# Basic Math Symbols and Formulas

## Basic Math Symbols

Symbols  Meaning  Definition/Example
square root square root of 9 is 3. 3 squared is 9, so a square root of 9 is 3
< less than 4 < 9 shows that 4 is less than 9
> greater than 9 > 4 shows that 9 is greater than 4
not equal one value is not equal to another a ≠ b
= equal The equality between A and B is written A = B
equivalent equivalent numbers are numbers that are written differently but represent the same amount
approximately x ≈ y means x is approximately equal to y
smaller or equal notation a ≤ b or a ≤ b means that a is less than or equal to b
bigger or equal notation a ≥ b or a ≥ b means that a is greater than or bigger to b
÷ division 20 is the dividend, five is the divisor, and four is the quotient
× multiplication 6 x 9 = 54, the numbers 6 and 9 are the factors, while the number 54 is the product.
+ addition we add 2 and 3 we get 5. We can write it like this: 2 + 3 = 5
subtraction Ex: 5 - 3 = 2
number 5 is the minuend
number 3 is the subtrahend
number 2 is the difference
angle angle measures the amount of turn
° degree Degrees are a unit of angle measure
π pi (3.14) Pi is a number - approximately 3.142
A area Area is the size of a two-dimensional surface
m slope of a line It is a number that measures its "steepness"
S.A. surface area The total area of the surface of a three-dimensional object.
L.A lateral area Lateral indicates the side of something
B area of base the area for the base of an object can be calculated
V volume Volume is a measure of how much space an object takes up
^ perpendicular Perpendicular lines are two lines that intersect
in such a way that they have a right angle
or a 90 degree angle, between them
fraction bar fraction bar separates the numerator and denominator of a fraction
right angle sign a right angle is an angle of exactly 90° (degrees)
% percent sign used to indicate a percentage, a number or ratio
± plus or minus sign indicates a choice of exactly two possible values
GCF greatest common factor greatest factor that divides two numbers
LCM least common multiple A common multiple is a number that is a multiple of two or more numbers
| divides splitting into equal parts or groups
a : b ratio how many times the a number contains the b number
xn x to the nth power nth power of x just means the product of n x's multiplied together
|| parallel lines Lines on a plane that never meet
|  | sign for absolute value absolute value. 6. = 6 means the absolute value of 6 is 6
() parentheses for grouping show where a group starts and ends
b base length The length between two points as drawn by a straight line
h height height can be defined the vertical distance from the top to the base of the object
p or P perimeter The perimeter is the length of the outline of a shape
l Length or slant height All regular pyramids also have a slant height
w width The words along, long, and length are all related
C circumference The distance around the edge of a circle
-a opposite of a Opposite number or additive inverse of any number (a)
d diameter or distance Diameter is a line segment that passes through the center
b1, b2 base lengths of a trapezoid
r rate or radius The radius of a circle is the distance
from the center of a circle to any point on the circle

# Algebra Formulas

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab + b2

a2 + b2 = (a + b)2 - 2ab

a2 + b2 = (a - b)2 + 2ab

(a + b)3 = a3 + b3 + 3ab(a + b)

(a - b)3 = a3 - b3 - 3ab(a - b)

a3 + b3 = (a + b)3 - 3ab(a + b)

a3 - b3 = (a - b)3 + 3ab(a - b)

a2 - b2 = (a + b)(a - b)

a3 - b3 = (a - b)(a2 + ab + b2)

a3 + b3= (a + b)(a2 - ab + b2)

a4 – b4 = (a2 – b2)(a2 + b2) = (a + b)(a + b)(a2 + b2)

a4 + b4 = (a2 + b2)2 – 2a2b2 = (a2 + √2ab + b2)(a2 – √2ab + b2)

a5 + b5 = (a + b)(a4 – a3b + a2b2 – ab3 + b4 )

a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)

an - bn = (a - b)(an-1 + an-2 b + an-3 b2 + . . . + bn-1n-1)

(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)

If a + b + c = 0, then the above identity reduces to a3 + b3 + c3 = 3abc

# Exponentiation Formulas

Multiplication

xa . xb = x a + b Add exponent

Example

53 * 54 = 53+4 = 57

Division

xa / xb = xa – b Subtract exponent

Example

x7 / x5 = x7-5 =x2

Power of power

(xa)b = xab Multiply exponent

Example

(32)3 = 32*3 = 36

Power of Product

(xy)a = xa y aMultiply exponent

Power of fraction

(x/y)a = xa / ya

Inverse

x-a = 1 / xa

Zero power :

x0 = 1 (x != 0)

A number raised to the 0 power is

x / x = 1
x / x = x1 / x1 = 1

x1 / x1 = x 1-1 = x0 = 1

X1/n = n√a

xm/n = n√am

# Root Formulas

Square Root :

If x2 = y then square root of y is x

can write as √y = x

So, √4 = 2, √36 = 6

Cube Root:

The cube root of a given number x is the number whose cube is x.

cube root of x by 3√x

√xy = √x * √y

√x/y = √x / √y = √x / √y x √y / √y = √xy / y.

