Basic Facts and Formula

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

a² - b² = (a + b)(a - b)

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

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

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

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

If (a + b + c) = 0 , then a³ + b³ + c³ = 3abc

(xⁿ - aⁿ) is divisible by (x - a) for all values of n

(xⁿ - aⁿ) is divisible by (x  + a ) for all EVEN values of n
(xⁿ + aⁿ) is divisible by (x  + a ) for all ODD values of n

(1 + 2 + 3 + ... + n) = [n(n + 1 )]/2
(1² + 2² + 3² + ... + n²) = [n(n + 1)(2n + 1)]/6
(1³ + 2³ + 3³ + ... + n³) = [n(n + 1)/2]²

Dividend = ( Divisor * Quotient ) + Remainder

Factor and Multiple: If a number "a" divides another number "b" with a remainder of zero then:
a is a factor of b
b is a multiple of a

Highest Common factor (HCF) or Greatest Common Divisor (GCD)
or Greatest Common Measure (GCM):
The HCF of two or more numbers is the greatest number that divides each of them exactly.

Least Common Multiple (LCM)
The least number which is exactly divisible by each one of the given numbers is called their LCM.

Product of two numbers = Product of their HCF and LCM

To check the methods of finding LCM and HCF and examples on it check the page:
http://arithmophobia-elixir.blogspot.in/2014/07/hcf-and-lcm.html

Fraction ( or Common Fraction or Vulgar Fraction ): A number of the form a/b where the number a is not divisible by b is called fraction.
Here an and b both are integers.
Here a is called numerator and b is called denominator

HCF of Fractions = ( HCF of Numerators ) / ( LCM of Denominators )
LCM of Fractions = ( LCM of Numerators ) / ( HCF of Denominators )

Decimal Fractions: Fractions in which Denominators are powers of 10.

Decimal representation: A decimal representation of a non-negative real number r is an expression of the form of a series, traditionally written as a sum

where a₀ is a nonnegative integer, and a₁, a₂, … are integers satisfying 0 ≤ a_i ≤ 9, called the digits of the decimal representation. The sequence of digits specified may be finite, in which case any further digits a_i are assumed to be 0.

Recurring Decimal:  The decimal representation of a number is said to be repeating if it becomes periodic (repeating its values at regular intervals) and the infinitely-repeated portion is not zero.
For example, the decimal representation of ⅓ becomes periodic just after the decimal point, repeating the single-digit sequence "3" forever, i.e. 0.333….
The repeated portion is expressed by putting a bar or a dot at its top.

Pure Recurring Decimal: A decimal representation in which all the figures after the decimal point are repeated.

Mixed Recurring Decimal: A decimal representation in which some figures do not repeat and some of them are repeated.

Representation of Recurring Decimal:

Fraction	Ellipsis	Vinculum	Dots	Parentheses
1/9	0.111…	0.1		0.(1)
1/3	0.333…	0.3		0.(3)
2/3	0.666…	0.6		0.(6)
9/11	0.8181…	0.81		0.(81)
7/12	0.58333…	0.583		0.58(3)
1/81	0.012345679…	0.012345679		0.(012345679)
22/7	3.142857142857…	3.142857		3.(142857)

Converting a Pure Recurring Decimal into Vulgar Fraction:
Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures.

Converting a Mixed Recurring decimal into Vulgar fraction:
In the numerator take the difference between the number formed by all the digits after decimal ( taking repeated digits only once) and that formed by the digits which are not repeated.
In the denominator take the number formed by as many zeros as is the number of non-repeating digits.

Algebraic Expression ( or Expression ) :  An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y) and operators (like add,subtract,multiply, and divide.

BODMAS rule: The correct sequence in which the operations are to be executed in order to find out the value of a given expression.
Here, (1) B = Bracket, (2) O = of , (3) D = Division, (4) M = Multiplication, (5) A = Addition, (6) S= Subtraction. In the same order of 1 to 5, we simplify an expression.
note: If an expression contains bar( or vinculum) then before applying "BODMAS" rule, simplify the expression under vinculum.

Modulus of a Real number a is defined as:
| a | = a, if a > 0
| a | = -a. if a < 0

Square root: If x² = y, we say that the square root of y is x and we write √y = x.

Cube root: The cube root of a given number x is the number whose cube is x. The cube root of x is denoted by  ³√x

Prime number: A number greater than 1 is called prime if it has exactly two factors the number 1 and the number itself.

Composite number: Numbers greater than 1 which are not prime are called composite number.

Total Prime numbers upto 100 are 25 in number.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,
43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Prime numbers greater than 100:
Let p be a given prime number greater than 100.
Method to find:
k > √p , where k is a whole number greater than square root of p
Test whether p is divisible by any prime number less than k.
Yes divisible, then p is no prime.
No not divisible, p is prime.

Example: Is 191 prime number?
Here p=191
Let's find a whole number greater than √191
Now we know, 13*13=168 and 14*14=196, that means square root of 191
is greater than 13 and less than 14.
i.e 14 > √191 (So lets's take the whole number as 14)
1.e k=14
Prime numbers less than 14 are 2, 3, 5, 7, 11 and 13
191 is not divisible by any of them.
Therefore, 191 is a prime number.

Co prime numbers: Two numbers are said to be co prime if the highest common factor of
them is the number 1.
Example: 2 and 3 are co prime numbers
               8 and 11 are co prime numbers

If a number is divisible by p as well as q, where p and q are co-primes,
then the given number is divisible by pq.
But if p and q are not co-primes then the given number need not be divisible by pq,
even if it is divisible by both p and q.

Average = ( Sum of Observations) / ( Total Number of Observations )

We may be add links to this page redirecting you to the other pages as we add more examples on above topics.

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Some of the keyboard strokes used for typing:
alt 0184 = ¼
alt 0189 = ½
alt 0175 = ¯
alt 251 = √
(alt + 0179) + (alt + 251) = ³√
alt 228 = Σ