Complete Detail About Rational Number

Rational Numbers

The numbers of the form \frac { p }{ q } , where p and q are integers and q \neq 0, are known as Rational Numbers.

Remarks-:       (i) 0 is a rational number, since we can write, 0=\frac { 0 }{ 1 }.

                        (ii) Every natural number is a rational number, since we can write, 1=\frac { 1 }{ 1 }, 2=\frac { 2 }{ 1 }, 3=\frac { 3 }{ 1 }, etc.

                       (iii) Every integer is a rational number, since an integer a can be written as \frac { a }{ 1 }.

Equivalent Rational Numbers

We know that \frac { 1 }{ 2 } =\frac { 2 }{ 4 } =\frac { 3 }{ 6 } =....\frac { 15 }{ 30 } =\frac { 16 }{ 32 } =....\frac { 144 }{ 288 } =…

These are known as equivalent rational numbers.

Simplest form of a Rational Number

A rational number \frac { p }{ q } is said to be in simplest form, if p and q are integers having no common factor other than 1 and q \neq 0.

Thus, the simplest form of each of \frac { 2 }{ 4 } ,\frac { 3 }{ 6 } ,\frac { 4 }{ 8 } ,\frac { 5 }{ 10 } ,etc.,is\frac { 1 }{ 2 }.

Similarly, the simplest form of \frac { 6 }{ 9 } is\frac { 2 }{ 3 } and\quad that\quad of\frac { 95 }{ 133 } is\frac { 5 }{ 7 }.

Representation of Rational Number on real line

Draw a line XY which extends endlessly in both the directions. Take a point O\quad on it and let it represent 0 (zero).

Taking a fixed length, called unit length, marks off OA = 1 unit.

The midpoint B of OA denotes the rational number \frac { 1 }{ 2 } . Starting from O, set off equal distance each equal to OB = \frac { 1 }{ 2 } unit.

line 1

From the pont O, on its right, the points at distance equal to OB, OB 2OB, 3OB, 4OB, etc., denote respectively the rational numbers \frac { 1 }{ 2 } ,\frac { 2 }{ 2 } ,\frac { 3 }{ 2 } ,\frac { 4 }{ 2 }, etc.

Similarly, from the pont O, on its left, the points at distances equal to OB, 2OB, 3OB, 4OB, etc., enote respectively the rational numbers \frac { -1 }{ 2 } ,\frac { -2 }{ 2 } ,\frac { -3 }{ 2 } ,\frac { -4 }{ 2 }, etc.

Thus, each rational number with 2 as its denominator can be represented by some point on the number line.

Next, draw the line XY. Take a point O on is representing 0. Let OA= 1 unit. Divide OA into three equal parts with OC as the first part.

Then C represents the rational number \frac { 1 }{ 3 } .

line 2

From the point O, set off equal distance, each equal to OC = \frac { 1 }{ 3 } unit on both sides of O.

The points at distances equal to OC, 2OC, 30C, 4OC,etc., from the point O on its right denote respectively the rational numbers \frac { 1 }{ 3 } ,\frac { 2 }{ 3 } ,\frac { 3 }{ 3 } ,\frac { 4 }{ 3 } ,etc.

Similarly, the points at distances equal to OC, 2OC, 30C, 4OC,etc., from the point O on its right denote respectively the rational numbers \frac { -1 }{ 3 } ,\frac { -2 }{ 3 } ,\frac { -3 }{ 3 } ,\frac { -4 }{ 3 } ,etc.

Thus, each rational number with 3 as its denominator can be represented by some point on the number line.

Proceeding in this manner, we can represent each and every rational number by some point on the line.

Example- Represent

(i) 2\frac { 3 }{ 8 } \quad and\quad (ii)\quad -1\frac { 5 }{ 7 }

Solution:- Draw a line XY and taking a fixed length as unit length, represent integers on this line.

line 3

(i) On the right of O, take OA=1 unit. Then, OB=2 units.

Divide the 3rd unit BC into 8 equal parts.

BP represents \frac { 3 }{ 8 } of a unit. Therefore, P represents 2\frac { 3 }{ 8 } .

(ii) On the left of O, take OD=1 unit.

Divide the 2nd unit DE into 7 equal parts.

DQ represents \frac { 5 }{ 7 } of a unit. Therefore, Q represents -1\frac { 5 }{ 7 } .

Two Important Results

(i) Let x and y be two rational numbers such that x < y.

Then, \frac { 1 }{ 2 }\left( x \right +\left y \right) is a rational number lying between x and y.