# Arithmetic Progressions Formulas

an = a + (n-1)d

sn = (a1 + an)n / 2

a1 = first term of the arithmetic progression

a2 = last term of the arithmetic progression

n = number of patterns

# Rate Formulas

a / b = c / d => ad = bc

=> a = bc / d

=> b = ad / c

a / c = b / d ; d / b = c / a ; d / c = b / a

a ± b / b = c ± d / d ;

a + b / a – b = c + d / c – d;

a ± b / a = c ± d / c ;

a / a ± b = c / c ± d; b / a ± b = d / c ± d

# Polynomial Formulas

## Different type of polynomial

Monomial

5x2

Binomial

2x + 5

Trinomial

3x – y + 4z

Polynomial

– 2x5 + 3x2 – x + 4

## Types of Polynomial Function

Degree->

0-> Constant—–> x = 2

1-> Linear ——-> x = 2y + 1

2-> Quadratic —> x = 2y2 + x – 1

3-> Cube ———> x = 2y3 + y2 + y – 1

4-> Quartic ——> x = 2y4 + 2y2 – 1

# Logarithm Properties

## Product Rule

loga (xy) = loga x + loga y

## Quotient Rule

loga (x/y) = loga x – loga y

i.e, log x X = 1

## Logarithm of 1 to any base Zero

i.e, loga 1 = 0

loga (xn) = n(loga x)

loga x = 1 / logx a

## Change of Base Rule

loga x = logb x / logb a = log x / log a

logb N = logb a . loga N, ( a > 0, a ≠ 1, N>0 )

logb a = 1 / loga b , l (a > 0, a ≠ 1)

logb 1 = 0

loga a = 1

logb 0 = { – ∞ , b > 1, + ∞ , b < 1 }

## Decimal Logarithm

log10 N = lgN ( b = 10)

lgN = x <=> 10x = N

## Natural Logarithm

loge N = InN

InN = x <=> ex = N

# Inequalities Formulas

The sign shows that it is a greater than suppose 9 > 6 which means 9 is bigger than 6.

Ex. 3>2, 8>6

<

The sign shows that it is a lesser than suppose 6 < 9 which means 6 is lesser than 9.

Ex. 3<8, 2<8

>

The sign = shows that both are equal also a is greater than b suppose a=b.

Ex. a > b

<

The sign = shows that both are equal also a is lesser than b suppose a=b.

Ex.a < b

## Types of Inequalities :

a ≤ b => -a≥b

a ≤ b => a ± c ≤ b ± c

a ≤ b, c > 0 => ac ≤ bc, a / c ≤ b / c;

a ≤ b , c < 0 => ac ≥ bc, a / c ≥ b / c

0 < a ≤ b => 1 / a ≥ 1 / b > 0

a ≤ b < 0 => 0 > 1 / a ≥ 1 / b

a < 0 < b => 1 / a < 0 < 1 / b

a ≤ b <=> an = bn, (n,a,b > 0)

a ≤ b <=> a-n ≤ b-n

a ≤ b <=> In a ≤ In b

ex ≥ 1 + x

xx ≥ (1/e)1/e,     x ≥ 1

Xxx ≥ x,     x ≥ 1

aa + bb ≥ ab + ba > 1,     a , b > 0

# Complex plane Formulas

The point M(a,b) represent the complex number a + bi

r = OM = a + bi = √(a2+b2) : modules

φ : argument

tan φ = b / a;

cos φ = a / √(a2+b2)

sin φ = b / √(a2+b2)

## Trigonometric Form of Complex Number

a + bi = r( cos φ + i sin φ )

[r(cos φ + i sin φ )]n = rn(cos φ + i sin φ )

# Complex Numbers Formulas

Definition

i = √-1 and i2 = -1, i3 = i2 .i = -i,

i4 = i3 . i = -i . i = 1,…i4n = 1,
i4n+1 = 1, i4n+2 = -1, i4n+3 = -i

Complex number is any number of the form a + bi and where as a and b are real number.

