The fractions can be classified in to four types like fractions, unlike fractions, Proper fraction & improper fraction. This differentiation is vital in various mathematical operations.
fractions having the same denominator are called as like fraction. For example, 2/4 and 3/4 are like fractions as the denominator 4 is same for both the fractions.
fractions having different denominators are called unlike fraction. For example, 3/4 and 1/7 are unlike-fractions as the denominator differs for both the fractions.
fractions having denominator greater than the numerator is called proper fraction. For example, 2/5 & 9/13 are proper fractions as the denominator is grater than numerator
fractions having numerator lesser than the denominator is called improper fraction. For example, 5/3 & 9/4 are improper fraction as the numerator is greater than denominator
Fractions are the rational numbers of the form p/q where q is a non zero number. For q = 0 the fraction would be undefined. Numbers for example 1/5, 2/7, 7/9 etc. represents the fraction and called as simple fractions.
Whereas the numbers like 5 3/4 are called as mixed fractions.
The highest common factor is a quantity obtained from the given quantities and which divides each of them without leaving a remainder.
Read Full post!The LCM is defined as that quantity which is divisible by the quantities of which it is the LCM without leaving the remainder.
Read Full post!Intervals which expands indefinitely in both the directions are known as unbounded intervals.
1. (a,infinity) is the set of all real numbers x such that a < x.
2. (–infinity, a) is the set of all real numbers x such that x < a.
3. [a, +infinity) is the set of all real numbers x such that a <= x.
4. (–infinity, a] is the set of all real numbers x such that x <= a.
Let X and Y be fixed real numbers such that X < Y on a number line. Various types of intervals as as fallows
1. Open Interval: open interval (x, y) with end points x and y as a set of all real numbers “n”, such that x < n < y. i.e., the real number n will be taking all the values between a and b. An vital point to consider in this case is the type of brackets used. Generally open intervals are denoted by ordinary brackets ( ).
2. The closed interval [x, y]: closed interval [x, y] with end points x and y as a set of all real numbers “n”, such that x <= n <= y. In this case the real number n will be taking all the values between x and y inclusive of the end points x and y. Generally closed intervals are denoted by [ ] brackets.
3. The half open interval [x, y): a half open interval [x, y) with end points x and y as a set of all real numbers “n”, such that x <= n < y. In this case the real number n will be taking all the values between x and y, inclusive of only x but not y. The half open interval in different (x, y].
If 0 < X < Y, then 0 < 1/X < 1/Y
The basic concept of inequalities is employed in-order to understand the intervals. Read Full post!
To put it simple, its an alternate way to express statements.
If it is given that a real number ‘p’ is not less than another real number ‘q’ , then either p should be equal q or p should be greater than q. The p and q now can be expressed as p=q or p > q or p >= q, such types of statements are called inequalities.
These are inequalities because p and q may not be equal in every case, if it is so then there would be an equation i.e. p = q.
In the equation the p or q could be algebraic expressions, bearing situation specific values.
According to this property any real number X if multiplied by zero would yield a zero.
For Example:
According to this property a constant quantity when present on both sides of the equation can be cancelled.
If A + P = A + Q then P = Q
If A . P = A. Q then P = Q
provided A is a non zero.
Addition – According to this property for every element A there exists another element –A such that addition of both returns zero value.
Example: A+0 = A
Example: B.1 = B
Read Full post!
For Addition – when 0 (identity element) is added to a real number it returns back the number itself
A + 0 = A
For Multiplication – when 1 (identity element) is multiplied to a real number it returns the same number
Associative property says elements can be grouped together in any manner, i.e the result of element will not change, no mater however its group. This property nullifies BODMAS rules.
For Example:
(X + Y) + Z = X + ( Y + Z )
(A .B) . C = A . ( B . C )
So, if an expression contains only the addition or multiplication sign, it can be grouped in any order. Combination of elements is not be applicable under this property
According to this property the addition and multiplication can be carried out in any order
A + B + C = B + A + C
A.B.C = C.B.A
So, the order of addition or multiplication will not affect the result in any way.
But for solving the expressions which contain more than one mathematical operator, order of solving becomes vital. Following the order, known as operational hierarchy or BODMAS, should be followed in solving the mathematical expression.
B All the brackets.
M/D Multiplication or Division.
A?S Addition or Subtraction.
Properties denote the basic characteristics of the real numbers. Without fail that must be followed in any mathematical processing.
It is very vital to know the basic properties which help in solving the expressions as well as adherence to the mathematical conventions.
Following Properties of the real numbers are explained for Addition and Multiplication
* Commutative property
* Associative property
* Distributive property
* Identity property
* Inverse property
* Cancellation property and
* Zero factor property for multiplication.
However, Real numbers does not include imaginary numbers.
Numbers can be classified in to two parts
1.Real numbers
2.Imaginary numbers
Imaginary numbers - Consider a number (-16)^1/2. The roots of (-16)^1/2 are +4i or -4i where i stands for an imaginary number.
Real numbers can be shown on number line. Number line shows the positive or negative real numbers.
All the number sets discussed in previous posts i.e., Natural numbers, Whole numbers, Integers and Rational numbers comprise the set of real numbers. So a set of natural/whole/integers numbers can be termed as a sub set of real numbers.
The numbers used to measure exact quantities such as length, area, volume, temperature; GNP, GDP, growth rate, inflation etc. are called as real numbers.
A Set of real numbers includes set of Natural numbers, Whole numbers, Integers and Rational numbers so it can be represented as "R = { N, W, I, Q }"
Even a set of integer is inadequate, because it does not include rational numbers like, 3/4, -8/9 etc.
Rational numbers are of the form x / y (p/q) where x (p) and y (q) are integers and second condition is that y (q) must be a non-zero otherwise that number would be undefined. For example, number 0/5 is a rational number but its not the same for the number 5/0 as it is an undefined number.
The set of rational numbers are denoted Q and expressed as
Q = (... -2/5, -1/4, 100/99, 15/7...)
In a set of rational numbers decimal part may be terminating or not terminating or/and repeating. Numbers whose decimals are non terminating and non repeating are included in a set of numbers called Irrational Numbers.
One of the type of Number System. natural numbers does not have a zero. This shortcoming is made good when we consider the set of whole numbers. It consists of zero as well over the natural numbers.
The set of a whole number is represented by W and is expressed as
W = {0, 1, 2,……….}
Natural numbers is first type of number system, in practice natural numbers are denoted by "N". The natural numbers are originated by adding 1 to the antecedent number starting from 1. i.e.. N = 1, 2, 3,…. to get an infinite series of the natural numbers.
It is always important to express such numbers as a set. Set is a collection of any well-defined objects. Each well defined individual object is also referred to as an element of that particular set. The concept of a set can be used and popularly used to represent infinite and finite number of elements.
For example: A Set of natural number is represented as
N = {1, 2, 3, 4, 5, 6,……} or {1, 2, 3, 4, 5}
