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Rational Numbers

Natural Numbers
The numbers 1, 2, 3… are called natural numbers or counting numbers.

Whole Numbers
All the whole numbers including 0, i.e. 0, 1, 2 ……. are whole numbers.

Integers (z)
All the whole numbers including positive as well as negative are integers.
Z = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

Rational Numbers
A number is called rational number if it can be expressed in the form of \(\frac{p}{q}\) where p and q are integers (q > 0)
Example: \(\frac{1}{2}\), \(\frac{4}{3}\), 1 etc.

Properties of Rational Numbers

Closure Property
The closure property states that when multiplication, subtraction etc.) on any two numbers in a certain type of number, the result of computation is another number in the same type of number.

 

Addition

Subtraction

Multiplication

Division

Rational Numbers

Yes

Yes

Yes

No

Integers

Yes

Yes

Yes

No

Whole Numbers

Yes

No

Yes

No

Natural Numbers

Yes

No

Yes

No

Commutative Property
The commutative property status that order does not matter. So, if you swap numbers in an operation (such as addition, multiplication, subtraction etc.) on any two numbers in a certain type of number, the result of the computation is same as the result without swap.

 

Addition

Subtraction

Multiplication

Division

Rational Numbers

Yes

No

Yes

No

Integers

Yes

No

Yes

No

Whole Numbers

Yes

No

Yes

No

Natural Numbers

Yes

No

Yes

No

Associative Property
The associative property states that it doesn't matter how we group the numbers. We can add parenthesis anywhere and we get the same answer.

 

Addition

Subtraction

Multiplication

Division

Rational Numbers

Yes

No

Yes

No

Integers

Yes

No

Yes

No

Whole Numbers

Yes

No

Yes

No

Natural Numbers

Yes

No

Yes

No

Additive Identity/ Role of Zero
a + 0 = 0 + a = a, where a is a whole number.
b + 0 = 0 + b = b, where b is an integer.
c + 0 = 0 + c = c, where c is a rational number.
Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well.

Multiplicative identity/ Role of one
We observe,
a \(\times\) 1 = a, where a is a whole number.
b \(\times\) 1 = b, where b is an integer.
c \(\times\) 1 = c, where c is a rational number.
1 is the multiplicative identity for rational numbers. It is the multiplicative identity for integers and whole numbers as well.

Additive Inverse
a + (-a) = 0, where a is a whole number.
b + (-b) = 0, where b is an integer.
(\(\frac{a}{b}\)) + (-\(\frac{a}{b}\)) = 0, where \(\frac{a}{b}\) is a rational number.
So, we say that (-\(\frac{a}{b}\)) is the additive inverse of \(\frac{a}{b}\) and \(\frac{a}{b}\) is the additive inverse of (-\(\frac{a}{b}\)).

Reciprocal or Multiplicative Inverse

  • The multiplicative inverse of any rational number \(\frac{a}{b}\) is defined as \(\frac{b}{a}\) so that (\(\frac{a}{b}\)) \(\times\) \(\frac{b}{a}\) = 1.
  • Zero does not have any reciprocal or multiplicative inverse.
  • Distributivity of multiplication over addition and subtraction

For all rational numbers a, b and c
- a(b + c) = ab + bc
- a(b – c) = ab – bc

  • How to represent rational numbers on the number line natural numbers?
    Natural Numbers

    Whole Numbers

    Integers

  • Rational numbers can also be represented on number line.
    Example: \(\frac{4}{a}\)
    In a rational number, denominator tells the number of equal parts into which first unit has been divided. The numerators tells 'how many' of these parts are considered. So, we need to divide the first unit into a parts and we need to move to 4th

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