Compare 0.1 and 0.1 repeating, 0.13 and 0.13 repeating, and 0.157 and 0.157 repeating when written as fractions. Make a conjecture about expressing repeating decimals like these as fractions.

0.1 and 0.11...

Ok, to compare them as a fractions we can write:

0.1 we multiply the numerator and denominator by the same number, 10^n where n depends on the number of digits after the decimal point, in this case only one, so n = 1.

0.1 = 0.1*(10/10) = 1/10.

now, for 0.1...

First, we only multiply by 10^n same as before, in this case n = 1.

0.11...*10 = 1.11...

Now, we can subtract the initial number:

1.11... - 0.111... = 1

So we have that:

0.11...*(10 - 1) = 0.11...*9 = 1

0.11... = 1/9

So the 0.1 is written as 1/10

and 0.1 repeating is written as 1/9.

So the denominator is one unt smaller in the case of the repeating decimals.

now, with 0.13 and 0.13...

Same as before, but in this case we have two digits after the decimal point, so we use n = 2 and 10^2 = 100.

0.13 = 0.13*(100/100) = 13/100.

and

0.13...*100 = 13.13

we subtract the initial number:

13.13... - 0.13... = 13

Then:

0.13...*(100 - 1) = 13

0.13...*99 = 13

0.13... = 13/99

So again, the difference between 0.13 and 0.13... is that in the fraction notation, the repeating one has a denominator that is one unit smaller.

And finally, for:

0.157 and 0.157...

We do the same as before, but here we have 3 digits after the decimal point, so n = 3.

10^3 = 1000.

then:

0.157 = 0.157*1000/1000 = 157/1000.

And for:

0.157...*1000 = 157.157...

we subtract the initial number.

157.157... - 0.157... = 157

then:

0.157...*(1000 - 1) = 157

0.157... = 157/999

And again, in the case of the repeating decimals the denominator is one unit smaller than in the other case.

a negative exponent is translated as follows:

so, we have