What is the difference between rational and irrational numbers?

[VashtiMaiden]

Can someone pls help me on "rational and Irrational numbers". Esp. on Decimals. I can't classify if it is rational or irrational.

 

[nicksauce]

If you can write z = x/y where x and y are integers, then z is rational. Otherwise z is irrational.

 

[VashtiMaiden]

is pi rational?

 

[VashtiMaiden]

is .66666... rational? why?

 

[nicksauce]

Is Pi rational? No. Not sure how to prove it though.
Is .66666... rational? Can you think of a fraction that gives .666666... ? I would hope you can.

 

[VashtiMaiden]

ok, thanks nicksauce

 

[lurflurf]

for decimal form a useful fact is
a real nummber x is rational if and only if its decimal expansion at some point repeats.
let () be repeat this sequence
1/9=.(1) so rational
8134808921309.2872918752801(29148991280409) so rational

pi has no such patern, though this is not obvious

 

[VashtiMaiden]

ah, ok,

 

[JohnDuck]

for decimal form a useful fact is
a real nummber x is rational if and only if its decimal expansion at some point repeats.

Little off-topic, but here goes: I'm curious, is this not true in some integer base?

 

[HallsofIvy]

If you mean "is it true in any integer base", yes.

 

[kts123]

The question has been answered, but maybe I can help you grasp this a little easier. "Irrational" means that it cannot be expressed as a ratio (NOT that it is 'irrational' in the sense of not being reasonable.) Hence "irrational," or "un-ratio-expressable" if you will. A rational number, on the other hand, CAN be expressed as a ratio. It's "rational," or "ratio-expressable." Since a repeating decimal is given by the 'ratio' of two numbers, it is indeed rational (i.e. 'expressable as a ratio.')

 

[lurflurf]

Little off-topic, but here goes: I'm curious, is this not true in some integer base?

Sorry no
in base pi
pi which is irrational=10
4 which is rational=10.220122021

a problem with algebraic bases
in base root-2
root 2=10
2=100

 

[HallsofIvy]

Did you miss the word "integer" in "integer base"?