Rational Number

[sunset_park]

A rational number is defined as the quotient of two integers. When written as a decimal, the decimal will either repeat or terminate. By looking at the denominator of the rational number, there is a way to tell in advance if its decimal representation will repeat or terminate. What is the best and fastest way to tell in advance if the decimal representation of a rational number will repeat or end?

 

[Dr.Peterson]

A rational number is defined as the quotient of two integers. When written as a decimal, the decimal will either repeat or terminate. By looking at the denominator of the rational number, there is a way to tell in advance if its decimal representation will repeat or terminate. What is the best and fastest way to tell in advance if the decimal representation of a rational number will repeat or end?

You imply that you know one way; what is it? Without knowing that, we can't say what is faster, if anything.

The main method I know involves first reducing the fraction to lowest terms; without doing that, you can't be sure just by looking at the original denominator.

Then, you have to factor the denominator (which you may have already done in simplifying the fraction). What do you look for in the factors?

 

[mathdad]

You imply that you know one way; what is it? Without knowing that, we can't say what is faster, if anything.

The main method I know involves first reducing the fraction to lowest terms; without doing that, you can't be sure just by looking at the original denominator.

Then, you have to factor the denominator (which you may have already done in simplifying the fraction). What do you look for in the factors?

Dr. Peterson,

Please, see attachment.

 

[blamocur]

Dr. Peterson,

Please, see attachment.

Not sure how this answers @Dr.Peterson's question. Do you want us to learn from your attachments, or are you asking for help in understanding this stuff? What are your thoughts/approaches?

READ BEFORE POSTING

Welcome to FreeMathHelp.com! Please take the time to read the following before you make your first post. It will help you to get your math questions answered promptly and in the most helpful manner. A summary is available here, but please read the complete guidelines below at your earliest...

www.freemathhelp.com

 

[mathdad]

Not sure how this answers @Dr.Peterson's question. Do you want us to learn from your attachments, or are you asking for help in understanding this stuff? What are your thoughts/approaches?

READ BEFORE POSTING

Welcome to FreeMathHelp.com! Please take the time to read the following before you make your first post. It will help you to get your math questions answered promptly and in the most helpful manner. A summary is available here, but please read the complete guidelines below at your earliest...

www.freemathhelp.com

I am asking for help with every question posted.

 

[blamocur]

I am asking for help with every question posted.

Here is one of the items in "Read before posting" I linked in my previous post:

Show your beginning work, or ask a specific question about the exercise, or explain why you're stuck.

 

[Dr.Peterson]

Dr. Peterson,

Please, see attachment.

What you've shown is just a likely source for the previously asked problem (which perhaps implies that this is the book you're studying?).

What it adds to the discussion is not a more specific question for us, or information about what you've tried, but more suggestions for you to try:

​

So suppose I had told you this. Now your task is to do what it says, and show us what you found! It may raise further questions you can ask (that's what it means to ask for help), or it may reveal to you the answer.

(Or, of course, you could search for information telling you the answer, which is probably not hard to find. After all, that's part of what they're asking you to do.)

 

[mathdad]

What you've shown is just a likely source for the previously asked problem (which perhaps implies that this is the book you're studying?).

What it adds to the discussion is not a more specific question for us, or information about what you've tried, but more suggestions for you to try:

View attachment 38534​

So suppose I had told you this. Now your task is to do what it says, and show us what you found! It may raise further questions you can ask (that's what it means to ask for help), or it may reveal to you the answer.

(Or, of course, you could search for information telling you the answer, which is probably not hard to find. After all, that's part of what they're asking you to do.)

I posted the question to see how members here would reply to this problem. It is not a homework problem for me. It is textbook question of curiosity.

 

[mathdad]

Here is one of the items in "Read before posting" I linked in my previous post:

Show your beginning work, or ask a specific question about the exercise, or explain why you're stuck.

Say we have the fraction 4/5. By taking a quick look, how can we determine if the number will repeat or terminate?

 

[Dr.Peterson]

Say we have the fraction 4/5. By taking a quick look, how can we determine if the number will repeat or terminate?

Please -- read the problem, and do what it says. You'll learn a lot better when you are willing to think for yourself.

I posted the question to see how members here would reply to this problem. It is not a homework problem for me. It is textbook question of curiosity.

You said before that you were asking for help. That's not what you're saying now. (It doesn't matter whether it is a homework problem or not; you still learn the same way.)

But the problem, as quoted in full by you, gives all the help you need in order to learn. Just do it.

 

[mathdad]

Please -- read the problem, and do what it says. You'll learn a lot better when you are willing to think for yourself.

You said before that you were asking for help. That's not what you're saying now. (It doesn't matter whether it is a homework problem or not; you still learn the same way.)

But the problem, as quoted in full by you, gives all the help you need in order to learn. Just do it.

I found the following online that is quite helpful.

Here’s the best and fastest way to determine if the decimal representation of a rational number will repeat or terminate:
Examine the prime factorization of the denominator after the fraction is in simplest form.

• If the prime factorization of the denominator contains only the primes 2 and/or 5, the decimal will terminate.
  • For example, the denominator of 3/20 simplifies to 2 x 2 x 5. Since the prime factors are only 2 and 5, the decimal representation of 3/20 (which is 0.15) will terminate.
• If the prime factorization of the denominator contains any prime factors other than 2 or 5, the decimal will repeat.
  • For example, the denominator of 2/15 simplifies to 3 x 5. Since the prime factorization contains a 3, the decimal representation of 2/15 (which is 0.13333…) will repeat.

 

[Harry_the_cat]

I found the following online that is quite helpful.

Here’s the best and fastest way to determine if the decimal representation of a rational number will repeat or terminate:
Examine the prime factorization of the denominator after the fraction is in simplest form.

• If the prime factorization of the denominator contains only the primes 2 and/or 5, the decimal will terminate.
  • For example, the denominator of 3/20 simplifies to 2 x 2 x 5. Since the prime factors are only 2 and 5, the decimal representation of 3/20 (which is 0.15) will terminate.
• If the prime factorization of the denominator contains any prime factors other than 2 or 5, the decimal will repeat.
  • For example, the denominator of 2/15 simplifies to 3 x 5. Since the prime factorization contains a 3, the decimal representation of 2/15 (which is 0.13333…) will repeat.

Yes that is correct. It is also important to understand WHY that works!!

 

[mathdad]

Yes that is correct. It is also important to understand WHY that works!!

Ok. Cool.