Number System- Lesson 3

Converting recurring number to p/q form

Non-terminating but recurring numbers are rational and hence can be expressed in the form p/q.

Now, let us see how can we find thee recurring form of .4444444……
Let x=0.44444…..
Therefore, 10x=4.4444…

Subtracting the two of them now we get

Try the same thing with 0.6666… you’ll get the p/q form as 2 / 3.

If x=.43434343…….
Then 100x= 43.43434343…….
Subtracting the two again we get
99x=43
=> x=43 / 99

Thus for a purely recurring number we can identify the procedure as
The p/q form = The recurring part written once / As many 9s as the number of digits in the recurring part

What if the number is like 0.12333333…
Let x=.1233333…….
& 100x=12.33333….
& 1000x=123.333….

subtracting, 900x=123-12
=> x=111 / 900

Cosider following examples for your practice now
1) 4.33333….
2) 0.126666…..

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