Mathematical notations and functions
1. Floor and Ceiling Functions:
If x is a real number, then it means that x lies between two integers which are called the floor and ceiling of x. i.e.
_x_ is called the floor of x. It is the greatest integer that is not greater than x.  x  is called the ceiling of x. It is the smallest integer that is not less than x.
If x is itself an integer, then _x_ =  x , otherwise _x_ + 1 =  x 
E.g.
_3.14_ = 3, _8.5_ = 9, _7_ = 7
 3.14 = 4,  8.5  = 8,  7 = 7
2. Remainder Function (Modular Arithmetic):
If k is any integer and M is a positive integer, then:
k (mod M)
gives the integer remainder when k is divided by M.
E.g.
25(mod 7) = 4 25(mod 5) = 0
3. Integer and Absolute Value Functions:
If x is a real number, then integer function INT(x) will convert x into integer and the fractional part is removed.
E.g.
INT (3.14) = 3 INT (8.5) = 8
The absolute function ABS(x) or  x  gives the absolute value of x i.e. it gives the positive value of x even if x is negative.
E.g.
ABS(15) = 15 or ABS  15 = 15 ABS(7) = 7 or ABS  7  = 7 ABS(3.33) = 3.33 or ABS  3.33  = 3.33
4. Summation Symbol (Sums):
The symbol which is used to denote summation is a Greek letter Sigma ?.
Let a1, a2, a3, ….. , an be a sequence of numbers. Then the sum a1 + a2 + a3 + ….. + an will be written as:
n ? aj j=1
where j is called the dummy index or dummy variable.
E.g.
n ? j = 1 + 2 + 3 +…..+ n j=1
5. Factorial Function:
n! denotes the product of the positive integers from 1 to n. n! is read as ‘n factorial’, i.e.
n! = 1 * 2 * 3 * ….. * (n2) * (n1) * n
E.g.
4! = 1 * 2 * 3 * 4 = 24 5! = 5 * 4! = 120
6. Permutations:
Let we have a set of n elements. A permutation of this set means the arrangement of the elements of the set in some order.
E.g.
Suppose the set contains a, b and c. The various permutations of these elements can be:
abc, acb, bac, bca, cab, cba.
If there are n elements in the set then there will be n! permutations of those elements. It means if the set has 3 elements then there will be 3! = 1 * 2 * 3 = 6 permutations of the elements.
7. Exponents and Logarithms:
Exponent means how many times a number is multiplied by itself. If m is a positive integer, then:
am = a * a * a * ….. * a (m times) and am = 1 / am E.g. 24 = 2 * 2 * 2 * 2 = 16 24 = 1 / 24 = 1 / 16
The concept of logarithms is related to exponents. If b is a positive number, then the logarithm of any positive number x to the base b is written as logbx. It represents the exponent to which b should be raised to get x i.e. y = logbx and by = x
E.g.
log_{2}8 = 3, since 2^{3}=8
log_{10}0.001 =  3, since 10^{3} = 0.001 log_{b}1 = 0, since b^{0} = 1 log_{b}b = 1, since b^{1} = b
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