## Important Notes and Formulas

### Numbers

 Type Definition Natural numbers All whole numbers except 0eg: 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 zeroeg: -1, -2, -3, 1, 2, 3... Prime number a natural number which has only 2 different factorseg: 2, 3, 5, 7, 11, 13... Composite number a natural number that has more than 2 different factorseg: 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 105
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 = 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

### 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/how-to/content/coordinate-geometry-formulas.html

### 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!

### 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

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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)
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

### Geometrical terms and relationships

Parallel Lines Perpendicular Lines

Right Angle

Acute Angles
: angles less than 90o

Obtuse Angles: angles between 90o and 190
o

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

#### Triangles

 Triangle Property Equilateral All sides of equal lengthAll angles are equalEach angle is 60o Isoceles 2 sides are equal2 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

 Quadrilateral Property Rectangle All sides meet at 90o Square All sides meet at 90oAll sides are of equal length Parallelogram 2 pairs of parallel lines Rhombus All sides are of equal length2 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 right-angled triangle are equal to the hypothenuse and the corresponding side of the other right-angled 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 Angles on a straight line add up to 180o2 angles are complementary is they add up to 90o2 angles are called supplementary if they add up to 180o 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 sum of interior angles of an n-sided polygon = (n-2) x 180o each interior angle of a regular n-sided polygon = (n-2) x 180o / n 10 Angle properties of polygons sum of exterior angles of an n-sided polygon is 360oeach exterior angle of a regular n-sided polygon = 360o / n ### 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 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:

#### Volume of Figures

 Cube  Cuboid V = l x b x hSA = 2bl + 2hb + 2hl Cylinder  Sphere  Prism V = base area x height Pyramid  Cone  ### Trigonometry

#### SINE RULE

To find an angle, can write as follows: #### 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.
4. Stem-and-leaf 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)