Elementary Math

Contents

1 Important Notes and Formulas

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^-5Multiplying 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

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/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 + b^2
-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 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)

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 90^o

Obtuse Angles: angles between 90^o and 190^oObtuse 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
Right-angled	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 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 180^o 2 angles are complementary is they add up to 90^o 2 angles are called supplementary if they add up to 180^o
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	sum of interior angles of an n-sided polygon = (n-2) x 180^o 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 360^o each exterior angle of a regular n-sided polygon = 360^o / n

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

Statistics

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. 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

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

Subpages (5): Emath formulas and notes Exam papers Number Sequence Questions Unclassified Similarity/Congruency & Ratios of length, area & volume

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