Math Intuition

Under construction

Updated mental models

Numbers aren’t just a count; a better viewpoint is a position on a line. This position can be negative (\(-1\)), between other numbers (\(\sqrt{2}\)), or in another dimension (\(i\)).

Arithmetic became a general way to transform a number. Addition is sliding along the number line (\(+ 3\) means slide 3 to the right) and multiplication is scaling ( \(\times 3\) means scale it up 3 times).

Mathematically, the exponent function does this:

\[ \mathrm{original} \times \mathrm{growth}^{\mathrm{duration}} = \mathrm{new} \]

or \[ \mathrm{growth}^{\mathrm{duration}} = \frac{\mathrm{new}}{\mathrm{original}} \]

Operation Old concept New concept
Addition Repeated counting Sliding
Multiplication Repeated addition Scaling
Exponents Repeated multiplication Growth for amount of time

Understanding e

The number e is the base amount of growth shared by all continually growing processes. The number e merges Rate and Time. When we write:

\[ e^x \]

the variable x is a combination of rate and time.

\[ x = \text{rate} \times \text{time} \]

So, our general formula becomes:

\[ \text{growth} = e^x = e^{r t} \]

The number e is about continuous growth. Intuitively, \(e^x\) means:

  • How much growth do I get after after x units of time (and 100% continuous growth)

Common statistical tests are linear models

https://lindeloev.github.io/tests-as-linear/

https://lindeloev.github.io/tests-as-linear/