How is Pi a Ratio if it Cannot be Written as a Fraction?

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Discussion Overview

The discussion revolves around the nature of pi as a ratio, specifically addressing the question of how pi can be considered a ratio of circumference to diameter if it cannot be expressed as a fraction. Participants explore the implications of pi being an irrational number and the definitions of ratios.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that pi is the ratio of circumference to diameter, questioning how it can be a ratio if it cannot be expressed as a fraction.
  • Others argue that a ratio does not need to be a rational number, and that real numbers, including irrational numbers like pi, can represent ratios.
  • One participant reflects on thinking of pi as an infinite sum rather than a real number.
  • It is noted that a rational number is defined as a ratio of integers, which pi is not, as it is irrational.
  • Some participants discuss the implications of measuring circumference and diameter, suggesting that for any integer diameter, the corresponding circumference will not be an integer.
  • There is a recurring question about the convention of using circumference/diameter to define pi, with various humorous suggestions for alternative names and definitions.
  • One participant claims that pi can be expressed as a decimal, suggesting that it can be written in forms other than a fraction.

Areas of Agreement / Disagreement

Participants express differing views on the nature of ratios and the implications of pi being irrational. There is no consensus on how to reconcile the definition of pi as a ratio with its irrationality, and the discussion remains unresolved.

Contextual Notes

Participants highlight the distinction between rational and irrational numbers, and the implications of these definitions on the concept of ratios. The discussion includes assumptions about the nature of measurements and conventions in mathematics.

cragar
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If pi is the ratio of its circumference to its diameter , But it cannot be written as a fraction
then how is it a ratio ?
 
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cragar said:
If pi is the ratio of its circumference to its diameter , But it cannot be written as a fraction then how is it a ratio ?
A ratio doesn't need to be a rational number. A real number is perfectly acceptable.

For example, a ratio could be a ratio between any two numbers, real numbers included. No reason I can't have a ratio of pi/e if I wanted. Still perfectly valid proportions.
 
i see , thanks for your answer , i guess i was thinking of pi as an infinite sum and not a real number.
 
A rational number is not merely a ratio of numbers, it is a ratio of integers.
 
Jimmy Snyder said:
A rational number is not merely a ratio of numbers, it is a ratio of integers.

yes but pi is irrational
 
cragar said:
If pi is the ratio of its circumference to its diameter , But it cannot be written as a fraction then how is it a ratio ?

Pick a nice circle with unit diameter. Now measure its circumference. The circumference is not in rational units. The fraction is not expressed as two rational numbers.
 
cragar said:
yes but pi is irrational
That means pi cannot be expressed as the ratio of two integers.
 
That means that for all integer value of diameter, there will never be a corresponding integer value of circumference.
 
Why does everybody use circumference/diameter and call it pi instead of using diameter/circumference and call it cake (or another Greek letter)?
 
  • #10
because if the diameter is rational then the circumference isn't, so the ratio is irrational
 
  • #11
Dr Lots-o'watts said:
Why does everybody use circumference/diameter and call it pi instead of using diameter/circumference and call it cake (or another Greek letter)?

Mainly because that's just how it happened...
 
  • #12
Dr Lots-o'watts said:
Why does everybody use circumference/diameter and call it pi instead of using diameter/circumference and call it cake (or another Greek letter)?

wiki sez said:
http://en.wikipedia.org/wiki/Pi"

so i guess, convention, getting there first/most often/something. could have just as easily been a chinese symbol for a dragon eating its tail or something.
 
Last edited by a moderator:
  • #13
Dr Lots-o'watts said:
Why does everybody use circumference/diameter and call it pi instead of using diameter/circumference and call it cake (or another Greek letter)?

Because the guy pushing the wheel had the final say?
 
  • #14
It can be written as a fraction. Just as easily as it can be written as a decimal of 3.1415926... How long do you want to continue carrying the numbers out in either form? Neither one will end.
 

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