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)(a^n-1 + a^n-2 b + a^n-3 b² + . . . + b^n-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

x^a . x^b = x ^{a + b} Add exponent

Example

5³ * 5⁴ = 5³⁺⁴ = 5⁷

Division

x^a / x^b = x^{a – b} Subtract exponent

Example

x⁷ / x⁵ = x^7-5 =x²

Power of power

(x^a)^b = x^ab Multiply exponent

Example

(3²)³ = 3^2*3 = 3⁶

Power of Product

(xy)^a = x^a y ^aMultiply exponent

Power of fraction

(x/y)^a = x^a / y^a

Inverse

x^-a = 1 / x^a

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

X^1/n = n√a

x^m/n = n√a^m

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

a_n = a + (n-1)d

s_n = (a₁ + a_n)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

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 (xⁿ) = 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

log₁₀ 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 <=> aⁿ = bⁿ, (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 = √(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.

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 + 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

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 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

Square Properties

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 = 180^o

Circle Properties

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

Pythagorean Theorem

a² + b² = c²

c = √a² + √b²

Circular Ring Properties

A = π (R² – r²)

Sphere Properties

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

Cube Properties

A = 6l²

V = l³

Cylinder Properties

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

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 + 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)