Like the word "run", the meaning depends on context:

- crawl / walk / run (movement)
- run a company (general operation)
- a run of good luck (sequence)
- and a dozen more definitions

Sticking with a single interpretation of "run" leads to confusion, and the same happens in math. Let's clarify how exponents are used.

We first learn that exponents like 3^{2} or a^{n} are repeated multiplication: multiply a, n times.

Like counting on your fingers, this breaks down beyond the positive integers. What does a fractional exponent mean? A negative one? Zero? (Since a^{0} = 1, we multiply zero times and get 1?)

*Common usage of a ^{n}*: Counting problems. If you flip a coin n times, you have 2

Let's say I have an exponent like 3^{4.5}. I mentally convert it to 1.0 · 3^{4.5}, and then 1.0 · g^{t}.

With the "growth microwave" analogy, an exponent grows our starting amount (1.0) by g for t units of time. (In this example, 3x growth applied for for 4.5 seconds.)

What values can t have?

- If t is positive, we go forward in time and get larger (assuming g > 1). Fractional time is ok -- I can run a microwave for 3.5 minutes, and get some effect between 3 minutes and 4 minutes.
- If t is negative, we go backwards in time and get smaller. If a regular microwave allowed negative time, it would cool down your food, right?
- If t is zero, we didn't use the machine at all! We're left with 1.0, our original amount.

The growth microwave interpretation helps with fractional powers (and resolves the t=0 issue), but it's not *flexible*. Doubling the rate and halving the time doesn't have the result we expect:

2 seconds of 3x growth isn't the same as 1 second of 6x growth. Ugh. I'm not a caveman, we need to mix rate and time! (Hold onto that thought.)

*Common usage of g ^{t}*: Man-made systems. If I agree to pay you 15% at a certain discrete interval (yearly), we can model the outcome as (1.15)

Aside: Let's prove g^{t} isn't flexible.

However, this shows the special case of 2^{2} = 4^{1} does work.

Regular readers know I think of e as a continuous growth engine:

Instead of waiting to grow at discrete intervals, we apply interest immediately and compound as fast as we can. A pleasant consequence of e's definition is that we merge rate and time into a single, interchangeable quantity:

Conveniently, 2 years of 50% growth is the same as 1 year of 100% growth. We doubled our rate, halved our time, and got the same result. (Practically, we may prefer the shorter time period but the final quantity is the same.)

The input x is the "growth fuel" that can be separated into "rate * time". The base, e, is a machine that just cares how much fuel you gave it. Drip the fuel over 50 time periods, or firehose it into a single one. Either way, the same total input x gets the same final result.

*Common usage of e ^{rt}*: Natural systems. Most laws of physics have continuous growth patterns (no delay between earning interest and using it). We may occasionally use the man-made version for our convenience, e.g., describing a radioactive half life of 20 years, even though the atoms are decaying on an instant-by-instant basis.

(Aside: Use the natural log to convert one exponent format to another. g^{t} = e^{ln(g}t))

We can treat e^{x} as a fancy mathematical function:

You may see e^{x} written exp(x), treated like any other function f(x). Here, x is just a numerical input to an intricate power series. Concepts like repeated counting, growth rate, and time fall into the background (though we can see them if we look).

Curiously, we're left with integer powers (x^{0}, x^{1}, x^{2}, x^{3}) and our "repeated multiplication" interpretation shows up again! The power of the exponent, x, switches from the number of multiplications to the base being multiplied. (The *ciiiircle* of life.)

*Common usage of exp(x)*: When we see e^{x} as just another function, a few properties emerge:

- Using calculus with exponents gets way easier, since we can take the derivative / integral of each term (and realize frac(d)(dx) e
^{x}= e^{x}). - Exponential approximations become easy: e
^{x}sim 1 + x for small values of x, since the higher-order powers become negligible. - Other math patterns click. Sine and cosine have expansions similar to e
^{x}, hinting that trig functions and exponents are connected (Euler's Formula). - e
^{x}looks like a polynomial of infinite degree, and will eventually surpass any finite polynomial. (While x^{2}+ 100 > e^{x}in the beginning, e^{x}will eventually exceed it.)

