# Math

Jeremy is playing a game called “Rational Round Up” where he has to collect all the numbers in a maze that are rational and then get to the end of the maze. When he collects a number, he must prove it is rational by writing it as a quotient of two integers. Help him determine how to prove that each of the following numbers is rational.

1. 2.4 = 12/5

2. 74 = 74/1 rational

3. 17.3333333… = 52/3 rational

4. π pi cant be expressed as a fraction so irrational.

5. 61/11 rational

6. –18 = -18/1 rational

7. = 6/10 rational

8. 87.125

9. –30 = -30/1 rational

10. –8.3

11. 58.25 = 58 ¼ rational

12. 121

13. 4.5

14. = 71/10 rational

I have completed a few i just need help on the rest.

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1. 8.
87.125 = 87125/1000 rational :)

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Damon
2. same for 83/10

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Damon
3. same for 121 / 1 :)

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Damon
4. and for 45 / 10

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Damon
5. Thanks! but could you also explain the process.

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6. I just said if you could show it as a ratio of two whole numbers it is rational.
If it is a decimal fraction, that does it unless the number of digits required is infinite.
like
1.00001 = 100001 * 10^-5 = 100,001/ 100,000
now if it is pi or e or something, you can never settle on how many digits or what they are. (if they repeat like 3.13131313 ..... that can be done
but if they are unpredictable like
3.14159 .... who knows what
then they can not be expressed exactly by a ratio of whole numbers
of course an approximation like 22/7 is possible, but not exact.

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Damon
7. Now maybe you are not there yet but for something like 3.13131313 ....
that is
3 + 13*10^-2 + 13*10^-4 + 13 * 10^-6
= 3 + geometric series a + ar + a r^2 + ar^3 ...
where a = 13*10^-2 and r = 10^-2
the sum of that series is a /(1-r)
13*10^-2 (1/(1-10^-2) = 13*10^-2/ 0.99
so 3 + 13/99 = (297+ 13)/99 = 310/99 =
which believe it or not is 3.13131313 ....

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Damon
8. so if you can write it as 310/99
it is rational
but if you simply can not write it as a ratio of whole numbers, it is not rational.

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Damon

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