1 = 0.9999….

Terminating decimals

Some fractions when expressed in decimals will terminate. For examples:

Repeating decimals

Other fractions when expressed in decimals have repeating (or recurring) decimals.

For examples:

We use dot(s) on the top of the numbers to express the repeating parts.

1 = 0.9999….

There are many proofs, but some need higher mathematics.

To begin with, let us try some easier ones.

Proof 1

x = 0.9999…. (1)

\ 10 x = 9.9999…. (2)

(2) – (1), 9 x = 9

\ x = 1

Proof 2

0.9999…. = 3 ´ 0.3333….

\ 0.9999…. = 1 (3)

From repeating decimals to fractions

Example 1

0.4444…. = ?

x = 0.4444…. (4)

10x = 4.4444…. (5)

(5) – (4), 9x = 4

Example 2

0.232323…. = ?

x = 0.232323…. (6)

100x = 23.232323….. (7)

(7) – (6), 99x = 23

Example 3

1.234234234…. = ?

x = 1.234234234…. (8)

1000x = 1234.234234… (9)

(9) – (8), 999x = 1233

Other way

If you accept (3), that is, 1 = 0.9999….

you can redo Examples 1 – 3 as simply as follows:

Example 1

0.4444….

Example 2

0.232323….

Example 3

1.234234234….

Remember the pattern

and so on. You have to cancel down the fractions.

With some twists, you can write:

(1)

(2)