# what is .64444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444infanent as a fraction???

Well if you look at a pattern

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

how could you summarize this concisely?

The simplest way is to say that if you shift it by one place it stays the same, exact for the boundaries.

Let's do that here. Let's only look at the 4's here:

Put x = 0.04444444444444444444444

Let's shift all the fours one place to the left. This corresponds to multiplication by 10.

10x = 0.444444444444444444444444444

10x looks very similar to x, but it isn't the same because of the first 4. Yo make it the same, you must get rid of that term, which means you must subtract 0.4.

So let's subtract the 0.4 from 10x:

10x - 0.4 =

0.044444444444444444444444444

which is exactly the same as x!

THis means that:

10x - 0.4 = x ---->

9x = 0.4 ---->

x = 0.4/9 = 4/90 = 2/45

If you add 0.6 you obtain the fraction you wrote down.

29/45

All repeating decimals, regardless of the period and length, are rational numbers. This simply means that it can be expressed as the quotient of two integers. A question that frequently arises is how to convert a repeating decimal, which we know to be rational, back to a fraction.

Rational decimal fractions may be converted to fractions as follows:

Given the decimal number N = 0.078078078...

Mulitply N by 1000 or 1000N = 78.078078078...

Subtracting N = .078078078...

.................................999N = 78 making N = 78/999 = 26/333

In your case:

N = .6444444...

10N = 6.444444...

-N = .6444444...

9N = 5.8 = 5 4/5 = 29/5

Therefore, N = (29/5)/9 = 29/45 = 29/45 = .6444444...