Elementary Math



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 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

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+zy = x-z
 x = y-zy = x+z
 x = yzy = x/z ; z = x/y
 x = y/zy = xz ; z = y/x
 wx = yzw = yz/x ; x=yz/w ; y = wx/z ;        z = wx/y
 x = y2y = +/-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 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 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


 TriangleProperty
 EquilateralAll sides of equal length
All angles are equal
Each angle is 60o
 Isoceles2 sides are equal
2 corresponding angles are equal
 Scalene    All sides are of unequal length
 AcuteAll 3 angles in the triangle are acute angles
 Obtuse1 of the 3 angles is obtuse
 Right-angled1 of the 3 angles is 90o

Quadrilaterals

 QuadrilateralProperty
 Rectangle    All sides meet at 90o
 Square All sides meet at 90o
All sides are of equal length
 Parallelogram2 pairs of parallel lines
 RhombusAll sides are of equal length
2 pairs of parallel lines
 TrapeziumExactly 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
 Square44
Yes
 Rectangle2
2
Yes
 Kite1
1
None
 Isosceles trapezium
11
None
 Parallelogram0
2
Yes
 Rhombus22
Yes
 Regular pentagon
5
5
Yes
 Regular hexagon
6
6
Yes

Angle properties

 No. PropertyExplanation
 Example
 1Angles on a straight line
  •  Angles on a straight line add up to 180o
  • 2 angles are complementary is they add up to 90o
  • 2 angles are called supplementary if they add up to 180o
 
 2Angles at a point
Angles at a point add up to 360o 
 3Vertically opposite angles
Vertically opposite angles are equal
 
 4Angles formed by parallel lines
Alternate interior angles are equal
 
 5Angles formed by parallel lines
Alternate exterior angles are equal 
 6Angles formed by parallel lines
Corresponding angles are equal
 
 7Angle properties of triangles
The sum of angles in a triangle adds up to 180o
 
 8Angle properties of triangles
The sum of 2 interior opposite angles is equal to the exterior angle
 
 9Angle 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
 
 10Angle properties of polygons
  • sum of exterior angles of an n-sided polygon is 360o
  • each exterior angle of a regular n-sided polygon = 360o / 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 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

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