Laws of Exponents
Law
Example
x1= x
61= 6
x0= 1
70= 1
x-1= 1/x
4-1= 1/4
xmxn= xm+n
x2x3= x2+3= x5
xm/xn= xm-n
x6/x2= x6-2= x4
(xm)n= xmn
( 2)3= 23=
(xy)n= xn yn
(xy)3= x3 y3
(x/y)n= xn /yn
(x/y)2= x2/y2
x-n= 1/xn
x-3= 1/x3
x6
fractional exponents :
Laws of Logarithms
Example : y=log4(1/4). Find y
4y = 4 log4(0.25) ; since 4 log4(0.25) =
4y= 4-1
y = -1
Properties of logarithm :
1. loga(mn) = logam + logan
the log of a multiplication is the sum of the logs
2. loga(m/n) = logam - logan
the log of a division is the difference of the logs
3. loga(1/n) = loga1 logan
= -logan
since loga(1) = 0
4. loga(mr) = r ( logam)
the log of m with an exponent r is r times the log of m
Conversion Factors
1. Length
Imperial Unit
Metric Unit
1 inch
1 foot ==> 12 inch
1 yard ==> 3 foot
1 mile ==> 1760 yard
1 nautical mile
1 league ==> 3 mile
2.54 cm or 25.4 mm
(12 X 2.54) cm
= 30.5 cm
(3 X 30.5) cm
= 91.5 cm
(0.915 X 1760) m = 1610 m = 1.6 km
1.85 km
4.8 km
Metric Unit
Imperial Unit
1 metre
1.1 yard
Imperial Unit
Metric Unit
1 ounce
28.3 g
Metric Unit
Imperial Unit
1 kg
2.2 pound
2. Mass
3. Time
1 minute ==> 60 second
1 hour ==> 60 minute = (60 X60) sec.
= 3600 seconds
1 day
==> 24 hour = (24 X 60) min.
= 1440 minutes
4. Temperature
Celsius
Fahrenheit
(multiply 9/5 + 32) ==> perform calculation in the order of multiply by 9/5 then add with 32 (constant)
Fahrenheit
Celsius
( - 32 multiply 5/9) ==> perform calculation in the order of minus 32 (constant) then multiply by 5/9
Celsius to Fahrenheit
Fahrenheit to Celsius
(C 9/5) + 32 = F
(F - 32) x 5/9 = C
Celsius to Fahrenheit C 1.8 + 32 = F
Fahrenheit to Celsius (F - 32) / 1.8 = C
Celsius
Kelvin
(add 273) ==> C + 273 = Kelvin, K
Kelvin
Celsius
(minus 273) ==> K 273 = C
Fahrenheit
Kelvin
(convert to Celsius first, then + 273)
Kelvin
==> (F - 32) / 1.8 = C
==> C + 273 = Kelvin, K
Fahrenheit
(convert to Celsius first, then convert to Fahrenheit) ==> K 273 = C
(minus 273)
(C 1.8 + 32 = F)
==> C 1.8 + 32 = F
5. Area
Imperial Unit
Metric Unit
1 inch X 1 inch
1 square inch (1 in2)
(2.54 cm X 2.54 cm)
(6.45) cm2 = 6.45 cm2
1 foot X 1 foot
1 square foot (1 ft2)
(0.305 m X 0.305 m)
0.093 m2
1 mile X 1 mile
1 square mile (1 mile2)
1.6 km X 1.6km
2.56 km2
Metric Unit
Imperial Unit
1 metre X 1 metre
1 square metre (1 m2)
1.1 yard X 1.1 yard
1.21 yard2
6. Volume
Imperial Unit
Metric Unit
1 cubic inch (in3)
1 gallon
(2.54 cm X 2.54 cm X 2.54 cm)
= 16.39 cm3
4.55 litre
Metric Unit
Imperial Unit
1 cubic metre
(1.1 yard X 1.1 yard X 1.1yard)
= 1.33 yard3
= 0.001 m3
= 1 dm3
= 1000 cm3
1.76 pint
1 litre
1 litre
7. Speed
Non SI Unit
SI unit
Knot = (nautical mile/ hour)
(1.85 km/hour)
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Example: Convert 1 cubic yard into cubic feet.
Answer : There are 27 cubic feet in a cubic yard.
