swiss-algebra-help.com # 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.

 Related Pages •Natural Numbers •Whole Numbers •Integer Numbers •Irrational Numbers from repeating decimals back to the homepage

### On the shore of a vast sea ###### Donate Free 24 hour pass to lynda.com.

Stay in touch with nature. It's full of mathematics! Enjoy This Site?
Then why not use the button below, to add us to your favorite bookmarking service?

| Homepage| About Us |Study Tips |Contact |

This Website is Powered by Site Build It! - and I would never use anything else.