Write # as a ratio of two integers

In summary, the conversation is about finding the ratio of two integers that equals to the decimal number 3.1415999999999... The first speaker shares a similar example and explains the solution using a geometric series. The second speaker attempts to use the same method but comes up with a slightly different result. The third speaker confirms that the result is still correct even though it is not exactly the same as the decimal number. The fourth speaker questions the correctness of the result, to which the fifth speaker clarifies that the result may be slightly different but it is still correct. The sixth speaker suggests another method of solving the problem.
  • #1
noboost4you
61
0
Problem: Write the number 3.1415999999999... as a ratio of two integers.

In my book, they have a similar example, but using 2.3171717... And this is how they solved that problem.

2.3171717... = 2.3 + (17/10^3) + (17/10^5) + (17/10^7) + ...

After the first term we have a geometric series with a = (17/10^3) and r = (1/10^2). Therefore:

2.3171717... = 2.3 + [(17/10^3) / (1 - (1/10^2))] = 2.3 + [(17/1000)/(99/100)] = (23/10) + (17/990) = 1147/495 == 2.3171717...

Thinking I could follow the similar steps with a different number, I thought it would work, but it really isn't.

This is what I did:

3.1415999999999... = 3.1415 + (99/10^6) + (99/10^8) + (99/10^10)

a = (99/10^6) and r = (1/10^2)

3.1415 + [(99/10^6) / (1 - (1/10^2))] = 3.1415 + [(99/1000000)/(99/100) = (31415/10000) + (1/10000) = (31416/10000) = 3.1416 which isn't 3.1415999999999...

What am I doing wrong?

Thanks
 
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  • #2
3.1416 which isn't 3.1415999999999...

Actually, it is.


P.S. any particular reason you were grouping the nines in pairs?
 
  • #3
technically, it is, but is that correct though? and no, there was no reason i paired them up.
 
  • #4
noboost4you said:
technically, it is, but is that correct though? and no, there was no reason i paired them up.

3.1416=3.141599999999... is very true. So any fractional representation of one is a representation of the other. In fact, that's how I would have solved this problem; I wouldn't have bothered with an infinite geometric series in this case.
 
  • #5
noboost4you said:
technically, it is, but is that correct though? and no, there was no reason i paired them up.


Techically it's true but is it correct? Is that what you are asking?

"True" is "true"- there is no "technically"! And if it's true, then it's correct.
 

1. What does it mean to write a number as a ratio of two integers?

Writing a number as a ratio of two integers means expressing the number as a fraction, where the top number (numerator) is divided by the bottom number (denominator). This allows us to compare the number to a whole or other numbers in a more simplified form.

2. How do you write a number as a ratio of two integers?

To write a number as a ratio of two integers, you need to first identify the numerator and denominator. The numerator is usually the number itself, and the denominator is typically 1. Then, you can simplify the fraction by dividing both the numerator and denominator by their greatest common factor. The resulting fraction is the number written as a ratio of two integers.

3. Can a number be written as a ratio of two integers if it has a decimal or fraction?

Yes, any number can be written as a ratio of two integers. For numbers with decimals, you can multiply both the numerator and denominator by 10, 100, or any power of 10 until the decimal is removed. For numbers with fractions, you can multiply both the numerator and denominator by the reciprocal of the fraction. This will give you a simplified fraction, which can then be written as a ratio of two integers.

4. What is the significance of writing a number as a ratio of two integers?

Writing a number as a ratio of two integers allows us to compare the number to other numbers in a more simplified form. It also helps us understand the relationship between the numerator and denominator, and how they work together to represent a fraction or proportion of a whole.

5. Can a number have multiple ratios of two integers?

Yes, a number can have multiple ratios of two integers. This is because there are multiple ways to express a number as a fraction or ratio. However, the ratios may not always be simplified or in their lowest form, so it is important to simplify the fraction to find the most accurate and useful ratio of two integers for that number.

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