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Arithmetic Fundamentals
Table of Squares
Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 20 25
Square 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 400 625
Table of Cubes
Number 2 3 4 5 10` 20 100
Cube 8 27 64 125 1000 8000 1000000
Commonly used Decimal, Percent and Fractions(Less than 1)
Percent 10% 20% 25% 30% 33% 40% 50% 60% 66% 75% 80% 90% 100%
Fractions 1
10
2
10
1
4
3
10
1
3
2
5
1
2
3
5
2
3
3
4
4
5
9
10 1
Decimals 0.1 0.2 0.25 0.3 0.33 0.4 0.5 0.6 0.66 0.75 0.8 0.9 1
Commonly used Decimal, Percent and Fractions (Greater than 1)
Percent 100% 125% 133.33% 150% 200%
Fractions 1 5
4
4
3
3
2 2
Decimals 1 1.25 1.33 1.5 2.0
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Divisibility Rule
Number Rule Example
2 If last digit is 0,2,4,6, or 8 22, 30, 50, 68, 1024
3 If sum of digits is divisible by 3 123 is divisible by 3 since 1 + 2 + 3 = 6 (and 6 is divisible by 3)
4 If number created by the last 2 digits is divisible by 4
864 is divisible by 4 since 64 is divisible by 4
5 If last digit is 0 or 5 5, 10, 15, 20, 25, 30, 35, 2335
6 If divisible by 2 & 3 522 is divisible by 6 since it is divisible by 2 & 3
9 If sum of digits is divisible by 9 621 is divisible by 9 since 6 + 2 + 1 = 9 (and 9 is divisible by 9)
10 If last digit is 0 10, 20, 30, 40, 50, 5550
Logarithms
Definition: π = πππππ β π = ππ
Example: ππππππ = π β ππ = ππ
Properties
πππππ = π πππππ = π
ππππππ = π ππππππ = π
ππππ ππ = ππππππ
ππππ π Γ π = πππππ + πππππ
ππππ π
π = πππππβ πππππ
Special Exponents
Reciprocal π
ππ= πβπ
Power 0 ππ = π
Power 1 ππ = π
Operations Involving Exponents
Multiplication ππ Γ ππ = π(π+π)
Division ππ Γ· ππ = π(πβπ)
Power (ππ)π = ππΓπ
Roots πππ = πππ
= ( ππ
)π
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Progressions
Arithmetic Progression: ππ‘π term of an
Arithmetic Progression
ππ = ππ + π β π π
Sum of n terms of an arithmetic expression
ππ =π ππ + ππ
π=
π πππ + π β π π
π
The first term is π1, the common difference
is π, and the number of terms is π.
Geometric Progression: ππ‘π term of a
Geometric Progression
ππ = πππ πβπ
Sum of n terms of a geometric progression:
ππ =π(ππ+π β π)
π β π
The first term is π1, the common ratio is π, and
the number of terms is π.
Infinite Geometric Progression
Sum of all terms in an infinite geometric series = ππ
(πβπ) π€ππππ β π < π < π
Roots of a Quadratic Equation
A quadratic equation of type ππ±π + ππ± + π has two solutions, called roots. These two solutions
may or may not be distinct. The roots are given by the quadratic formula: βbΒ± b2β4ac
2a where
the Β± sign indicates that both βbβ b2β4ac
2a and
βb+ b2β4ac
2a are solutions
Common Factoring Formulas
1. ππ β ππ = π β π Γ π + π
2. ππ + πππ + ππ = (π + π)π
3. ππ β πππ + ππ = (π β π)π
4. ππ + ππππ + ππππ + ππ = (π + π)π
5. ππ β ππππ + ππππ β ππ = (π β π)π
Binomial Theorem
The coefficient of π₯(πβπ)π¦π in (π₯ + π¦)π is:
πΆππ =
π!
π! (π β π)!
Applies for any real or complex numbers x and y, and any non-negative integer n.
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Summary of counting methods
Order matters Order doesnβt matter
With Replacement
If π objects are taken from a set of π objects, in a specific order with replacement, how many different
samples are possible? ππ
N/A
Without Replacement
Permutation Rule: If π objects are
taken from a set of π objects without replacement, in a specific order, how many different samples are possible?
πππ =
π!
(π β π)!
