Most Used Math Symbols and Formulas
Basic Math Symbols and Formulas
Algebra Formulas  Complex Numbers Formulas  Exponentiation Formulas  Trigonometric Formulas  Inequalities Formulas
Unit Conversion Formulas  Complex plane Formulas  Logarithm Properties  Polynomial Formulas  Geometry Formulas
Arithmetic Progressions Formulas  Rate Formulas  Root Formulas Math Symbols
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 twodimensional 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 threedimensional 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  
x^{n}  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  
b_{1}, b_{2}  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} = a^{2} + 2ab + b^{2}
(a  b)^{2} = a^{2}  2ab + b^{2}
a^{2} + b^{2} = (a + b)^{2}  2ab
a^{2} + b^{2} = (a  b)^{2} + 2ab
(a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)
(a  b)^{3} = a^{3}  b^{3}  3ab(a  b)
a^{3} + b^{3} = (a + b)^{3}  3ab(a + b)
a^{3}  b^{3} = (a  b)^{3} + 3ab(a  b)
a^{2}  b^{2} = (a + b)(a  b)
a^{3}  b^{3} = (a  b)(a^{2} + ab + b^{2})
a^{3} + b^{3}= (a + b)(a^{2}  ab + b^{2})
a^{4} – b^{4} = (a^{2} – b^{2})(a^{2} + b^{2}) = (a + b)(a + b)(a^{2} + b^{2})
a^{4} + b^{4} = (a^{2} + b^{2})^{2} – 2a^{2}b^{2} = (a^{2} + √2ab + b^{2})(a^{2} – √2ab + b^{2})
a^{5} + b^{5} = (a + b)(a^{4} – a^{3}b + a^{2}b^{2} – ab^{3} + b^{4} )
a^{5} – b^{5} = (a – b)(a^{4} + a^{3}b + a^{2}b^{2} + ab^{3} + b^{4})
a^{n}  b^{n} = (a  b)(a^{n1} + a^{n2} b + a^{n3} b^{2} + . . . + b^{n1}n1)
(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2(ab + bc + ca)
a^{3} + b^{3} + c^{3} – 3abc = (a + b + c)(a^{2} + b^{2} + c^{2} – ab – bc – ca)
If a + b + c = 0, then the above identity reduces to a^{3} + b^{3} + c^{3} = 3abc
Exponentiation Formulas
Multiplication
x^{a} . x^{b} = x ^{a + b } Add exponent
Example
5^{3} * 5^{4} = 5^{3+4} = 5^{7}
Division
x^{a} / x^{b} = x^{a – b }Subtract exponent
Example
x^{7} / x^{5} = x^{75} =x^{2}
Power of power
(x^{a})^{b} = x^{ab} Multiply exponent
Example
(3^{2})^{3} = 3^{2*3} = 3^{6}
Power of Product
(xy)^{a} = x^{a} y ^{a}Multiply exponent
Power of fraction
(x/y)^{a} = x^{a} / y^{a}
Inverse
x^{a} = 1 / x^{a}
Zero power :
x^{0} = 1 (x != 0)
A number raised to the 0 power is
x / x = 1
x / x = x^{1} / x^{1} = 1
x^{1} / x^{1} = x ^{11} = x^{0} = 1
X^{1/n} = n√a
x^{m/n} = n√a^{m}
Root Formulas
Square Root :
If x^{2} = 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
a_{n} = a + (n1)d
s_{n} = (a_{1} + a_{n})n / 2
a_{1} = first term of the arithmetic progression
a_{2} = 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
5x^{2}
Binomial
2x + 5
Trinomial
3x – y + 4z
Polynomial
– 2x^{5} + 3x^{2} – x + 4
Types of Polynomial Function
Degree>
0> Constant—–> x = 2
