How to convert decimal numbers to p by q form?
We have different 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:
Non terminating repeating
Non terminating non-repeating
a) Terminating decimals
The rational numbers with finite decimal parts for which the long division terminates or ends after a definite number of steps are known as finite or terminating decimals. In the case of terminating decimals, we remove the decimal point of the number and divide it by 10n, where n is the number of digits after the decimal point.
Example: 1.44 = 144/100
b) Non-Terminating decimals
Non-terminating decimals are those which keep on continuing after the 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 a bar and equal to x. (Bar is for digits that repeating in the same pattern)
3) Determine the number of digits having a bar on their heads or the 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 is 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