(a + bi) + (c + di) = (a + c) + (b + d)i

Subtraction

(a + bi) – (c + di) = (a – c) + (b – d)i

Multiplication

(a + bi)(c + di) = ac + adi + bci + bdi2 = (ac – bd) + (ad + bc)i

Multiplying Conjugates

(a + bi)(a – bi) = A2 + b2

Division

a + bi / c + di = a + bi / c + di x c – di / c – di = ac + bd / c2 + d2 + (bc – ad / c2 + d2)i

# Unit Conversion Formulas

## LENGTH

km = Kilometer

m = Meter

cm = centimeter

1 km = 1000 m

1 m = 100 cm

1 cm = 0.01 m = 10-2m

1 mm = 10-3 m

1 µm = 10-6 m

1 mµ = 1 nm = 10-9 m

a angstrom (A) = 10-10 m

1 inch (in) = 2.54 cm

1 foot (ft) = 30.48 cm

1 cm = 0.3937 in

1 m = 39.37 in

1 Km = 0.6214 mi (mile)

1 yard = 0.9144 m

1.6 m = 5.24 ft

1.8 m = 5.9 ft

1 mile = 1.6.9 Km

1 nautical mile (NM) = 1.852 Km

 10-1 m dm decimeter 101 m dam decameter 10-2 m cm centimeter 102 m hm hectometer 10-3 m mm millimeter 103 m km kilometer 10-6 m µm micrometer 106 m Mm megameter 10-9 m nm nanometer 109 m Gm gigameter 10-12 m pm picometer 1012 m Tm terameter 10-15 m fm femtometer 1015 m Pm petameter 10-18 m am attometer 1018 m Em exameter 10-21 m zm zeptometer 1021 m Zm zettameter 10-24 m ym yoctometer 1024 m Ym yottameter

## VOLUME

1 liter(l) = 1000 cm3 = 1.057 quart(qt) = 61.02 in3 = 0.03532 ft3

1 m3 = 1000 l = 35.32 ft3

1 ft3 = 7.481 U.S. gal = 0.02832 m3 = 2832 l

1 U.S. gallon(gal) = 231 in3 = 3.785 l

## TIME

1 hour = 60 minutes = 3600 seconds

1 day = 24 hours

1 month ≈ 30 days

1 year ≈ 365 days ≈ 52 weeks = 12 months

## MASS

1 ton = 1000 kg

1 Kilogram (kg) = 2.2 pounds (lb) = 0.0685 slug

1 lb = 453.6 gm = 0.031 slug

1 slug = 32.174 lb = 14.59 kg

1 lb = 16 ounce (oz)

1 troy ounce = 31.1034768 gram

## SPEED

Km/h =Kilometer per hour

1 km/h = 0.2778 m/sec = 0.6214 mi/h = 0.9113 ft/sec

1 mi/h = 1.609 Km/h = 1.467 ft/sec = 0.4470 m/sec

1 knot = 1 nautical mile / hour = 1.852 km/h

## DENSITY

1 lb/ft3 = 0.01602 gm/cm3

1 slug/ft3 = 0.5154 gm/cm3

1 gm/cm3 = 103 kg/m3 = 62.43 lb/ft3

## FORCE

1 long ton = 2240 lbwt

1 metric ton = 2205 lbwt

1 newton(nt) = 105 dynes = 0.1020 kgwt = 0.5548 lbwt

1 pound weight (lbwt) = 4.448nt = 0.4536 kgwt = 32.17 poundals

1 kilogram weight (kgwt) = 2.205 lbwt = 9.807 nt

## TEMPERATURE

0o = 32o F = 273 K

20o C = 68o F

## ENERGY

1 electron volt (ev) = 1.602 x 10-19 joule

1 Kilowatt hour (kw hr) = 3.60 x 106 joules = 860.0 kcal = 3413 Btu

1Btu (British thermal unit) = 778 ft lbwt = 1055 joules = 0.293 watt hr

1 joule = 1 ny m = 107 ergs = 0.7376 ft lbwt = 0.2389 cal = 9.481 x 10-4 Btu

1 ft lbwt = 1.356 joules = 0.3239 cal = 1.285 x 10-3 Btu

## PRESSURE

1 nt/m2 = 10dynes/cm2 = 90869 x 10-6 atmosphere = 2.089 x 10-2 lbwt/ft2

1 atm = 1.013 x 105 nt/m2

= 1.013 x 106 dynes/cm2

1470 lbwt/in2

76 cm mercury

= 406.8 in water

1 lbwt/in2 = 6895 nt/m2 = 5.171 cm mercury

= 27.68 in water

# Geometry Formulas

P = Perimeter

A = Area

S = Side

d = diameter

P = 4 x s

A = S2

d = a x √2

## Rectangle Properties

P = Perimeter

A = Area

d = diameter

P = 2 x ( a + b )