1> Linear ——> x = 2y + 1
2> Quadratic —> x = 2y^{2} + x – 1
3> Cube ———> x = 2y^{3} + y^{2} + y – 1
4> Quartic ——> x = 2y^{4} + 2y^{2} – 1
Logarithm Properties
Product Rule
log_{a} (xy) = log_{a} x + log_{a} y
Quotient Rule
log_{a} (x/y) = log_{a} x – log_{a} y
Logarithm of any quantity same base is unity
i.e, log x _{X} = 1
Logarithm of 1 to any base Zero
i.e, log_{a} 1 = 0
log_{a} (x^{n}) = n(log_{a} ^{x})
log_{a} ^{x} = 1 / log_{x} a
Change of Base Rule
log_{a} ^{x} = log_{b} ^{x} / log_{b} ^{a} = log x / log a
log_{b} N = log_{b} a . log_{a} N, ( a > 0, a ≠ 1, N>0 )
log_{b} a = 1 / log_{a} b , l (a > 0, a ≠ 1)
log_{b} 1 = 0
log_{a} a = 1
log_{b} 0 = { – ∞ , b > 1, + ∞ , b < 1 }
Decimal Logarithm
log_{10} N = lgN ( b = 10)
lgN = x <=> 10^{x} = N
Natural Logarithm
log_{e} N = InN
InN = x <=> e^{x} = 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 <=> a^{n} = b^{n}, (n,a,b > 0)
a ≤ b <=> a^{n} ≤ b^{n}
a ≤ b <=> In a ≤ In b
e^{x} ≥ 1 + x
x^{x} ≥ (1/e)^{1/e}, x ≥ 1
X^{xx} ≥ x, x ≥ 1
a^{a} + b^{b} ≥ a^{b} + b^{a} > 1, a , b > 0
Complex plane Formulas
The point M(a,b) represent the complex number a + bi
r = OM = a + bi = √(a^{2}+b^{2}) : modules
φ : argument
tan φ = b / a;
cos φ = a / √(a^{2}+b^{2})
sin φ = b / √(a^{2}+b^{2})
Trigonometric Form of Complex Number
a + bi = r( cos φ + i sin φ )
[r(cos φ + i sin φ )]^{n} = r^{n}(cos φ + i sin φ )
Complex Numbers Formulas
Definition
i = √1 and i^{2} = 1, i^{3} = i^{2 }.i = i,
i^{4} = i^{3} . i = i . i = 1,…i^{4n} = 1,
i^{4n+1} = 1, i^{4n+2} = 1, i^{4n+3} = i
Complex number is any number of the form a + bi and where as a and b are real number.
Addition
(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 + bdi^{2} = (ac – bd) + (ad + bc)i
Multiplying Conjugates
(a + bi)(a – bi) = A^{2} + b^{2}
Division
a + bi / c + di = a + bi / c + di x c – di / c – di = ac + bd / c^{2} + d^{2} + (bc – ad / c^{2} + d^{2})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^{2}m
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  10^{1} m  dam  decameter 
10^{2} m  cm  centimeter  10^{2} m  hm  hectometer 
10^{3} m  mm  millimeter  10^{3} m  km  kilometer 
10^{6} m  µm  micrometer  10^{6} m  Mm  megameter 
10^{9} m  nm  nanometer  10^{9} m  Gm  gigameter 
10^{12} m  pm  picometer  10^{12} m  Tm  terameter 
10^{15} m  fm  femtometer  10^{15} m  Pm  petameter 
10^{18} m  am  attometer  10^{18} m  Em  exameter 
10^{21} m  zm  zeptometer  102^{1} m  Zm  zettameter 
10^{24} m  ym  yoctometer  10^{24} m  Ym  yottameter 
VOLUME
1 liter(l) = 1000 cm^{3} = 1.057 quart(qt) = 61.02 in^{3} = 0.03532 ft^{3}
1 m^{3} = 1000 l = 35.32 ft^{3 }
1 ft^{3} = 7.481 U.S. gal = 0.02832 m^{3} = 2832 l
1 U.S. gallon(gal) = 231 in^{3} = 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/ft^{3} = 0.01602 gm/cm^{3}
