# 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
Homework Helper
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
Homework Helper
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
Homework Helper
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,

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

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
Homework Helper
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