While it is easier to think of large numbers in terms of logarithms, cause it makes unfathomly large numbers much more palatable, I disagree that it makes scales more intuitive.
1 billion is a very large number, and thinking of it as 10^9 make it seem smaller.
1 trillion is "just" 3 orders of magnitude above 1 billion, and "only" 9 orders of magnitude less than the number of atoms in a mole.
I don't know the answer to making the mind understand scale. I don't think things like "it 's about 2000 football pitches" help either. I don't think "a billion is the number of cubic milimeter in a cubic meter" work either. I don't think the logarithm based "zoom visualization" work either. I just think the brain just cannot picture what those numbers mean. We're not wired to understand those things very well, just as we aren't wired to work with 4+ dimensions
(Author of Mag World here) I'm here to attest that it's worked for me! I see now that I had a fractured collections of number lines in my head (thousands, millionths, billions, and of course all the various units). Since using mag notation and integrating it into my thinking, it has absolutely helped create a grounded "magnitude line" in my head. So I'm not proposing this idly. Notation is a tool of thought (as per Ken Iverson), and using mag notation consistently, turned the advanced method of "logarithms" into a basic perspective of magnitude for me.
Maybe give it a few weeks or months and see if it gets easier for you too.
This scale sometimes goes to 11, for example the size of the solar system is ↑11 meters, but even the size of the entire universe is only ↑27 meters. Which should indicate both how “universal” this scale is, and also how big the universe is!
This feels like the problem of understanding scale gets pushed around syntactically without actually aiding understanding.
"Got it, so the universe is 3x bigger than the solar system."
It feels like that at first glance because you haven't done the work to internalize the notation. This is like someone who uses Roman Numerals being introduced to decimal notation, and getting confused that "22" isn't just twice "2". Yet no one these days would say that decimal notation is merely syntactic shuffling.
The notation helps, but you still have to do the work.
This is uncharitably condescending because I myself am not confused by the notation. Obviously I "did the work."
It absolutely is syntactic shuffling of scientific notation and does nothing to solve the problem of intuitive scale that scientific notation doesn't already handle. The problem is that, unlike 2 and 22, 10^16 is simply not an intuitive quantity, and pretending it is because of Knuth notation / whatever is only displacing the misunderstanding.
Sorry it felt condescending. I'm sure you think it's obvious that you "did the work" because you know about scientific notation and understand the notation. I thought I had done it too, but more and more I see how my internal number line is/was still fragmented, and gets better when I think in terms of exponents first/foremost.
Focusing on the magnitude instead of the value of a number, changes the perspective when we're talking or thinking in these scales.
From the site:
> The universe is very large, but it is not infinite. All quantities in the universe (distance, time, energy, mass, etc) exist within 50 to 100 orders of magnitude.
> The human species interacts with only 25 of these magnitudes.
While it is easier to think of large numbers in terms of logarithms, cause it makes unfathomly large numbers much more palatable, I disagree that it makes scales more intuitive.
1 billion is a very large number, and thinking of it as 10^9 make it seem smaller.
1 trillion is "just" 3 orders of magnitude above 1 billion, and "only" 9 orders of magnitude less than the number of atoms in a mole.
I don't know the answer to making the mind understand scale. I don't think things like "it 's about 2000 football pitches" help either. I don't think "a billion is the number of cubic milimeter in a cubic meter" work either. I don't think the logarithm based "zoom visualization" work either. I just think the brain just cannot picture what those numbers mean. We're not wired to understand those things very well, just as we aren't wired to work with 4+ dimensions
(Author of Mag World here) I'm here to attest that it's worked for me! I see now that I had a fractured collections of number lines in my head (thousands, millionths, billions, and of course all the various units). Since using mag notation and integrating it into my thinking, it has absolutely helped create a grounded "magnitude line" in my head. So I'm not proposing this idly. Notation is a tool of thought (as per Ken Iverson), and using mag notation consistently, turned the advanced method of "logarithms" into a basic perspective of magnitude for me.
Maybe give it a few weeks or months and see if it gets easier for you too.
"Got it, so the universe is 3x bigger than the solar system."
"No no, remember these are base 10 logarithms."
"Ah... so it's 16x!"
"No, 10^16x."
"Ah... so it's totally incomprehensible."
It feels like that at first glance because you haven't done the work to internalize the notation. This is like someone who uses Roman Numerals being introduced to decimal notation, and getting confused that "22" isn't just twice "2". Yet no one these days would say that decimal notation is merely syntactic shuffling.
The notation helps, but you still have to do the work.
This is uncharitably condescending because I myself am not confused by the notation. Obviously I "did the work."
It absolutely is syntactic shuffling of scientific notation and does nothing to solve the problem of intuitive scale that scientific notation doesn't already handle. The problem is that, unlike 2 and 22, 10^16 is simply not an intuitive quantity, and pretending it is because of Knuth notation / whatever is only displacing the misunderstanding.
Sorry it felt condescending. I'm sure you think it's obvious that you "did the work" because you know about scientific notation and understand the notation. I thought I had done it too, but more and more I see how my internal number line is/was still fragmented, and gets better when I think in terms of exponents first/foremost.
Reminds me of the work of Neal Agarwal, e.g.:
https://neal.fun/size-of-space/
https://neal.fun/size-of-life/
https://neal.fun/space-elevator/
https://neal.fun/deep-sea/
Very interesting.
Focusing on the magnitude instead of the value of a number, changes the perspective when we're talking or thinking in these scales.
From the site:
> The universe is very large, but it is not infinite. All quantities in the universe (distance, time, energy, mass, etc) exist within 50 to 100 orders of magnitude.
> The human species interacts with only 25 of these magnitudes.
> The universe is very large, but it is not infinite.
We actually don't know this, it's still an open question.
s/universe/observable universe/