1 slug/ft^{3} = 0.5154 gm/cm^{3}
1 gm/cm^{3} = 10^{3} kg/m^{3} = 62.43 lb/ft^{3}
FORCE
1 long ton = 2240 lbwt
1 metric ton = 2205 lbwt
1 newton(nt) = 10^{5} 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
0^{o} = 32^{o} F = 273 K
20^{o} C = 68^{o} F
ENERGY
1 electron volt (ev) = 1.602 x 10^{19} joule
1 Kilowatt hour (kw hr) = 3.60 x 10^{6} joules = 860.0 kcal = 3413 Btu
1Btu (British thermal unit) = 778 ft lbwt = 1055 joules = 0.293 watt hr
1 joule = 1 ny m = 10^{7} 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/m^{2} = 10dynes/cm^{2} = 90869 x 10^{6} atmosphere = 2.089 x 10^{2} lbwt/ft^{2}
1 atm = 1.013 x 10^{5} nt/m^{2}
= 1.013 x 10^{6} dynes/cm^{2}
1470 lbwt/in^{2}
76 cm mercury
= 406.8 in water
1 lbwt/in^{2} = 6895 nt/m^{2} = 5.171 cm mercury
= 27.68 in water
Geometry Formulas
Square Properties
P = Perimeter
A = Area
S = Side
d = diameter
P = 4 x s
A = S^{2}
d = a x √2
Rectangle Properties
P = Perimeter
A = Area
d = diameter
P = 2 x ( a + b )
A = a x b
d = √a^{2} + b^{2}
Triangle Properties
P = Perimeter
A = Area
P = a + b + c
A = b x h / 2
A = √s(sa)(sb)(sc);
s = a + b + c / 2 = p / 2.
a + ß + γ = 180^{o}
Circle Properties
P = Perimeter
A = Area
P = 2πr
A = πr^{2}
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 = πr^{2} θ/360 ^{0}
Pythagorean Theorem
a^{2} + b^{2} = c^{2}
c = √a^{2} + √b^{2}
Circular Ring Properties
A = π (R^{2} – r^{2})
Sphere Properties
S = 4πr^{2}
V = 4πr^{2} / 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 = πr^{2} + πrs
S = √r^{2} +√h^{2}
V = 1 x πr^{2} h / 3
Cube Properties
A = 6l^{2}
V = l^{3}
Cylinder Properties
A = 2πr( r + h)
V = πr^{2} h
Frustum of a Cone Properties
V = 1 x πh (r^{2} + rR + R^{2}) / 3
Trigonometric Formulas
sin^{2} α + cos^{2} 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 + tan^{2} α = 1 / cos^{2} α = sec^{2} α
1 + cot tan^{2} α = 1 / sin^{2} α = cos sec^{2} α
Trigonometric Table
α  0^{0}  30^{0}  45^{0}  60^{0}  90^{0}  120^{0}  180^{0}  270^{0}  360^{0} 
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  ∞ 
CoRatios Table
sin  cos  tan  cot  
α  sin α  +cos α  tan α  cot α 
90^{0} – α  +cos α  +sin α  +cot α  +tan α 
90^{0} + α  +cos a  sin α  cot α  tan α 
180^{0} – α  +sin α  cos α  tan α  cot α 
180^{0} + α  sin α  cos α  +tan α  +cot α 
270^{0} – α  cos α  sin α  +cot α  +tan α 
270^{0} + α  cos α  +sin α  cot α  tan α 
360^{0}k – α  sin α  +cos α  tan α  cot α 
360^{0}k – α  +sin α  +cos α  +tan α  +cot α 
Trigonometry Addition Formulas
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 + t^{2}
cos x = 1 – t^{2} / 1 + t^{2}
tan x = 2t / 1 – t^{2}
cot x = 1 – t^{2} / 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)