Rational and Irrational

Can someone pls help me on "rational and Irrational numbers". Esp. on Decimals. I cant classify if it is rational or irrational.
 

nicksauce

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If you can write z = x/y where x and y are integers, then z is rational. Otherwise z is irrational.
 
is pi rational?
 
is .66666..... rational? why?
 

nicksauce

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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.
 
ok, thanks nicksauce
 

lurflurf

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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
 
ah, ok,
 
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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

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

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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

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

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