Combination Rule: If π objects are
taken from a set of π objects without
replacement and disregarding order,
how many different samples are
possible?
πΆππ =
π !
π!(πβπ)!
Probability
The probability of an event A, π π΄ is defined as
π π΄ =ππ’ππππ ππ ππ’π‘πππππ π‘πππ‘ ππππ’π ππ ππ£πππ‘ π΄
πππ‘ππ ππ’ππππ ππ ππππππ¦ ππ’π‘πππππ
Independent Events: If A and B are independent events, then the probability of A happening and the probability of B happening is:
π π΄ πππ π΅ = π(π΄) Γ π(π΅)
Dependent Events: If A and B are dependent events, then the probability of A happening and the probability of B happening, given A, is:
π π΄ πππ π΅ = π(π΄) Γ π(π΅ | π΄)
Conditional Probability: The probability of an event occurring given that another event has already occurred e.g. what is the probability that B will occur after A
π π΅ π΄ =π(π΄ πππ π΅)
π(π΄)
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Geometry Fundamentals
The sum of angles around a point will always
be 360 degrees.
In the adjacent figure
π + π + π + π = πππΒ°
Vertical angles are equal to each other.
In the adjacent
figure π = π and
π = π
The sum of angles on a straight line is 180Λ.
In the adjacent
figure sum of
angles and b is
180 i.e. π + π =
πππΒ°
When a line intersects a pair of parallel lines,
the corresponding angles are formed are
equal to each other
In the figure above
π = π
When a line intersects a pair of parallel lines,
the alternate interior and exterior angles
formed are equal to each other
In the adjacent
figure alternate
interior angles
a=b and
alternate exterior
angles c=d
Any point on the perpendicular bisector of a
line is equidistant from both ends of the line.
In the figure, k
is the
perpendicular
bisector of
segment AB
and c=d, e=f
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Triangle Properties
1. Sum of all internal angles of a
triangle is 180 degrees i.e.
π + π + π = πππΒ°
2. Sum of any two sides of a triangle
is greater than the third i.e.
π + π > π or π + π > π or π + π > π
3. The largest interior angle is
opposite the largest side; the
smallest interior angle is opposite
the smallest side i.e. if π© > πͺ β
π > π
4. The exterior angle is supplemental
to the adjoining interior angle i.e.
π + π = πππΒ°. Since π + π + π =
πππΒ° it follows that π = π + π
5. The internal bisector of an angle
bisects the opposite side in the
ratio of the other two sides. In the
adjoining figure BD
DC=
AB
BC
Pythagoras Theorem
ππ = ππ + ππ
Commonly Used Pythagorean Triples
Height, Base Hypotenuse
3,4 or 4,3 5
6,8 or 8,6 10
5, 12 or 12,5 13
7, 24 or 24,7 25
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Special Right Triangles
The lengths of the
sides of a 45- 45-
90 triangle are in
the ratio of π:π: π
The lengths of the sides
of a 30- 60- 90
triangle are in the ratio
of π: π:π
Test of Acute and Obtuse Triangles
If ππ < ππ + ππ then it is an acute-angled triangle, i.e. the angle facing side c is an acute angle.
If ππ > ππ + ππthen it is an obtuse-angled triangle, i.e. the angle facing side c is an obtuse angle.
Polygons
Sum of interior angles of a quadrilateral is 360Λ
π + π + π + π = 360Β°
Sum of interior angles of a pentagon is 540Λ
π + π + π + π + π = 540Β°
If n is the number of sides of the polygon then, sum of interior angles = (n - 2)180Β°
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Properties of a Circle
The angle at the centre of a circle is twice any
angle at the circumference subtended by the
same arc.
In the adjacent figure
angle π = ππ
Every angle subtended at the circumference
by the diameter of a circle is a right angle
(90Λ).
In a cyclic quadrilateral, the opposite angles are
supplementary i.e. they add up to 180Λ.
In the figure
above angle
π + π =
πππΒ° and
π + π = πππΒ°
The angles at the circumference subtended
by the same arc are equal.
In the
adjacent
figure angle
π = π
A chord is a straight line joining 2 points on the
circumference of a circle.
A radius that
is
perpendicular
to a chord
bisects the
chord into
two equal
parts and
vice versa.
