How can I convince someone that $1/3$ is a real number?

Question

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?

Answer 1

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.

Answer 2

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

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 $\frac13$.

Answer 3

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.