What Are Rational Numbers What Are Rational Numbers Our life revolves around numbers. In real life we find that no job could be ever done without dealing with numbers. Numbers are of different types. Here we will learn What is a Rational Number. If we define Rational numbers, we say Rational numbers are all the set of numbers which can take the form of p/q where p and q are the integers and q <> 0. So we observe that rational numbers includes natural numbers, Whole numbers, integers and even the fraction numbers. Some of the examples of rational numbers will be ( 4) a natural number, ( -3) as an integer, (2/5) as a fraction, 1.4 as a real number all satisfy the property of a rational number, so they all are included in the set of Rational numbers. Thus we conclude that all the properties of rational numbers are also satisfied by the set of these numbers. Let us look at some common properties of rational numbers:
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1. Closure property of rational numbers: Closure property holds true for all mathematical operators, which means that if p1/q1 , p2/ q2 are any two rational numbers then a)
their sum ( p1/q1) + ( p2/ q2) is a rational number
b)
their difference ( p1/q1) - ( p2/ q2) is a rational number
c)
their product ( p1/q1) * ( p2/ q2) is a rational number
d)
their quotient ( p1/q1) ÷ ( p2/ q2) is a rational number
2. Power of Zero: In the set of rational numbers, there exists a number zero such that if the number zero is added to an number, it does not change the value of the rational number, i.e. p1/q1 + 0 = p1/q1. On another hand if this rational number zero is multiplied to any of the rational number, then it results to zero ,i.e. ( p1/q1) * 0 = 0 Eg : ( 3/5 ) + 0 = ( 3/ 5) And ( 3/5 ) * 0 = 0 3. Multiplication by 1: 1 is called the Multiplicative identity of the rational numbers, which means that if any rational number p1/q1 is multiplied to 1, the result is unchanged, i.e. p1/q1 * 1 = p1/q1. For example: ( 3/5 ) * 1 = 3/5 4. Rational numbers are dense: It indicates that there exist infinite and uncountable sets of rational numbers. This also means that the number of rational numbers is endless. Between any given two rational numbers, there exist again uncountable rational numbers So the set of rational numbers is called a dense set.
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We must remember the following facts about rational numbers: A)
All negative rational Numbers are always less than 0.
B)
All positive rational Numbers are always greater than 0.
C)
All negative rational Numbers are always less than any positive rational numbers.
D) In order to compare two rational numbers, which are if both negative or both positive rational numbers, we need to make the denominators same and then compare them.
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