In English, we often drop the subject of a phrase:

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

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 $i = \sqrt{-1}$)

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:

## Exponents

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.

## Imaginary Exponents

Let's combine insights. What does a strange exponent like $e^{i \pi}$ 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 ($\pi$).

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

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.