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MATHEMATICS
Rational Nos. 4/9, -3/4, 5/3.....
Integers -1, -2, -3, -4, .....
Whole Nos. 0....
Natural Nos. 1, 2, 3, 4, ....
Irrational Nos. √2 , √15
Divisibility
2 if it is even
3 if the sum of digits is a multiple of 3
4 if the last two digits is a multiple of 4
5 if it ends with 0 or 5
6 if it is divisible by both 2 and 3
7 if the difference between twice the last digit and the numbers formed
by the remaining digits is a multiple of 7
8 if the last three digits is a multiple of 8
9 if the sum of digits is a multiple of 9
10 if it ends with 0
11 if the difference between sums of alternate digits is 0 or a multiple
of 11
12 if it is divisible by both 3 and 4
Prime Factors
Prime Factorization
Greatest Common Factor
Least Common Multiple
Ex. Find the Greatest Common Factor and Least Common Multiple of 12 & 18
12 = 2 x 2 x 3
18 = 2 x 3 x 3
2 x 3 = 6 GCF
12 = 2 x 2 x 3
18 = 2 x 3 x 3
2 x 2 x 3 x 3 = 36 LCM
Laws of Exponents
|
Product Rule
 |
Quotient Rule
 |
Power Rule
 |
|
Zero Exponent
 |
Negative Exponent
 |
Power of a Product
 |
Power of a Quotient
 |
Absolute Value Inequality
|a| < b , -b < a < b
|a| > b , a < -b or a > b
Radicals and Rational Exponents
√a =
a ½ 3√b = b1/3
3√c2 = c2/3
√d5 = d5/2
Properties of Radicals
Products of Radicals with the Same Index
Quotient of Radicals with the Same Index
Radical Radicand
Addition of Radicals
√50
+ √8
= √(25)(2)
+ √(9)(2)
= 5√2
+ 3√2
= 8√2
Multiplication of
Radicals
√8 x √3
= √24
= √(4)(6)
= 2√6
Simplifying Radicals
__ _____
_
√50 = √(25)(2) = 5√2
___ _____
_
√108 = √(36)(6) = 6√3
Equation of a Line
The standard form of a
line is 
The slope-intercept
form is 
The two-point form
is 
The intercept form
is 
Slope of a Line
 
The graph of the line
leans to the right if 
The graph of the line
leans to the left if 
The slope of a
horizontal line is
The slope of a vertical
line is 
Parallel Lines
The slope of two or
more parallel lines are equal.
Perpendicular Lines
Slope
of a line perpendicular to another line is the negative reciprocal of
the other.
Special Products
Square of a Binomial
Difference of Squares
a2 - b2 = (a+b)(a-b)
Sum and Difference of Cubes
a3 + b3 = (a + b)(a2 – ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)
Quadratic Formula
ax2 + bx + c = 0
Discriminant
two different rational roots
perfect
square
two different irrational roots
not a perfect
square
one rational solution
two different imaginary
solutions (not real)
Variations
à
Direct Variation y = kx
à
Inverse Variation y = k/x
à
Joint Variation z = kwz
Y
w, x, y & z are variables while k is the constant of variations
Arithmetic Sequence
where
is the  th
term
is the first term
is the common difference
is the total number of terms
Sum of an Arithmetic Series
 
Geometric Sequence
where
is the th
term
is the first term
is the common ratio
is the total number of terms
Sum of a Finite Geometric Series
Sum of an Infinite Geometric Series
 if
Pythagorean Theorem
where c is
the length of the hypotenuse
 ,
are the lengths of the two other sides of the right triangle
c
b
a
Plane Coordinate Geometry
The Distance Formula
distance
Midpoint of a Line Segment
midpoint
Conic Section Equation
Parabola y = ax2
+ bx + c or
y = a(x - h)2 + k where (h,k) is the
vertex of the parabola
Ellipse (x - h)2 + (y
- k)2
a2 b2
where (h,k) is the center of the ellipse
Hyperbola (x - h)2
- (y - k)2
a2
b2
(y - k)2
- (x - h)2
a2
b2
Circle (x – h)2 + (y - k)2
= r2
Where r is the radius
Exponential Functions
where
is the base
Logarithmic Functions
where
is the base
To change from one base
to another…
Properties of Logarithm
à
à
à
à
à
Kinds of Logarithm
Common Logarithm
à
if 
then 
Natural Logarithm
à
if 
then 
Circular Functions
Law of Sines
Law of Cosines
à
à
à
Pythagorean Identities
à
à
à
Angle Addition/Subtraction Identities
à
cos (α + β)
= cosα
cosβ
- sinα sinβ
à
cos (α - β)
= cosα
cosβ
+ sinα sinβ
à
sin (α
+ β) = sinα
cosβ
+ cosα
sinβ
à
sin (α
- β) = sinα
cosβ
- cosα
sinβ
à
tan (α + β) =
tanα + tanβ
1- tanα
tanβ
à
tan (α - β) =
tanα - tanβ
1+ tanα
tanβ
Double Angle Identities
à 
à 
à 
à 
Half Angle Identities
à 
à 
à 
à 
Product to Sum Identities
à 
à 
à 
à 
Sum to Product Identities
Sum to Product Identities
à 
à 
à 
à 
à
à
à
à
Complex Numbers
Using the
definition 
The rectangular form
is 
and the polar form is
The conjugate
is 
or
Operations on Rectangular Coordinates
Addition
Subtraction
Multiplication
Division
Operations on Conjugate Pairs
Addition
Subtraction
Multiplication
Division
No. of diagonals of a
polygon = n (n – 3)
2
Sum of interior angels
of a polygon = (n – 2) 180
where n is the number of sides
Powers of i
i1
= i
i2
= -1
i3
= -i
i4
= +1
the pattern repeats every 4th power |
