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 10^{n}, 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 10^{5}0.0000987 written as standard form is 9.87 x 10^{5}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 = 1cm^{2} : 0.04km^{2}
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
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
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/howto/content/coordinategeometryformulas.html
Algebraic Manipulation
x = y+z  y = xz 
x = yz  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 = y^{2}  y = +/sqrt.x 
x = sqrt.y  y = x^{2} 
x = y^{3}  y = cuberoot.x 
x = cuberoot.y 
y = x^{3 } 
ax + bx = x(a+b)
ax + bx + kay + kby = x(a+b) + ky(a+b) = (a+b)(x+ky)
(a+b)^{2} = a^{2} + 2ab + b^{2}(ab)^{2 }= a^{2}  2ab + b^{2}a^{2}  b^{2} = (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 it
Step 2: Divide the square of the number by 4
Eg. y = x^{2} + 6x  11
y = x^{2} + 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 yintercept
 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)
3. Find yintercept
 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
Geometrical terms and relationships
Parallel Lines
Perpendicular Lines
Right Angle
Acute Angles: angles less than 90^{o}
Obtuse Angles: angles between 90^{o} and 190^{o}
^{}Obtuse Angles: angles between 180^{o} and 360^{o}
^{ }
^{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 
Triangles
Triangle  Property 
Equilateral  All sides of equal length All angles are equal Each angle is 60^{o} 
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 
Rightangled  1 of the 3 angles is 90^{o} 
Quadrilaterals
Quadrilateral  Property 
Rectangle  All sides meet at 90^{o} 
Square  All sides meet at 90^{o} 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.
b. SAS property: 2 given sides and a given angle of one triangle are equal to the corresponding sides and angle 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.
d. RHS property: The hypothenuse and a given side of a rightangled triangle are equal to the hypothenuse and the corresponding side of the other rightangled 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 360^{o}  
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 180^{o}  
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/mathformulas/mensurationformulas/
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 360^{o}
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.
4. Stemandleaf diagram: a diagram that summarises while maintaining the individual data point. The stem is a column of the unique elements of data after removing the last digit. The final digits (leaves) of each column are then placed in a row next to the appropriate column and sorted in numerical order.
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.
6. Quartiles
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)