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Repeating decimals

Repeating decimals are those rational numbers

with infinitely recurrent digits after the point.

These numbers are written by overlining the recurring digit(s) (but here, I underline them because the computer keyboard doesn't allow me to put a line over a number)

for example

1/3=0.3333333333333......= 0.3

1/7 = 0.142857142857....= 0.142857

1/9= 0.11111111111......= 0.1

1/11=0.09090909090...= 0.09

Why is 0.9 (repeating) equal to 1?

Explanation number 1
let's consider 1/3
We know that 1/3 = 0.3 (repeating)
then 3/3= 1 = 1/3 x3 = 0.3 * 3= 0.9 (repeating)= 1!

Explanation number 2
Suppose you have 9 candies and you give 1 to your friends. You gave him 1/9 of you candies. And he has got 1/9 of the 9 candies:
9 x 1/9 = 1 candy!
And because 1/9 = 0.1 we have
9 x 0.1 = 0.9 = 1 candy!

Explanation number 3
1/9 + 8/9 = 9/9 = 1
but this is the same to say that
0.1 + 0.8 = 0.9 = 1!

simple? No!

What do you think is 0.9999999999.... (repeating infinitely) equal to?

Well that's equal 1!

I repeat:

0.9 (repeating) is equal to 1

and so al other 9-ending recurrent decimals

1.9 = 2

2.9 = 3

3.9 = 4

.....

567.9 = 568

and so forth.

So remember when you use those decimals! Convert them into what they really mean: a whole number, an integer!

The confusion is rooted in the fact that we don't have a proper way to represent infinity and hence repeating decimals are approximated in the way above

As a result repeating decimals loose "touch" with reality and we don't see anymore that 0.9= 1.

The way out is to use fractions, like in explanation 3 of the box above.

Click here for an overview diagramm about fractions and decimals

•Natural Numbers

•Whole Numbers