Can all decimals be converted to fractions?

Question

Explain difference between terminating and repeating in terms of whether or not they both can be done and how.

Answer 1

Yes, both terminating and repeating decimals can always be expressed as a fraction.

If you have a terminating decimal, such as 0.12345 you can easily write it as 12345/100000. Just take the numbers after the decimal for the numerator, and 1 followed by X zeros, where X is the number of decimal places in your decimal).

If you have a repeating decimal, such as 0.33333333..., you can also write it as a fraction (1/3 in this case). Determining what the fraction is for repeating decimals is a little trickier. Some are obvious (like .66666666... = 2/3), whereas others are not (.09090909... = 1/11).

For a repeating decimal, you need to look at when the repeating occurs. In 0.333333... it repeats every 1 digit, in .09090909... it repeats every 2 digits. So take the number of digits when it repeats (call it n), and let's call your repeating decimal x.

Now take 10^n times your original number x, and minus your x. Let's use .090909... as an example.

10^2 * .090909... = 9.090909...
9.090909... - .090909.... = 10^2 (.090909...) - 1 (.090909...) = 99(.090909...)

But 9.090909.. - .090909... = 9

So, 99(.090909...) = 9

So, .090909... = 9/99 = 1/11

Tricky huh?

Feel free to email me if you have any more questions on repeating decimals!

~ ♥ ~

Answer 2

Every decimal is actually a fraction. 0.2 is 2 tenths, which can also be expressed as 2/10. And so on with hundredths, etc. Repeating decimals would have to be expressed using a symbol (a line over the last number? I can't remember) in the numerator, with the denominator being wherever you stop on the repeat, for example 333/1000.

I think repeating decimals only come into play when you're already starting with a fraction, right?

Answer 3

Short answer: No. Terminating decimals are real numbers with finite representation of numbers in base 10. Repeating decimals are NOT real numbers which have an infinite representation in base 10 that takes the form of some easily recognizable pattern.

Long Answer: Most mathematicians will tell you that repeating decimals are real but this is false. These repeating decimals are understood by most to represent the number defined by the limit of the partial sums.For example, 0.333... has a limit of 1/3. It is strictly less than 1/3. However, in base 10 there is no way of representing 1/3 in a finite form. The approximation 0.333 (without ellipsis) is used. It does not make any difference in the real world because our arithmetic which was designed to be finite always takes the form of approximation in such cases. Consider pi, e and other irrational numbers - these are always approximations no matter what radix (base) system is used. The conversion 'trick' you are taught in elementary school is a way of finding the upper bound of the partial sums or limit as most would say today. In the case of:

x = 0.333....
10x = 3.333...
=> 9x = 3
=> x = 1/3

This means that the sum 3/10+3/100+3/1000+... is bounded above by 1/3. The reason you are taught this method is because arithmetic on fractions is accurate because it can be performed finitely. Furthermore, even though 0.333... is strictly less than 1/3, it is a reasonable approximation just like 0.3, 0.33, 0.333 and 0.3333 would be. For more accurate results in radix arithemtic (example decimal system), we simply use more repeating digits.

Beware of articles on Wikipedia or other websites that tell you 0.999... = 1. This is false.

Answer 4

All About Decimals

Answer 5

decimals can be converted into fractions
terminating decimals stop and dont repeat themselves.
repeating decimals never end the just repeat themselves time and time again

Answer 6

It can be done, as long as it is not irrational. Pi for example, can't be converted into a fraction.

Answer 7

But those that don't have a period (a sequence of digits that repeats infinitely) can't.

Answer 8

yes they are just really big fractions