Math Intuition
Mathematics
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 \(e\) (it’s about growth)
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 actually 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)
100% continuous growth means that, at any given time, the rate of change is always equal to our current value. Or more mathy: (d/dx)*(e^x) = e^x
Understanding \(ln()\) (it’s about time)
- \(e^x\) lets us plug in time and get growth.
- \(ln(x)\) lets us plug in growth and get the time it would take.
Statistics
- Probability is starting with an animal, and figuring out what footprints it will make.
- Statistics is seeing a footprint, and guessing the animal.
Common statistical tests are linear models