Simple geometrical constructions
1. Triangle
- has three sides and three angles
Name of Triangle
Equilateral Triangle
--- Right Angle Triangle --
Scalene right angled triangle Isosceles right angled triangle
Shape of Triangle
Description
a which has :
- three equal angles of 60
- and three equal sides
Name of Triangle Acute Triangle
a which has :
- one right angle
- two unequal angles
-all three sides have different
lengths (no equal sides)
a which has :
- one right angle
- two equal angles of 45
- and two equal sides
(two sides equal in length)
Right angled triangle
Obtuse triangle
a which has :
- a right angle (90)
a which has :
- an angle more than 90
Shape of Triangle
Description
a which has :
- all angles that are less
than 90
Area
Area of Triangle = X base X height (measured at right angles to the base)
2. Quadrilateral (four sides : quad means four, lateral means side)
- any shapes with 4 sides
Types of Quadrilaterals:
Name of
Quadrilateral
Rectangle
Rhombus
Square
Shape of
Quadrilateral
Characteristic of
Quadrilateral
Name of
Quadrilateral
- every angle is a
right angle (90)
- opposites side are
parallel and same
length
Parallelogram
- all sides have equal length
- opposite sides are parallel and
opposite angles are equal
- two sets of parallel sides
- NO right angles (90)
Trapezium (UK)
Trapezoid (US)
- all sides have equal length and
every angle is a right angle (90)
- opposite sides are parallel
Kite
Shape of
Quadrilateral
Characteristic of
Quadrilateral
- opposite sides are
parallel and equal in
length
- opposite angles are
equal
- one pair of opposite sides parallel
(or two parallel sides)
- a trapezoid (in US - trapezium) is a
quadrilateral with NO parallel sides
(NOTE)
NOTE: Square, Rectangle and Rhombus are all Parallelograms
- has two pairs of adjacent
sides that are equal in length
- Diagonals intersect at right
angles
(has NO parallel sides & all the
sides are NOT equal)
3. Circle
- set of all points on a plane (two dimensional) that are a fixed distance from the centre.
Radius
- distance from the centre to the edge
Diameter
- twice the length of Radius
Circumference - distance around the edge of the circle.
- representation of locus points plotted equidistant from a central point
- Diameter X 3.142
Pi ()
- The ratio of the Circumference to the Diameter of a Circle
Area of a circle multiply the square of the radius by (radius2 X 3.142)
Lines of a Circle
Chord A straight line which goes from one point on the
circumference to another without passing through the centre of a
circle
Diameter - A straight line which goes from one point on the
circumference to another THAT passes through the centre of a circle.
Polygon
- a plane (2-dimensional) shape made of straight lines (no curve) with all the straight lines connected. (closed shape)
- Interior Angles of Polygons - an angle inside a shape (polygon)
- Exterior Angle of Polygons - add up to 360
Sum of Interior Angles
= (n-2) 180
Each Interior Angle (of a Regular Polygon) = (n-2) 180/n
; where n = number of sides
Angles
Measurement of Angles using :
1. Degree
- One full rotation/ revolution (one complete turn/ full circle) = 360 degrees (360 )
2. Radian
- a measure based on the radius of the circle
- is the angle made by taking the radius and wrapping it along the circumference of a circle:
- One full rotation/ revolution (one complete turn/ full circle) = 2 radians
- 1 Radian is about 57.2958 degrees (180/ = 57.2958)
Types of Angle
Acute Angle
angle that is less than 90
Right Angle
angle that is equal to 90 (/2 radian)
Obtuse Angle
angle that is greater than 90 but less than 180
Straight Angle
angle which is 180 ()
Reflex Angle
angle that is more than 180
Complementary Angles
angles whose sum is 90
Tips : Think of "C" from Complementary stands for "Corner"
(i.e. Right Angle)
Supplementary Angles
angles whose sum is 180
e.g : 2 angles added together that make a straight line (180)
Tips : Think of "S" from Supplementary stands for "Straight" (i.e. 180 degrees is a straight line)
Appendix
Trigonometry
The three main functions of trigonometry : Sine, Cosine and Tangent
Sine, Cosine and Tangent of 30 & 60
Sine, Cosine and Tangent of 45
sin 30 = 1/ 2
= 0.5
sin 45 = 1/ 2
= 0.707
cos 30 = 3/ 2
= 0.866
tan 30 = 1/ 3
= 0.577
sin 60 = 3/ 2 cos 60 = 1/ 2
= 0.866
= 0.5
tan 60 = 3/ 1
= 1.732
sin 0
=0
cos 0 = 1
tan 0 = 0
sin 90 = 1
cos 90 = 0
tan 90 = positive infinity
Plot of the Sine Function - starts at 0
Plot of the Cosine Function - starts at 1
cos 45 = 1/ 2
= 0.707
tan 45 = 1/1
=1
Plot of the Tangent Function - it goes between negative and positive Infinity crossing through 0 (every radians, or 180)
At /2 radians, or 90 (and -/2, 3/2, etc) the function is undefined, because it could be positive Infinity or negative
Infinity.
Glossary of Terms
Prime Number number that can be divided by 1 or itself ONLY without remainder (basic building blocks of all numbers)
Irrational Number number that cannot be rationed. e.g. = 3.1415926
Oblique sloping at an angle (an angle that is not 90 degree)
References
: Pierce, Rod. (8 Nov 2014). "Math Is Fun Citation". Math Is Fun. Retrieved 7 Nov 2014 from
http://www.mathsisfun.com/citation.php
