h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

0.9 recurring is an easy way of saying 0.9 followed by an infinite number of 9's. Since it includes infinity which is an undefineably large number 0.9 rec is also an undefineably large number. However since it includes an infinite number of 9's 0.9 rec is logically numerically equivalent to 1. It's one of thouse interesting ideas in maths like infinity and the number i which is the square root of -1 (since you can't square root negative numbers).

0.999999... = 1

This is a hard concept. It is slightly easier when you notice that
1 - 0.9 = 0.1
1 - 0.99 = 0.01
.
1 - 0.999... = 0.000... = 0 since you never get to the 1.

Because of this fact, you can see that 0.999.. (like all recurring decimals) is a rational number.

At school I was shown a really clever way of converting recurring decimals to numbers of the form x/y. I would write an entry on it if I thought anyone would be in the least bit interested.

Yeah, I'd be quite interested Jay!

OK - here it is:

Converting Decimals Into Fractions
http://www.h2g2.com/A353224

Enjoy

Jay

Ah! Really simple when you know how!

sorry, I never had anyone explain it to me, I was just writeing some of my thoughts. Personally I think that 1-0.9rec should be 0.0rec with a one on the end, but I'm not about to argue with the rest of the world... well may be just a little bit.

Converting a recurring decimal to fraction.
E.g: 1/3 will give 0.3333333...... where 3 is recurring number.

Method:
a) Separate the recurring number from the decimal fraction.
b) devide by '9's as many times as the length of the recurring number.
c) Reduce the fraction to its lowest terms.

Example: Consider 0.34343434.....
Step a: The recurring number is 34
Step b: 34/99 [the number 34 is of length 2 so we have added two nines]
Step c: Reducing it to lowest terms : 34/99 [it can not be reduced further].

How to Convert a mixed-recurring decimal to vulgar fraction ? A decimal with both recurring and non-recurring value is called mixed recurring decimal.
E.g: 28/25 will give 1.1199999999...... where 11 is non-recurring number and 9 is recurring number.

Method:
a) Separate the recurring number, non recurring number from the decimal fraction.
b) Round the decimal after point to the first recurring value.
c) Result of step b - non recurring number.
d) Add as many '9' as recurring number length and append 9's with as many '0' as non-recurring number length.
e) Step c / Step d
f) Add the fraction with the number before decimal point.
Example: Consider 1.11999999...

Step a: The recurring number is 11, non-recurring number is 9
Step b and c: 119-11 [rounded value of number after decimal point - non recurring value]
Step d: 900
Step e: 108/900 [c/d]
Step f: 1+108/900 [adding with number before decimal point ]
Reducing it to lowest terms : 900+108 / 900 = 1008/900 = 28/25


following the above method:

1 - 0.999999...
= 1 - 9/9
= 1 - 1
= 0