First off, e was discovered, not chosen. Think of the speed of light, c. It wasn't decided to be 299,792,458 m/s -- we did measurements and realized that under ideal, universal conditions (in a vacuum), this was *the fastest anything could possibly move*.

Ok -- let's look at ideal growth and ask *under ideal, universal conditions, what's the fastest something can possibly grow?* This means:

- Growing by the unit rate (100%)
- Growing for the unit time (1 time period)
- Growing perfectly, without any delay (continuous)

Applying these assumptions, we get the formula:

And plugging in bigger values for n (for a better approximation) yields e ~ 2.71828:

**Objection: But 13.74 ^{x} can model exponential growth just like e^{x} can!**

Sure. But what assumptions did you make to get 13.74? They probably weren't "unit rate, unit time, perfectly compounded". (You can pick k as the speed of light through Kool Aid too -- but why?)

Arguably, 2^{n} is also universal ("the discrete e"), because you have zero compounding (n is an integer like 0, 1, 2, 3). Instead of perfectly continuous, it's perfectly non-continuous (discrete), and we take growth step-by-step.

So, I'd say either 2^{n} (in discrete systems) or e^{x} (in continuous systems) are "universal".

**Objection: But things can grow faster than e ^{x}, which is just 2.71828^{x} -- what about 13.74^{x}?**

What is it with you and 13.74? Yes, you can beat e^{x} in an exponential footrace, if you grow for longer or at a faster rate than 100%. 13.74^{x} is really [e^{ln(13.74})]^{x}. Because ln(13.74) ~ 2.6, you are assuming a 260% continuous interest rate, more than the 100% e^{x} uses. (Alternatively, you are growing for 260% of the unit time period that e^{x} uses.)

Related:

]]>What goes through our head the first time we see this?

Ok, y is usually the final value in an equation. It's based on two parts... mx and b. If x goes up by 1 let's say, we only change y by... m.

Hrm, let's try an example. If m=.3, and x goes from 0 to 10, we go up by 10 * .3 = 3. Ok. Got that part.

Now what does b do? It adds something on top of whatever we just figured out. If x = 0, I guess we just have y = b.

There's a lot of Math <> English translation. In math jargon, we say the slope (m) determines the direct rate of change between x and y, and the y-intercept (b) adds a starting value.

Ugh. It's abstract and doesn't resonate. Here's a more natural wording:

Makes sense, right? Start somewhere, make a change, get to the new location.

Zooming in, we realize a change has two parts: how fast are we changing, and for how long?

Ah. I can see the role of each of the pieces. Turning this into succint math notation:

"Start at b, add the rate of change (m) times the duration (x), and you end up at y."

We aren't forced to graph the equation and describe b as the "y intercept". Who thinks (or talks) like that?

In plain English:

- b is our starting point
- x is the change we intend to make
- m is the percentage (rate) of the change we actually see

Boom. Later, if we *decide* to graph our line (fun fun), then b is the height on the y-axis when x=0. But thinking in terms of our starting value (the **b**ias) just clicks better.

The graphed line is a list of all possible outcomes under some set of circumstances. We might not wait at all (x =0), or wait for a bit (x=1), wait for a long time (x = 100), or look back in time (x = -3). The line shows what would have happened in each of those scenarios.

People claim they don't think in terms of math. Sure, you might not describe things as slope/intercept form, but I'd guess "original + change = new" is pretty familiar to you. Even the simplest equations can use an intuitive wording.

**Q: Find the equation for a line between (3, 4) and (7, 12)**

Let's think about this intuitively. We know ultimately we want an equation of the form:

`position = start + change`

For the two points we have, let's look at the change:

- Change in x: 7 - 3 = 4
- Change in y: 12 - 4 = 8

Between those points, x changed by 4 and y changed by 8. The rate was 8/4 = 2, so the equation should look like:

`position = start + x * 2`

We don't know where we started, but whenever x changes, y changes by 2 times that amount.

Ok. Now what was the starting position, when x = 0? Let's see the change between our default value of (0, ?) to (3, 4).

- Change in x: 3 - 0 = 3
- Change in y: 4 - ? [unknown]

Well, if x changed by 3, and the rate of change is 2, then y changed by 3*2 = 6. The original value was therefore 4 - 6 = -2.

- Original point is (0, -2)

The final equation is then:

`position = start + change`

`position = start + x * 2`

`position = -2 + x * 2`

and written in more standard y = mx + b notation,

`y = 2x - 2`

Let's double check. We start at (0, -2) and go 3 units on the x coordinate. Our final position is `2*3 - 2 = 6 -2 = 4`

, which matches up with (3, 4).

