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The naturals: $\{1, 2, 3, ... \}$

$2 - 1 = 1$ and $\frac{2}{1} = 2$

$3 - 2 = 1$ and $\frac{3}{2} = 1.5$

$4 - 3 = 1$ and $\frac{4}{3} = 1.333...$

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$167834632 - 167834631 = 1$ and $\frac{167834632}{167834631} = 1.000000006...$

$\displaystyle \lim_{n \to \infty} (n + 1) - n = 1$

But ...

$\displaystyle \lim_{n \to \infty} \frac{n + 1}{n} = 1$

We've only considered a difference of $1$ (consecutive numbers) but look at the following example:

$10000001000 - 10000000000 = 1000$

$\frac{10000001000}{10000000000} = 1.0000001$

Observation
Ratio fails to distinguish or is bad at distinguishing one very large number from another very large number, but Difference can.
Ratios create an illusion of equality, but differences expose the druj (deception).

Question
Is this a known mathematical fact? What are views/opinions of mathematicians on the matter?

Agent Smith
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    It's a bit unclear what kind of answer you're looking for. I think it's rather intuitive that $$ \lim_{n \to \infty} \frac{n+C}{n} = 1 \qquad \text{for any constant }C $$ and it's definitely a "known thing", as far as I'm considered. – Matti P. Oct 18 '23 at 11:17
  • Mathematical Illusions vis-à-vis ratios. – Agent Smith Oct 18 '23 at 11:26

2 Answers2

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Ratios and differences have different meanings and measure different things. Your assertion that differences are superior because they are intuitively better in one application is very short sighted.

For example, the distance between the Moon and the Earth is about $400{,}000$ kilometers. Now, you tell me, which of these two sentences conveys information better?

  1. The sun is about $4000$ times farther from the Earth than the Moon
  2. The sun is about $150$ million kilometers farther from the Earth than the Moon.
5xum
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  • I don't know. There's our brain, our senses, biology. Are human beings better at discerning scaling (multiplication) or addition/subtraction? – Agent Smith Oct 19 '23 at 02:28
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Illusion? Deception?

The ratio and the difference constitute distinct ways to think about or compare pairs of numbers. Either might be more informative—or appealing—depending on context and what features of the comparison are of interest.

Consider the old joke about the museum guard who told a visitor, “This fossil skeleton is sixty-six million twenty-one years old.” The visitor responded, “Wow! How do we know that?” to which the guard replied, “When I started here I was told it was sixty-six million years old, and I have been here twenty-one years.”

Paul Tanenbaum
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  • Si, but $\displaystyle \lim_{x \to \infty} \frac{x + C}{x} = 1$. What is $1$? That, mon ami, is the question. – Agent Smith Oct 18 '23 at 11:48
  • If that were indeed the question, then you should have simply asked that in the OP. What is it that you sincerely wish to know? – Paul Tanenbaum Oct 18 '23 at 12:07
  • The multiplicative identity ($1$) corresponds to the additive identity ($0$). $a - a = 0 \implies \frac{a}{a} = 1$ and vice versa – Agent Smith Oct 19 '23 at 02:24
  • Ok, @AgentSmith, and that’s why they’re both called the same thing, identities. But my query remains, What is it that you sincerely wish to know? – Paul Tanenbaum Oct 19 '23 at 13:17