(ii) let x and y be two rational numbers such that x< y.

     Suppose we want to find n rational numbers between x and y.

    Let d=\frac { \left( { y }-{ x } \right)  }{ \left( { n }+{ 1 } \right)  } 

Then, n rational numbers lying between x and y are:

      \left( { x }+{ d } \right) ,\left( { x }+2{ d } \right) ,\left( { x }+{ 3d } \right) ,...,\left( { x }+{ nd } \right) .

TERMINATING AND RECURRING DECIMALS

TERMINATING DECIMAL-:

Every fraction \frac { p }{ q } can be expressed as a decimal. If the decimal expression of \frac { p }{ q }  terminates, i.e., comes to an end, then the decimal so obtained is called a terminating decimal.

Examples-: We have: (i)\frac { 1 }{ 4 } =0.25, (ii) \frac { 5 }{ 8 } =0.625, (iii) 2\frac { 3 }{ 5 } =\frac { 13 }{ 5 } =2.6 .

Thus, each of the number \frac { 1 }{ 4 }, \frac { 5 }{ 8 } and 2\frac { 3 }{ 5 } can be expressed in the form of a terminating decimal.

An Important Observation-: A fraction \frac { p }{ q }  is a terminating decimal only, when prime factors of q are 2 and 5 only.

Example-: Each one of the fractions \frac { 1 }{ 2 } ,\frac { 3 }{ 4 } ,\frac { 7 }{ 20 } ,\frac { 13 }{ 25 } is a terminating decimal, since the denominator of each has no prime factor other than 2 and 5.

Repeating (or recurring) Decimals

A decimal in which a digit or a set of digits repeats periodically, is called a repeating or a recurring decimal.

In a recurring decimal, we place a bar over the first block of the repeating part and omit the other repeating block.

Examples-: We have:

                     (i) \frac { 2 }{ 3 } =0.666...=\bar { 0.6. } 

                    (ii) \frac { 3 }{ 11 } =0.272727...=\bar { 0.27. } 

                   (iii) \frac { 15 }{ 7 } =2.142857142857...=\bar { 2.142857. } 

                   (iv) \frac { 11 }{ 6 } =1.18333...=\bar { 1.183. }

Length of a period

Repeated number of decimal places in a rational number is called the length of its period.

Example-\frac { 15 }{ 7 } =\bar { 2.14257. }.

So, the length of its period is 6.

Special Characteristics of Rational Numbers

(i) Every rational number is expressible either as a terminating decimal or as a repeating decimal.

(ii) Every terminating decimal is a rational number.

(iii) Every repeating decimal is a rational number.

How to find rational number between two integral numbers?

Suppose we have to find 4 rational numbers between 2 and 3.

Method:

  1. Write 2 and 3 with denominator (4+1)
  2. 2= \frac { 2\times (4+1) }{ (4+1) } =\frac { 10 }{ 5 } \quad and\quad \frac { 3\times (4+1) }{ (4+1) } =\frac { 15 }{ 5 } 
  3.  So, the four required numbers are: \frac { 11 }{ 5 } ,\frac { 12 }{ 5 } ,\frac { 13 }{ 5 } ,\frac { 14 }{ 5 } 
\Rightarrow \frac { 11 }{ 5 } <\frac { 12 }{ 5 } <\frac { 13 }{ 5 } <\frac { 14 }{ 5 } 

Insertion of Two or More Rational Numbers Between any Two Rational Numbers

THEOREM-: Prove that between any two distinct rational numbers a and b, there exists another rational number.

Proof-: Let a and b be the two rational numbers, such that a < b.

Now,                      a < b

\Rightarrow      a+a,b+a

\Rightarrow      2a<a+b

\Rightarrow      a<\frac { a+b }{ 2 }

Again,               a<b

\Rightarrow      a+b<b+b

\Rightarrow      a+b<2b

\Rightarrow      \frac { a+b }{ 2 } <b

Combining (1) and (2), a<\frac { a+b }{ 2 } <b

Since a, b and 2 are non-zero rational numbers, therefore \frac { a+b }{ 2 } is a rational number.

Hence, between two distinct rational numbers a and b, there exists another rational number  \frac { a+b }{ 2 }.

Final Wordings:

So Guys, You have made it to the end of our complete detail about Rational Numbers. And hope you like it. If yes, then share it with your friends using the sharing tools at your end. If you have questions and comments ask us by using the form at the bottom, or send us a mail.

Source-: Ts aggarwal & R.L Arora References


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