This approach is slower than a plug-and-chug shortcut, but the idea is to figure out exactly what you're doing:

- Figure out the rate of change between two points (m)
- Using that rate of change, work backwards to find our starting point (b)

Happy math.

]]>Who are these signs written for? It's really *You, stop* or *You, yield* or *You, be alert for bears* (I'm not doing it).

After internalizing a language, we can take hints without explicit instructions. But, to put it politely, math isn't usually well-internalized.

Let's get clear about who the "math signs" are referring to.

Imaginary numbers are often defined as i^{2} = -1, and written this way they're utterly baffling.

A better restatement is:

which is still confusing. But what about this:

It's getting clearer. The instructions are: "Start with 1, multiply by i, multiply by i again, and (somehow) end up at -1".

What could do that? Realizing we *start at 1 and end up at -1* helps us visualize something like this:

Aha! i is a change (visualized as a rotation) that moves us from positive to negative in two steps (More about imaginary numbers.)

Writing i^{2} = -1 without a clear subject is confusing. (Don't get me started with i = √(-1))

Missing the implied subject of "1" in 1· i · i caused me years of confusion. I wish this sign was hanging on the classroom wall:

Why is x^{0} = 1 for any value of x? How do we ask for 0 of something and get 1 back out?

Again, let's break it down with a simple example. Here's a typical exponent:

But it's missing a subject. It's written better as:

We start at 1 (our default multiplicative scaling factor), scale by x, then scale by x again, ending up at x^{2}. The size of the exponent (2) tells us the number of times to use our "times x" scaling machine.

Stepping back, multiplication is about scaling: 3 is really "1 * 3", or the unit quantity enlarged 3 times.

If we want to scale by x (just once), we write:

What if we don't plan on using our scaling machine at all?

The notation is a bit weird, but I'm using empty parenthesis to indicate a lack of action. See, the zero in x^{0} is that of indifference -- taking no action -- and not obliteration. "Using" x^{0} means we haven't scaled our original quantity at all.

Subtle, right?

We can take this "growth machine" idea further with the Expand-o-tron 3000.

Let's combine insights. What does a strange exponent like e^{i π} represent?

With our new "implicitly start at 1" perspective, it's really:

Start at 1 and then apply the growth engine. Here, growth is aimed sideways (i) with enough fuel to last for half a circle (π).

The essence of Euler's Identity is that we are starting at 1 and transforming it with a spin. We aren't creating a negative number out of seemingly positive exponents directly. (See article and video.)

Calculus has numerous notational shortcuts. When we write:

it really means:

which really means:

Here's the tricky part. There isn't a single "dx", there's a whole chain of them along the number line. The sentence is something like:

*"Hey everyone on the number line! You're all spaced "dx" apart. Take your current position and square it. Then I'll come by and add you all up."*

The integral addresses not a single "you" like 1.0, but "them", the countless positions on the number line.

Find the implied subject in an equation, then work to shorten it (*Bears!*).

Happy math.

]]>"Argh, why couldn't they have explained it like this the first time?"

The difference between a semester of pain and instant understanding was one stupid, missing analogy. It still riles me up thinking about how close I came to missing the key concept (and disliking math).

In class there's the lesson about a specific formula, sure, but the meta-lesson is *how well the experience went*.

What worked? What didn't? How can we get more of the first and less of the second?

Over time, I realized individual topics were chances to explore what truly worked when learning. Not what a learning theorist or book said (*Flashcards! Mnemonics! Just study harder!*), but what *actually* worked for you.

A few of my scattered meta-lessons:

- Analogies, while imperfect, are a huge jump start. It's motivating to get the ball rolling and course correct along the way, vs. waiting to line things up perfectly.
- Nearly every explanation is improved with a visual or diagram.
- Humor and empathy put the reader at ease so they can tell you when they're
*actually*confused (vs. mindless head nodding). - Share the gotchas. The Wise Teacher hiding the 14 mistakes he made when learning the topic does students a disservice.

Every lesson is a chance to silently wonder "How well did that work?".

These days, I use ADEPT as a running checklist for how I get things to click:

This was pulled from actual frustrations (*Why can't they share a plain-English version first? A diagram?*) and I'm sure you'll have modifications of your own.

Don't just take a single lesson away from a lecture, article, or video. Think *How well did that work for me?* and build your learning approach around the best parts.

Happy math.

P.S. My buddy Nasos runs the excellent MetaLearn podcast and we have several chats about learning, a previous interview is below:

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