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 |
|_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.
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.
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.
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:
where j is called the dummy index or dummy variable.
? j = 1 + 2 + 3 +…..+ n
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 * ….. * (n-2) * (n-1) * n
4! = 1 * 2 * 3 * 4 = 24
5! = 5 * 4! = 120
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.
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)
a-m = 1 / am
24 = 2 * 2 * 2 * 2 = 16
2-4 = 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
log28 = 3, since 23=8
log100.001 = - 3, since 10-3
logb1 = 0, since b0
logbb = 1, since b1