There is very likely some step to sit on 🤣. To empty the water you just need a hose and do the same trick people use to steal gasoline (or a pump if you want to be fast and fancy).

Nice. Haven’t thought about that 🤣

It doesn’t matter if you divide ln(2) or x by three, it’s the same thing.

By tweaking a few parameters you can turn every base into any other base for exponentials. Just use e^(ln(b)*x)

PS: The formula here would be e^(ln(2)/3*X) and x is the number of months. So the behavior it’s exponential in nature.

Original post is not linearly interpolating but exponentially.

Doubling every three months is an exponential interpolation and not a linear one!

“when in doubt” is a bit broad but left to right is a great default for operations with the same priority. There is actually a way to calculate in any order if divisions are converted to multiplications (by using the reciprocal value) and subtractions are converted to additions (by negating the value) that requires at least a little bit of math knowledge and experience so it’s typically not taught until later to prevent even more confusion.

For example this: 6 / 2 * 3 can also be rewritten as 6 * 2⁻¹ * 3 and because multiplication is commutative you can now do it in any order for example like 3 * 6 * 2⁻¹

You can also “rearrange” the order without changing the meaning if you move the correct operation (left to the number) with it (should only be done with explicit multiplication)

6 / 2 * 3 into 6 * 3 / 2 (note that I moved the division with the 2)

You can even bring the two to the front. Just remember that left to the six is an “imaginary” (don’t quote me ^^) multiplication. And because we can’t just move “/2” to the beginning we have to insert a one (empty product - check Wikipedia) like so:

1 / 2 * 6 * 3

This also works for addition and subtraction

7 + 8 - 5

You can move them around if you take the operation left to the number with it. With addition the “imaginary” operation at the beginning is a plus sign and the implicit number you use is zero (empty sum - check Wikipedia)

8 - 5 + 7

or like this

0 - 5 + 8 + 7

because with negative numbers you can use the minus sign to indicate negative numbers you can even drop the leading zero like this

-5 + 8 + 7

That’s not really possible with multiplication because “/2” is not a valid notation for “1/2”

👍 That was actually one of the reasons why I wrote this blog post. I wanted to compile a list of points that show as clear as humanity possible that there is no consensus here, even amongst experts.

That probably won’t convince everybody but if that won’t probably nothing will.

In a scientific context it’s actually very rare to run into that issue because divisions are mostly written as fractions which will completely mitigate the issue.

The strong implicit multiplication will only cause ambiguity after a division with inline notation. Once you use fractions the ambiguity vanishes.

In practice you also rarely see implicit multiplications between numbers but mostly between variables or variables and their coefficients.

I’m not sure if I’d call it the “scientific” one. I’d actually say that the weak juxtaposition is just the simple one schools use because they don’t want to confuse everyone. Scientist actually use both and make sure to prevent ambiguity. IMHO the main takeaway is that there is no consensus and one has to be careful to not write ambiguous expressions.

You should read the part about WolframAlpha in the blog.

https://www.wolframalpha.com/input?i=6%2Fxy+where+x%3D2%2C+y%3D3

I’d really like to know if and how your view on that matter would change once you read the full post. I know it’s very long and a lot of people won’t read it because they “already know” the answer but I’m pretty sure it would shift your perception at least a bit if you find the time to read it.