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How can I convince someone that $1/3$ is a real number? I've tried explaining to them that you can draw it on a number line but they argue that (and I quote) "there is no number there, what's the number?" when I reply back that it can be written as $0.333...$ in decimal, they say "that's not a number".

I've also tried explaining that you can write the number in a different base system (say, base 12), and it won't have a repeating decimal, but then they go on about numbers must have a base 10 form or something along those lines. They also say that $0.999...$ is not a number and it does not equal $1$ ("there's a bunch of nines, each nine is getting closer, and the difference is $0.000...1$" they say).

Is there a clear and straightforward argument for this, or is it impossible?

Mauro ALLEGRANZA
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yusef
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    What is his or her definition of a real number? – rogerl Dec 07 '19 at 21:23
  • https://math.stackexchange.com/questions/11/is-it-true-that-0-999999999-dots-1 – Maximilian Janisch Dec 07 '19 at 21:24
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    You have to consider the possibility the one you are teaching is intentionally acting dumb. – achille hui Dec 07 '19 at 21:26
  • Give them a practical exercise that forces them to divide by three in a general setup (like: divide something in such a way that one person has twice the other) and demote them if they fail. –  Dec 07 '19 at 21:27
  • @rogel They believe that 0.5 is a number, 0.25, etc., just nothing with a repeating decimal when written decimally. – yusef Dec 07 '19 at 21:30
  • Ask them what a real number is. Why isn't $\frac 13$ a real number? Then work with them on what the actual definitions is. It seems you are both working without a definition and both think a number is "something you can see on a number line" which is circular at best (your interpretation) and wrong at worse (your friend who thinks that means only integers). – fleablood Dec 07 '19 at 21:36
  • With this person's belief of what a number is, they will not accept that $1/3$ is a number. Just like people who don't vaccinate their children, there is nothing you can do to change their mind. – Matt Samuel Dec 07 '19 at 21:39
  • One third is a value. You can cut things into three equal parts. The real numbers are every possible value. So it has to be a number. If someone thinks $\frac 13$ is not a real number then either 1) they think "real number" has a special meaning it doesn't or 2) they just are refusing to think. If the former, explain the misconception, If 2) then ... while I like B. Goddards advice about teaching pigs to sing. – fleablood Dec 07 '19 at 21:41
  • @yusef Okay... can you just try to introduce them to base 3? In base 3, $1/3$ has a finite decimale expansion. – WoolierThanThou Dec 07 '19 at 21:41
  • What does bases and finite decimals have to do with anything? Decimals aren't magic. Being able to write something in decimal has nothing to do with whether something is a number. Fractions are numbers. Period. Fractions are numbers. Decimals are one (incomplete) way of writing numbers but they do not define numbers and not every number can be written as a finite decimal. (They can be written as an infinite decimal but that is a hard and advanced concept to be proven--- it is NOT a definition) And that is the mistake. People think decimals DEFINE numbers. They don't. – fleablood Dec 07 '19 at 21:45
  • Hit them with a two-by-four, and shout in their ear "Decimals do not make numbers; decimals are a tool to describe numbers; but they don't make numbers." – fleablood Dec 07 '19 at 22:04

3 Answers3

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You can prove that the existence of $\frac13$ is as valid as the existence of $1$ with a compass and straightedge.

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Let $AB=1$. (As your friend has indicated that $1$ is a valid point on the number line, you can ask what such a representative distance would be.) On a separate ray, construct points $C,D,E$ such that $AC=CD=DE$. Construct a line parallel to $\overline{EB}$ that passes through $C$. That line will intersect $\overline{AB}$ at $F$.

$\angle AFC\cong\angle ABE$ and $\angle ACF\cong\angle AEB$, because these are corresponding angles of parallel lines cut by a transversal. Therefore, $\triangle ACF\sim\triangle AEB$, and therefore, their sides are in proportion. This means that $\frac{AB}{AF}=\frac{AE}{AC}=3$.

So we have a physical line segment $\overline{AF}$ that has a length that satisfies $3AF=1$. How long is it? If one believes that there is not an "intrinsic" number that signifies its length, hopefully one would acknowledge that this is a sufficiently relevant situation that we would need to artificially construct such a number. In either event, we will call that number $\frac13$.

  • This is much cooler than what I expected from answers to this question – Maximilian Janisch Dec 07 '19 at 22:10
  • I think you will need to explicitly state $\overline AB$ is the abstract (but in the friends mind very real) "number line" and that $A$ is zero and $B$ is $1$. The friend is likely to think this is mumbo jumbo and makes no sense but... it is a point an the line. – fleablood Dec 07 '19 at 22:11
  • @fleablood Agreed. If the friend acknowledges that answering the question "Where is it on the number line?" will satisfy them, and especially if they acknowledge that they are bisecting the number line to find the dyadic rationals, then a fair-minded skeptic ought to be persuaded by this argument. (And, by contraposition, anyone not persuaded by this argument is not being fair-minded. ^_^) –  Dec 07 '19 at 22:21
  • @MaximilianJanisch This is essentially how the Greeks came to terms with $\sqrt2$ being a number even though they knew it couldn't be the ratio of two natural numbers. –  Dec 07 '19 at 22:23
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I've had this issue too. I like to pose the question in the opposite direction. I would ask them, what numbers ARE real, then? They might answer the integers, or the naturals. But then, it should be possible to argue that the naturals are just as arbitrary as the rationals, they just happen to be more intuitive.

The only valid stances on the issue, in my opinion, is the stance that all numbers are "real", or none of them are.

Rushabh Mehta
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I think your friend has gotten it through their head that a "real number" is DEFINED to be anything that can be written as a decimal.

This is simply wrong. That is not the definition. It is backwards. Eventually (but not now) it will be proven that if all numbers can be written as a decimal if you allow infinite decimals. DON'T worry about teaching your friend that step yet. It will just confuse them.

Instead decimals were invented (not discovered, invented) as a way to express numbers in a way that we can compare sizes of numbers that are not whole numbers.

So what IS a real number? It's any value. Period. That's all.

And $\frac 13$ is a value. You can divide things evenly. HOW you divide things evenly is another issue but that isn't relevant. $\frac 13$ is a very obvious value.

Now it is interesting that we can express $\frac 13$ as a fraction but not as a finite decimal. But that is not relevant. Decimals do not define numbers. Not being able to express $\frac 13$ didn't make $\frac 13$ disappear out of the universe. We can still talk about $\frac 13$. What is $\frac 13$ then if it isn't a number? A rabbit? A non-number? Ah, I know!.... A "fraction"; a magical beast like a chimera gryphon that doesn't actually exist.

Sorry. It is a real value. Hence it is a "number"

Decimals do not define numbers. They are ONE way to (inefficiently and incompletely[*]) describe numbers but they don't magically turn some values into numbers and magically turn other values into chimeras.

All values are real numbers. What else could they be?

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[*] Unless you allow infinite decimals. But that's a lesson for later.

fleablood
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