determining an irrational/rational number

[preston123]

Is 24.33315633156 a rational or irrational number? Thanks for your help.

I think it is. the repeating decimal is 33156.

I have used higher math to investigate further, but this is 6-7th grade math.

24.33315633156

10^6 n =

24333156.33156000000 10n = 243.3315633156 subract both sides 24332912.99999670000 / 999990 = 24.33315633156 By multiplying by the divisor and dividend by 10^7 , you would have a fraction for the number.

So are we able to say this is a rational nubmer by looking at the

24.33315633156, and saying this is a repeating decimal?

 

[srmichael]

Is 24.33315633156 a rational or irrational number? Thanks for your help.

I think it is. the repeating decimal is 33156.

I have used higher math to investigate further, but this is 6-7th grade math.

24.33315633156

10^6 n =

24333156.33156000000 10n = 243.3315633156 subract both sides 24332912.99999670000 / 999990 = 24.33315633156 By multiplying by the divisor and dividend by 10^7 , you would have a fraction for the number.

So are we able to say this is a rational nubmer by looking at the

24.33315633156, and saying this is a repeating decimal?

If this is a non-terminating number with a repeating pattern of numbers, then, yes it is rational. If the non-terminating list of numbers has no pattern will it be irrational i.e. pi, e, √3, etc.

 

[preston123]

This is the response from the teacher. Number 21 should be marked irrational, as the patter does not repeat. 24.33315633156

 

[srmichael]

This is the response from the teacher. Number 21 should be marked irrational, as the patter does not repeat. 24.33315633156

HA! I misread it. I thought it said 24.333156333156. However, is this a terminating number? In other words is 24.33315633156 the number or is it really 24.33315633156... where the ... means the numbers continue on infinitely? If it is a terminating number i.e. 24.33315633156, then it is rational.

 

[lookagain]

> > > If this is a non-terminating number
with a repeating pattern of numbers [digits], then, yes it is rational. < < <

If the non-terminating list of numbers [digits]
has no pattern it will be irrational i.e. pi, e, √3, etc.

That is not true in general.


\(\displaystyle 0.101001000100001000001...\) is a number that is a non-terminating

decimal number, it has a pattern, but it is irrational.



For the highlighted portion above, it should be closer to:

If the number has a non-terminating list of digits after the decimal point,
but there is a continuous pattern of a repeating block of digits,
then the number is rational.

 

[Mitchell]

Is -23.75 rational?

 

[Deleted member 4993]

Is -23.75 rational?

Yes

-23.75 = -2375/100

 

[stapel]

This is the response from the teacher. Number 21 should be marked irrational, as the patter does not repeat. 24.33315633156

Unless the question somehow indicated that the decimal expansion continued in some manner, like with an ellipsis (the "dot, dot, dot" indicator) at the end, this number "terminates" at the "6" at the far right. But it terminates only in the helpful sense of not requiring more digits to be written. Technically, it continues with a repeating zero:

. . . . .24.3331563315600000000000000000000...

Thus, this is rational, AS WRITTEN. (If, on the other hand, the instructions said to "assume the number continues" or something, then the instructor might have a point.)

Also, I agree with your logic (and I would have accepted it, had I been your instructor). You have clearly indicated the portion of the expansion which you propose to be the "repeating" part, and it does indeed repeat.

But whether or not to appeal the instructor's grading will depend upon the instructor. Some don't know math well enough (or have sufficiently secure self-images) to be able to deal with this sort of thing.