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What is the difference between fractions and rational numbers? is $\frac{\pi}{1}$ a fraction?

rain
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    Take a look at: Is $\frac{1}{\sqrt{2}}$ a fraction or not? for some comments regarding the terminology 'fraction' <-> 'rational number'. – StackTD Feb 27 '18 at 13:31
  • The question is a bit like: what is the difference between the word 'pizza' and pizza? Answer: the word 'pizza' represents the food, while actual pizza is the food. You can only eat one of them. – Ittay Weiss Feb 27 '18 at 15:20
  • It seems like there isn't a standard definition of what a "fraction" is, but just for one example usage: in Landau's "Foundations of Analysis", he defines a fraction as an ordered pair of positive integers (this is early in the book, before he develops zero or negative numbers), and defines a rational number as an equivalence class of fractions. – Joe Nov 04 '21 at 20:37

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A rational number is a number which can be expressed as a fraction of two integer values.

$\cfrac \pi1$ is a fraction, but it does not evaluate to a rational expression.

A fraction which evaluates to a rational expression is defined by an expression that can be expressed as fraction of two integer values.

  • So any number is a fraction... that can sound quite misleading when first heard. – Mr Pie Feb 27 '18 at 13:46
  • In the set of real numbers, yes. Please provide a counterexample if you don't believe so @user477343, because I did not explicitly state that "any number is a fraction". –  Feb 27 '18 at 13:53
  • Ah yes, I read through this a bit too quickly, for I didn't notice the part $\longrightarrow$ that can be. – Mr Pie Feb 27 '18 at 13:55
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Even without touching on irrational numbers, there is a huge difference between rational numbers and fractions. The difference is somewhat analoguous to the difference between a natural number and the decimal system.

Rational numbers are points on the number line. They are quantities. They are numbers. They exist (as far as any number can exist) even without us humans to study them and write about them. If an alien species landed on earth and talked to us, they would almost certainly know what a rational number is (although they may not use the same word).

A fraction is one way (of several) that we humans have invented / discovered that lets us describe rational numbers in a nice way. It's a piece of terminology. The aliens may use fractions (possibly written in a different way) or they may not. Maybe they like to work in terms of repeating decimal expansions, and have never even thought of using division as a main way to describe numbers.

Finally, I'll leave with this: $\frac12$ and $\frac24$ are different fractions, but they represent the same rational number.

Arthur
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  • I don't buy this explanation one bit. Pi is also a point on the number line, somewhere between 3.14159 and 3.14160. It is a numerical quantity. It exists quite naturally in geometry and many other fields, and would do so without humans to study it. If an alien species made it here without knowing what pi was, I'd be impressed. – Nuclear Hoagie Feb 27 '18 at 13:40
  • @NuclearWang When did I say anything about $\pi$? I didn't say that irrational numbers don't exist. My answer is about the difference between fractions and rational numbers, which is the actiual question asked. It's not about the difference between rational and irrational. It doesn't take much thought after reading this answer to see that I would regard $\frac\pi 1$ as a fraction, but not as a rational number. So how would you say that I haven't answered what was asked? – Arthur Feb 27 '18 at 13:42
  • I don't understand why you got a downvote. Unfortunately though, I have reached my daily voting limit, for otherwise I would have upvoted. – Mr Pie Feb 27 '18 at 13:48
  • It seems like @Nuclear assumed that when you stated that "rational numbers are points on the number line" that you were defining rational numbers as points on the number line, and used pi as a counter example, trying to show that everything that you said about rational numbers is also true about pi, which isn't a rational number. Of course, that doesn't contradict anything you said, because your point was that rational numbers are *numbers, whereas fractions are particular representations* of numbers. – Joe Nov 04 '21 at 21:59