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Rational Numbers

Properties of Rational Numbers

Properties of Rational Numbers: Every one of us knows what natural numbers are.  The number of pages in a book, the fingers on your hand or the number of students in your classroom. These numbers are rational numbers. Now let us study in detail about the properties of rational numbers.

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Introduction to Natural and Whole Numbers
Introduction to rational numbers
Properties of rational numbers
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Properties of Rational Numbers

The major properties of rational numbers are:

  • Closure Property
  • Commutativity Property
  • Associative Property
  • Distributive Property

Let us now study these properties in detail.

Closure Property

Properties of Rational Numbers

                                                                                                                     Source: Solving math problems

1) Addition of Rational Numbers

The closure property states that for any two rational numbers a and b, a + b is also a rational number.

\( \frac{1}{2} \) + \( \frac{3}{4} \)
= \( \frac{4 + 6}{8} \)
= \( \frac{10}{8} \)
Or, =  \( \frac{5}{4} \)

The result is a rational number. So we say that rational numbers are closed under addition.

2) Subtraction of Rational Numbers

The closure property states that for any two rational numbers a and b, a – b is also a rational number.

\( \frac{1}{2} \) – \( \frac{3}{4} \)
= \( \frac{4 – 6}{8} \)
= \( \frac{-2}{8} \)
Or, =  \( \frac{-1}{4} \)

The result is a rational number. So the rational numbers are closed under subtraction.

3) Multiplication of Rational Numbers

The closure property states that for any two rational numbers a and b, a × b is also a rational number.

\( \frac{1}{2} \) × \( \frac{3}{4} \)
=  \( \frac{6}{8} \)

The result is a rational number. So rational numbers are closed under multiplication.

4) Division of Rational Numbers

The closure property states that for any two rational numbers a and b, a ÷ b is also a rational number.

\( \frac{1}{2} \) ÷ \( \frac{3}{4} \)
= \( \frac{1 ×4}{2 ×3} \)
=  \( \frac{2}{3} \)

The result is a rational number. But we know that any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division. But if we exclude 0, then all the rational numbers are closed under division.

Commutative Property

1. Addition

For any two rational numbers a and b, a + b = b+ a

\( \frac{-2}{3} \)+  \( \frac{5}{7} \) and \( \frac{5}{7} \)+  \( \frac{-2}{3} \) = \( \frac{1}{21} \)
so, \( \frac{-2}{3} \)+  \( \frac{5}{7} \) = \( \frac{5}{7} \)+  \( \frac{-2}{3} \)

We see that the two rational numbers can be added in any order. So addition is commutative for rational numbers.

2. Subtraction

For any two rational numbers a and b, a – b ≠ b –  a. Given are the two rational numbers \( \frac{5}{3} \) and  \( \frac{1}{4} \),

\( \frac{5}{3} \) – \( \frac{1}{4} \) = \( \frac{20-3}{12} \)
= \( \frac{17}{12} \)
But, \( \frac{1}{4} \) – \( \frac{5}{3} \) = \( \frac{3-20}{12} \)
= \( \frac{-17}{12} \)

So subtraction is not commutative for ratioanl numbers.

3. Multiplication

For any two rational numbers a and b, a × b = b × a

\( \frac{-7}{3} \)+  \( \frac{6}{5} \) = \( \frac{6}{5} \)+  \( \frac{-7}{3} \)
= \( \frac{-42}{15} \) =  \( \frac{-42}{15} \)

We see that the two ratrional numbers can be multiplied in any order. So multiplication is commutative for ratioanl numbers.

4. Division

For any two rational numbers a and b, a ÷ b ≠ b ÷ a. Given are the two rational numbers \( \frac{5}{3} \) and  \( \frac{1}{4} \)

\( \frac{5}{3} \) ÷ \( \frac{1}{4} \) = \( \frac{5×4}{3×1} \)
= \( \frac{20}{3} \)
But, \( \frac{1}{4} \) ÷ \( \frac{5}{3} \) = \( \frac{1×3}{4×5} \)
= \( \frac{3}{20} \)

We see that the expressions on both the sides are not equal. So divsion is not commutative for ratioanal numbers.

Associative Property

Take any three rational numbers a, b and c. Firstly add a and b and then add c to the sum. (a + b) + c. Now again add b and c and then a to the sum, a + (b + c). Is (a + b) + c and a + (b + c) same? Yes and this is how associative property works. It states that you can add or multiply numbers regardless of how they are grouped.

For example, given numbers are 5, -6 and \( \frac{2}{3} \)
( 5 – 6 ) + \( \frac{2}{3} \)
= -1 + \( \frac{2}{3} \)
= – \( \frac{1}{3} \)
Now, 5 + ( -6 + \( \frac{2}{3} \) )
=  – \( \frac{1}{3} \)

In both the groups the sum is the same.

  • Addition and multiplication are associative for rational numbers.
  • Subtraction and division are not associative for rational numbers.

Distributive Property

Distributive property states that for any three numbers x, y and z we have

  x × ( y + z ) = (x × y) +( x × z)

Solved Examples for You

Question 1: …………….. are not associative for rational numbers.

  1. Addition and multiplication
  2. Subtraction and multiplication
  3. Subtraction and division
  4. Addition and division

Answer : C. When all three rational numbers are subtracted or divided in an order, the result obtained will change if the order is changed. So, subtraction and division are not associative for rational numbers.

Question 2: What is the distributive property of rational numbers?

Answer: Distributive property says that when we multiply a sum of variables by a number equals to the same result when we multiply each variable by the number and then add the products together. In other words, it is the distributive property of multiplication by making use of variables or numbers.

Question 3: What is the use of irrational numbers?

Answer: It has many uses in engineering as this field includes Signal Processing, Force Calculations, Speedometer and more. Moreover, Calculus and other mathematical domains also make use of irrational numbers, also indirectly sometimes.

Question 4: Is 0 rational or irrational?

Answer: 0 may be represented as a ratio of two integers in addition to the ratio of itself and an irrational number such that zero is not dividend in any case. We say that 0 is rational as it is an integer.

Question 5: What is the closure property of rational number?

Answer: The closure property says that for any two rational numbers x and y, x – y is also a rational number. Thus, a result is a rational number. Consequently, the rational numbers are closed under subtraction.

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