division over 1/7

Hannibal lecter · 2025-12-07 22:55:00

hi, 1/7 is a ecurring decimal mean it's move forver infinitiy right?
if I want to represt 1.5 as a length like 1.5 meter or 2 meter or 0.5 meter
it's easy

but how to represent the 1.7 as meter let say there is a stick its length is 1.7 meter! is that possible?

and can I divive the recuring decimal over other numbers? like 1/7 divided by 3

can exmplain more pls about how to treat these ecurring decimal numbers or is there textbooks explain these to me

Last edited by Hannibal lecter (2025-12-07 23:31:21)

Jai Ganesh · 2025-12-07 23:32:42

Hi,

1/7 is a recurring decimal.

.

See the links.

Converting fractions to decimals.

Recurring Decimals.

Bob · 2025-12-08 06:44:31

Every recurring decimal can be represented by a fraction. I'll show how by doing an example:

Firstly you need to separate any decimal figures that are not part of the recurring part. Those are easily dealt with separately.

Then you can use a 'sum to infinity' trick on the recurring part.

Finally combine the two parts to make the final fraction.

eg. 0.741212121212......

Split into 0.74 and 0.001212121212.....

0.74 is 74/100 so that's easily sorted.

let a/b - 0.00121212121212.......... (1)

As there are 2 recurring digits multiply this by 100. (n recurring digits multiply by 10^n)

100a/b = 0.1212121212121212..... (2)

Subtract (1) from (2). This removes the recurring bit completely.

99a/b = 0.12 Convert this to an integer only equation.

9900a/b = 12

This means that a/b = 12/9900

Now recombine the two fractions:

74/100 + 12/9900 = ( 7326 + 12 )/9900 = 7338/9900

Bob

Phrzby Phil · 2025-12-09 13:44:15

I never thought about this problem Bob - you learn something every day if you're lucky.

Thanks.

Bob · 2025-12-10 00:26:52

hi Phrzby Phil

In that case, there's more.

A fraction either terminates or it recurs. A decimal that terminates can be shown to be a fraction, and the above shows that a recurring one can too.

So the set of all fractions (the rationals) is also the set of all terminating or recurring decimals.

So what else could there be?

Well it's fairly easy to construct a decimal that neither terminates nor recurs.

eg. 0.10010001000010000010000001..................... I'm adding an extra zero in each group of zeros before another 1.

It's possible to show that pi neither terminates nor recurs. I was shown the proof at University but I didn't really follow it then and I don't know how now.

e and the golden ratio are also such numbers. They are called the irrationals. Between every two rationals you can construct another; and also an irrational.

Between every two irrationals you can construct another and also a rational. So there are an infinite number of rational and irrationals.

But Cantor showed that the infinity of irrationals is bigger than the infinity of rationals. Whoops; that's a bit weird isn't it. But I do know his proof if you're interested.

Bob

Phrzby Phil · 2025-12-10 03:35:25

I know the Cantor work very well, having taken two graduate level math courses while getting my MS in Computer Science at Wisconsin-Madison in 1971: Foundation of Mathematics and Set Theory.

However, my first encounter was in George Gamow's charming 1947 book "One Two Three ... Infinity."

I have Paul Cohen's 1963 book proving results about the Continuum Hypothesis, but it's too complicated.

Math Is Fun Forum

#1 2025-12-07 22:55:00

division over 1/7

#2 2025-12-07 23:32:42

Re: division over 1/7

#3 2025-12-08 06:44:31

Re: division over 1/7

#4 2025-12-09 13:44:15

Re: division over 1/7

#5 2025-12-10 00:26:52

Re: division over 1/7

#6 2025-12-10 03:35:25

Re: division over 1/7

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