A Rational Number as a Decimal


Question: How do you write a rational number as a decimal?

A rational number is the one that can be written as a ratio of two integers a and b, where b is not zero.

Answer: We write a rational number as decimal; Terminating and Non-Terminating Repeating

While dividing a number a by b, if we get zero as the remainder, the decimal expansion of such a number is called terminating.

While dividing a number, if the decimal expansion continues and the remainder does not become zero, it is called non-terminating.

Explanation:

Below are the ways how we convert a rational number as decimals

There are two types of decimal representations for rational numbers

  • Terminating 
  • Non-Terminating Repeating

Conversion of Terminating rational number to decimal:

The rational number with a finite decimal part  for which the long division terminates or ends after a definite number of steps are known as finite or terminating decimals.

Example: 2/5 when converted to decimal gives 0.4 , 7/100 when converted to decimal gives 0.007

Conversion of Non Terminating rational number to decimal:

Non terminating decimals are those which keep on continuing after decimal point (i.e. they go on forever).

1) Assume the repeating decimal to be equal to some variable x
2) Write the number without using bar and equal to x
3) Determine the number of digits having bar on their heads or number of digits before the bar for mixed recurring decimal.
4) If the repeating number is the same digit after decimal such as 0.2222... then multiply by 10, if repetition of the digits are in pairs of two numbers such as 0.7878... then multiply by 100 and so on.
5) Subtract the equation formed by step 2 and step 4.
6) Then find the value of x in the simplest form.

Let's take an example of a repeating number 0.6666...

Let , x = 0.666...    -------------- (1)

Multiplying 10 on both the sides we get,

10x = 6.666.. ----------------- (2)  (This has to be chosen in such a way that on subtracting we get rid of the decimal)

Subtracting the two equations

10x - x = 6.666 - 0.666

9x = 6

x = 6/9 or 2/3

Let's take another example to understand this

Let x = 0.6565...  --------------------- (1)

Multiplying 100 on both the sides

100x = 65.6565... -------------------- (2)

Subtracting the above equations

100x - x = 65.6565 - 0.6565

99x = 65

x = 65/99

Thus, we have learnt to convert rational numbers in terms of decimal.