# DECIMAL REPRESENTATION OF RATIONAL NUMBERS

Let us consider the rational number p/q, where both p and q are integers and  ≠  0.

We can get the decimal representation of the rational number p/q by long division division.

When we divide p by q using long division method either the remainder becomes zero or the remainder never becomes zero and we get a repeating string of remainders.

Case 1 (Remainder = 0) :

Let us express 7/16  in decimal form. Then 7/16  =  0.4375.

In this example, we observe that the remainder becomes zero after a few steps.

Also the decimal expansion of 7/16 terminates.

Similarly, using long division method we can express the following rational numbers in decimal form as

1/2 = 0.5

7/5 = 1.5

-8/25 = -0.32

In the above examples, the decimal expansion terminates or ends after a finite number of steps.

Key Concept (Terminating Decimal) :

When the decimal expansion of p/q, q ≠ 0 terminates (i,e., comes to an end), the decimal expansion is called terminating.

Case 2 ( Remainder ≠ 0) :

Does every rational number has a terminating decimal expansion?

Before answering the question, let us express 5/11 and 7/6 in decimal form. Thus, the decimal expansion of a rational number need not terminate.

In the above examples, we observe that the remainders never become zero. Also we note that the remainders repeat after some steps. So, we have a repeating (recurring) block of digits in the quotient.

Key Concept (Non-terminating and Recurring) :

In the decimal expansion of p/q, q ≠ 0 when the remainder never becomes zero, we have a repeating (recurring) block of digits in the quotient. In this case, the decimal expansion is called non-terminating and recurring.

To simplify the notation, we place a bar over the first block of the repeating (recurring) part and omit the remaining blocks.

So, we can write the expansion of 5/11 and 7/6 as follows..

The following table shows decimal representation of the reciprocals of the first ten natural numbers. We know that the reciprocal of a number n is 1/n. Obviously, the reciprocals of natural numbers are rational numbers.

Number

Reciprocal

Type of Decimal

1

1.0

Terminating

2

0.5

Terminating

3

0.333.......

Non-terminating and recurring

4

0.25

Terminating

5

0.2

Terminating

6

0.1616.......

Non-terminating and recurring

7

0.142857142857.......

Non-terminating and recurring

8

0.125

Terminating

9

0.111.......

Non-terminating and recurring

10

0.1

Terminating

Thus we see that,

A rational number can be expressed by either a terminating or a non-terminating and recurring (repeating) decimal expansion.

The converse of this statement is also true.

That is, if the decimal expansion of a number is terminating or non-terminating and recurring (repeating), then the number is a rational number.

## Practice Questions

Express the following rational numbers as decimal numbers.

1)  3/4

2)  5/8

3)  9/16

4)  7/25

5)  47/99

6)  1/999

7)  26/45

8)  27/110

9)  2/3

10)  14/9

1)  3/4  =  0.75

2)  5/8  =  0.625

3)  9/16  =  0.5625

4)  7/25  =  0.0.28

5)  47/99  =  0.474747.........

6)  1/999  =  0.001001001........

7)  26/45  =  0.577777........

8)  27/110  =  0.2454545.........

9)  2/3  =  0.6666........

10)  14/9  =  1.5555..........

Kindly mail your feedback to v4formath@gmail.com

WORD PROBLEMS

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and Venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6

## Recent Articles 1. ### Exponential vs Linear Growth Worksheet

May 23, 22 01:59 AM

Exponential vs Linear Growth Worksheet

2. ### Linear vs Exponential Growth

May 23, 22 01:59 AM

Linear vs Exponential Growth - Concept - Examples