In the adjacent figure PW=PZ
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Area and Perimeter of Common Geometrical Figures
Rectangle
Area π¨ = π Γ π Perimeter π· = π Γ (π + π)
Square
Area π¨ = ππ
Perimeter π· = ππ
Triangle
Area π¨ =π
πΓ π Γ π
Circle
Area π¨ = π ππ Perimeter π· = ππ π
Equilateral Triangle
Area π¨ = π
πΓ ππ
Perimeter π· = ππ
Altitude π = π π
π
Trapezoid
Area π¨ =π
ππ (ππ + ππ)
Parallelogram
Area π¨ = π Γ π Perimeter π· = π Γ (π + π)
Ring
Area π¨ = π (πΉπ β ππ)
Sector
π in degrees
Area π¨ = π ππ Γ π
πππ
Length of Arc π³ = π π« Γ π
πππ
Perimeter π· = π³ + ππ
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Volume and Surface Area of 3 Dimensional Figures
Cube
Volume π = π Γ π Γ π Surface Area
π¨ = π(ππ + ππ + ππ)
Sphere
Volume π =π
ππ ππ
Surface Area π¨ = ππ ππ
Right Circular Cylinder
Volume π = π πππ‘
Surface Area π¨ = ππ π«π‘
Pyramid
Volume π =π
ππ©π
Surface Area = π¨ = π© +ππ
π
B is the area of the base
π is the slant height
Right Circular Cone
Volume π =π
ππ πππ‘
Surface Area π¨ = π π(π + π)
Frustum
Volume
π =π
ππ π‘(π«π + π«π + ππ)
Coordinate Geometry
Line: Equation of a line π = ππ + π
In the adjoining
figure, c is the
intercept of the
line on Y-axis i.e.
c=2, m is the
slope
Slope of a line π =ππβππ
ππβππ
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The slopes of parallel lines are equal
In the
adjoining
figure, the two
lines are
parallel to
each other i.e.
ππ = ππ
The slopes of perpendicular lines are opposite
reciprocals of one another.
In the
adjoining the
two lines are
perpendicular
to each other
i.e. ππ =βπ
ππ
Midpoint between two points (ππ,ππ) and
(ππ,ππ) in a x-y plane:
ππ,ππ = ππ β ππ
π,ππ β ππ
π
Distance between two points (ππ,ππ) and
(ππ,ππ) in a x-y plane:
π = (ππ β ππ)π + (ππ β ππ)π
Horizontal and Vertical line
Equation of line parallel to x-axis (line p) with
intercept on y-axis at (0, b) is π = π. Equation
of line parallel to y-axis (line q) with intercept
on x-axis at (a,0) is π = π
Equation of a circle with center (h,k) and
radius r
(π β π)π + (π β π)π = ππ
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Trigonometry Basics
Definition: Right Triangle definition for angle ΞΈ
such that 0 < ΞΈ < 90Β°
π πππ =ππππππ‘
ππ¦πππ‘πππ’π π , ππ ππ =
1
π πππ
πππ π =πππ π
ππ¦πππ‘πππ’π π, π πππ =
1
πππ π
π‘πππ =ππππππ‘
πππ π, πππ‘π =
1
π‘πππ
Pythagorean Relationships
π ππ2π + πππ 2π = 1
π‘ππ2π + 1 = π ππ2π
πππ‘2π + 1 = ππ π2π
Tan and Cot
π‘πππ =π πππ
πππ π,
πππ‘π =πππ π
π πππ
Half Angle Formulas
π ππ2π =1
2(1 β cos 2π )
πππ 2π =1
2 1 + cos 2π
π‘ππ2π =(1 β cos 2π )
(1 + cos 2π )
Even/Odd
sin βπ = βπ πππ
cos βπ = πππ π
tan βπ = βπ‘πππ
sin 2π = 2π ππππππ π, cos 2π = πππ 2π β π ππ2π
Periodic Formula: Where n is an integer
sin π + 2ππ = sinπ β sin π + 180Β° = π πππ
cos(π + 2ππ) = cosπ β cos(ΞΈ + 180Β°) = πππ π
tan π + ππ = tan π β tan ΞΈ + 90Β° = tanΞΈ
0Β° 30Β° 45Β° 60Β° 90Β°
Sin 0 1
2
1
2
3
2
1
Cos 1 3
2
1
2
1
2
0
Tan 0 1
3
1 3 β
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