What is the difference between rational and irrational numbers?

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Discussion Overview

The discussion focuses on the classification of rational and irrational numbers, particularly in relation to their decimal representations. Participants explore definitions, examples, and implications of these classifications, with some emphasis on the nature of decimals and their repeating patterns.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that a number is rational if it can be expressed as a fraction of two integers (x/y), while others question specific examples like pi and repeating decimals.
  • There is a discussion about whether pi is rational, with one participant stating it is not and expressing uncertainty about how to prove this.
  • Participants discuss the nature of repeating decimals, with one noting that .66666... can be expressed as a fraction, thus classifying it as rational.
  • One participant mentions that a real number is rational if its decimal expansion eventually repeats, providing examples to illustrate this point.
  • There is a side discussion about whether the property of rationality in decimal representation holds in different integer bases, with some participants agreeing that it does.
  • Another participant clarifies that "irrational" refers to numbers that cannot be expressed as a ratio, contrasting this with rational numbers.
  • One participant raises a question about the applicability of these concepts in non-integer bases, leading to further clarification and examples.

Areas of Agreement / Disagreement

Participants generally agree on the definitions of rational and irrational numbers, but there is disagreement regarding specific examples like pi and the implications of decimal representations in various bases. The discussion remains unresolved on some points, particularly regarding the nature of irrational numbers in different bases.

Contextual Notes

Some statements depend on the definitions of rationality and the properties of decimal expansions, which may not be universally accepted. The discussion includes unresolved questions about the nature of irrational numbers in non-integer bases.

VashtiMaiden
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Can someone pls help me on "rational and Irrational numbers". Esp. on Decimals. I can't classify if it is rational or irrational.
 
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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?
 
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
 
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,
 
lurflurf said:
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?
 
  • #10
If you mean "is it true in any integer base", yes.
 
  • #11
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.')
 
  • #12
JohnDuck said:
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
 
  • #13
Did you miss the word "integer" in "integer base"?
 

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