
But here's a quick reference for the important ones...
Area of Figures
| Triangle |  |  |
| Trapezium |  |  |
| Parallelogram |  | A=b x h |
| Circle |  |  |
| Sector |  |  |
Radian Measure
- Radian is another common unit to measure angles.
- A radian is a measure of the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.
- To convert radians to degrees and vice versa, use these formulas:
- π rad = 180º
- 1 rad = 180º/π
- 1º = π/180 rad
Volume of Figures
| Cube |  |  |
| Cuboid |  | V = l x b x h SA = 2bl + 2hb + 2hl |
| Cylinder |  |  |
| Sphere |  |  |
| Prism |  | V = base area x height |
| Pyramid |  |  |
| Cone |  |  |
Trigonometry
Pythagora's theorem
Trigonometrical Ratio
SINE RULE
To find an angle, can write as follows:
COSINE RULE
Area of Triangle
Mean
Mode
The mode is the most frequent value.Median
The median of a group of numbers is the number in the middle, when the numbers are in order of magnitude (in increasing order).If you have n numbers in a group, the median in:
Types of Chart
1. Bar chart: the heights of the bars represent the frequency. The data is discrete.
2. Pie chart: the angles formed by each part adds up to 360o
3. Histogram: it is a vertical bar graph with no gaps between the bars. The area of each bar is proportional to the frequency it represents.
5. Simple frequency distribution and frequency polygons: a plot of the cumulative frequency against the upper class boundary with the points joined by line segments.
Probability is the likelihood of an event happening
- The probability that a certain event happening is 1
- The probability that a certain event cannot happen is 0
- The probability that a certain event not happening is 1 minus he probability that it will happen
2 events are independent if the outcome of one of the events does not affect the outcome of another
2 events are dependent if the outcome of one of the events depends on the outcome of another
- If 2 events A and B are independent of each other, then the probability of both A and B occurring is found by P(A) x P(B)
- If it is impossible for both events A and B to occur, then the probability of A or B occurring is P(A) and P(B)
Set Notation
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