Logical fallacy that suggests $3/9=3/10$

Question

I have seen that $3/9 = 1/3$ can be written as $0.3$. However, $0.3=3/10$. Does this mean that $3/9=3/10$, or am I confused?

Why the downvote? Is there a rule that only PhDs in mathematics are allowed to ask questions? — DanielV, Jun 06 '18 at 02:24
Please don't say things that are just going to mislead @David — DanielV, Jun 06 '18 at 02:26
@DanielV While maybe it wasn’t phrased in the most robust/explicative manner, I think the comment was a good contribution. — gen-ℤ ready to perish, Jun 06 '18 at 03:07
I am going to take the liberty of significantly rewording this post without changing its gist in an effort to keep it from being closed, because I happen to think it has some merit to it $\ddot\smile$ — gen-ℤ ready to perish, Jun 06 '18 at 03:08
@David I'm sorry I don't understand why you have linked Principle of Explosion here, and as a reply to what — Harveen Bhatia, Jun 06 '18 at 03:19
Since the question has been changed the comment is no longer relevant. — David, Jun 06 '18 at 03:41

score 8 · Accepted Answer · answered Jun 06 '18 at 02:22

8

The error is that $1/3\ne 0.3$. So you find

$$\frac39=0.333\dots > 0.3 = \frac3{10}$$

which makes sense, since you're dividing by less. That is, $9$ is smaller than $10$, so $3/9 > 3/10$.

answered Jun 06 '18 at 02:22

zahbaz

2

Lo and behold, an actual honest answer. Thanks. – Randall Jun 06 '18 at 02:27
1

So 0.3 and 0.3 repeating are different values? – Harveen Bhatia Jun 06 '18 at 02:38
1

Yes, for sure. You might see some people round down, but that loses some accuracy and it's not the same number anymore. – Randall Jun 06 '18 at 02:43

score 3 · Answer 2 · answered Jun 06 '18 at 03:19

The issue here is the difference between equality $=$ and approximation $\approx$. Here is what’s actually happening:

$$\frac39 = \frac13 = 0.33333\cdots \approx 0.3$$

The approximation comes when we round down.

Because we know that $0.3=3/10$, we can just replace $0.3$ with $3/10$ on the righthand side, giving us

$$\frac39 = \frac13 = 0.33333\cdots \approx \frac3{10}$$

We don’t need three names for the same value, so let’s take out the second two “equal to” terms:

$$\frac39 \approx \frac3{10}$$

Voilà! If you decide to use $=$ instead, that’s you’re prerogative, but in most instances it’s slightly wrong $\ddot\smile$

2 Answers2