How to convert decimal numbers to p by q form?


Question:  How to convert decimal numbers to p by q form?

We have diferent types of decimals such as terminating, non terminating repeating, etc.

Answer: Conversion of decimal numbers to p/q depends upon the type of decimal we have

We have three types of decimals:

  • Terminating decimal
  • Non terminating repeating
  • Non terminating non repeating

Explanation:

a) Terminating decimals

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. In case of terminating decimals, we remove the decimal point of the number and divide it by a number having an equal number of zeros to the number of places after the decimal point.

Example: 1.44 = 144/100

b) Non Terminating decimals

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

c) Non terminating non repeating:

Non terminating non repeating decimals are irrational numbers and hence, cannot be expressed in the form of p/q

Thus, conversion of decimal numbers to p/q depends upon the type of decimal we have