A = a x b

d = √a2 + b2

## Triangle Properties

P = Perimeter

A = Area

P = a + b + c

A = b x h / 2

A = √s(s-a)(s-b)(s-c);

s = a + b + c / 2 = p / 2.

a + ß + γ = 180o

P = Perimeter

A = Area

P = 2πr

A = πr2

p = 3.14

## Parallelogram Properties

P = (a + b) x 2

P = 2a + 2b

A = bh = ab sin a

## Circular Sector Properties

L = πr = θ / 180 0

A = πr2 θ/360 0

a2 + b2 = c2

c = √a2 + √b2

A = π (R2 – r2)

S = 4πr2

V = 4πr2 / 3

## Trapezoid Properties

P = a + b + c + d

A = h x a + b / 2

## Rectangular Box Properties

A = 2ab + 2ac + 2bc

V = abc

## Right Circular Cone

A = πr2 + πrs

S = √r2 +√h2

V = 1 x πr2 h / 3

A = 6l2

V = l3

A = 2πr( r + h)

V = πr2 h

## Frustum of a Cone Properties

V = 1 x πh (r2 + rR + R2) / 3

# Trigonometric Formulas

sin2 α + cos2 α = 1

tan α . cot tan α = 1

tan α = sin α / cos α = 1 / cot tan α

cot tan α = cos α / sin α = 1 / tan α

1 + tan2 α = 1 / cos2 α = sec2 α

1 + cot tan2 α = 1 / sin2 α = cos sec2 α

## Trigonometric Table

 α 00 300 450 600 900 1200 1800 2700 3600 sin α 0 1/2 √2/2 √3/2 1 √3/2 0 -1 0 cos α 1 √3/2 √2/2 1/2 0 -1/2 -1 0 1 tan α 0 1/√3 1 √3 ∞ -√3 0 ∞ 0 cot α ∞ √3 1 1/√3 0 -1/√3 ∞ 0 ∞ sec α 1 2/√3 √2 2 ∞ -2 -1 ∞ 1 cosec α ∞ 2 √2 2/√3 1 2/√3 ∞ -1 ∞

## Co-Ratios Table

 sin cos tan cot -α -sin α +cos α -tan α -cot α 900 – α +cos α +sin α +cot α +tan α 900 + α +cos a -sin α -cot α -tan α 1800 – α +sin α -cos α -tan α -cot α 1800 + α -sin α -cos α +tan α +cot α 2700 – α -cos α -sin α +cot α +tan α 2700 + α -cos α +sin α -cot α -tan α 3600k – α -sin α +cos α -tan α -cot α 3600k – α +sin α +cos α +tan α +cot α

sin(A + B) = sinA cosB + cosA sinB

sin(A – B) = sinA cosB – cosA sinB

cos(A + B) = cosA cosB – sinA sinB

cos(A – B) = cosA cosB + sinA sinB

tan (A + B) = tanA + tanB / 1 – tanA tanB

tan(A – B) = tanA – tanB / 1 + tanA tanB

cot (A+ B) = cotA cotB – 1 / cotA + cotB

## Product of Trigonometric Functions

sin α cos ß = 1/2 [ sin (α + ß) + sin(α – ß)]

cos α cos ß = 1/2 [ sin (α + ß) + sin(α – ß)]

cos α cos ß = 1/2 [ cos (α + ß) + cos(α – ß)]

sin α sin ß = 1/2 [ cos (α – ß) + cos(α + ß)]

tan α tan ß = tan α + tan ß / cot tan α + cot tanß = – tanα – tan ß / cot tan α – cot tan ß

## Trigonometric Formula with t = tan(x/2)

sinx = 2t / 1 + t2

cos x = 1 – t2 / 1 + t2

tan x = 2t / 1 – t2

cot x = 1 – t2 / 2t

## Angle of a Plane Triangle

A, B, C are 3 angles of a triangle

sin A + sin B + sin c = 4 cos(A / 2) cos(B/2) cos(C/2)

cosA + cos B + cos C = 4 sin(A/2) sin(B/2) sin(C/2) + 1

sinA + sinB – sinC = 4sin (A/2) sin (B/2) cos (C/2)