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Tag: Math

July 29, 2021

A Logarithm is Just the Number of Digits

If you don’t have a good intuition for how logarithms work, (e.g. what does it mean that happiness scales with the log of income? Or why does an algorithm that runs in log(log(x)) steps always run in 5 steps or fewer?) here is how you can explain it to a third grader: It’s the number of digits (roughly). For example it’s the number of zeros in these:

log_{10}(100) = 2

log_{10}(1000) =3

log_{10}(0.01) = -2

log_{10}(0.001) = -3

For this to work out we have to transition through zero: log(1)=0

If it bothers you that I’m off by one, because clearly 100 has 3 digits, not 2, you can get the actual number of digits by rounding away from 0 to the next integer.

Where it gets complicated is when you have a number between two round numbers:

log_{10}(123) = ?

We can’t just count the number of digits. It’s got to be somewhere between the other two numbers, coming out to roughly 2.09. But lets not worry about those odd cases for now. Where this really helps is when trying to build an intuition for the logarithm rules:

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July 23, 2020

Partial Scaling – How to do Half a Multiplication

Programmers have a habit of over-generalizing things, and so it happened that I found myself writing more generalized versions of rotation, translation and scaling, the three most common operations that you’d want to do to 3D objects. These were more generalized in that they had a parameter for how much you want to do a certain operation. Like “translate by (10, 0, 0), but only apply the operation 20%”. This is easy to do for translation: Just multiply by the percentage and only translate by (2, 0, 0). Rotations are also easy in many representations: If the angles are explicit, like in Euler angles, you can just multiply those by the percentage; if you’re using quaternions, you can slerp.

But scaling is more complicated. Internally scaling is just multiplication, but how do you do half a multiplication? What does it mean to say “multiply by 4, but only apply the operation 50%”? My first approach was to multiply by the power, so you’d get “multiply by 4^{0.5}=2” (or 4^{0.1} if you only want to apply 10% of the operation) and that seems to work when you’re close to 1, but the further away you are from 1, the more wrong it gets. The answer ends up being to take the median of multiplication, division and exponentiation, but let me further explain the problem first:

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