You probably guessed it: it depends, though the interpretations are listed from most to least common for a general audience.

If a formula doesn't make sense, try switching versions. Life's too short to have only a single interpretation of exponents.

Happy math.

]]>Using f(ax + b) + c instead of f(x) has a few effects:

- a seems to squash the function
- b slides us left/right
- c lifts us up/down
- Interactive example on Desmos

What's going on? Well, we're describing the visual result on the graph, but aren't describing that underlying process that made the change. Let's take a look at the root cause.

The "a" in f(ax) is a fast-forward factor. Normally, we experience time as "1 minute per minute" -- for every minute we wait, the world advances one minute into the future.

What if we saw life on fast forward?

Here, 1 minute passing to us (in our "x" timeline) means 2 minutes passed in the real world. Or 10 minutes, an hour, or a year.

On our timeline (x), time passes as normal. But our function, which determines the results we see, is being fed a modified timeline. While we leisurely stroll from x=1 to x=2, f has to jump from f(1) to f(2) to f(3) ... up to f(20). Here, f needs to graph 10 minutes of events while we casually waited one minute. Cramming more data points into the same time period is a squashed, sped-up graph.

Intuitively, "Squashing the graph" really means "running time faster".

A simpler version of time travel is a basic shift. If things happen *ahead* of schedule, what does that mean?

Imagine a German/Japanese city where the trains run an hour ahead of schedule.

The 4pm train arrives at 3pm. The 11am flight takes off at 10am. In other words, a dystopian hellscape.

If I managed not to burst into flames upon arriving, I'd describe the situation like this:

- Actual time: 10am
- Flight that leaves: 11am
- Flight = actual + 1

In other words, f(x + 1) means things run ahead of schedule. We think it's 3pm (x = 3), but the 4pm events are happening [f(4) is happening].

Again, this can be tricky: doesn't it seem like we remove time to make things happen earlier? This is our visual intuition fooling us: if it's 3pm but f is running the 4pm events, it's going ahead of schedule. Note that we *add* time to our watch in order to arrive early.)

See "slide to the left" as "ahead of schedule".

Adding a value to a function moves it vertically.

What's happening? Unbeknownst to f(x), we take take the final result and make it larger.

Intuitively, we have a "bias". When f(x) = 0, it's telling us "don't change, stay at 0". Except our default value isn't zero, it's `c`

. When f(x) says don't change, that means "use your default value, c".

See "sliding up the function" as "changing the default value".

(In neural networks, you might have a default value if there's zero input. A "default bias" is a nice way to describe this, vs. "vertically sliding the function".)

In Calculus, the chain rule lets us compose functions. (Fancy phrase for cramming one function inside another.)

When we cram 2x inside of sin, and take the derivative, we get:

The chain rule tells us to take the derivative of the outer function (sin(2x) => cos(2x)) and multiply by the derivative of the inner function (2x => 2).

What's going on? Using the "derivative = slope" interpretation (not my favorite but good for graphing), we see this:

If we pick a point on the cycle (such as x = 1 radian), we find the slope there as

In other words, at x = 1, sin(x) has a nice upward slope of about 54%. Ok, great.

Now, what happens if we run sin(x) at twice the speed? Eat, eat, eat!

Well... the derivative (slope we see) should double! Compared to the original, sin(2x) runs through changes twice as fast as we do. 1 minute in our world means sin(2x) has chomped through 2 minutes of changes.

At the *corresponding point* in the cycle, we should expect double the slope. Let's make sure.

Instead of asking for x = 1, we know that same point on sin(2x)'s timeline is now x = 0.5. So let's ask for the derivative there:

Yay, the math worked!

We can mechanically describe the chain rule as a way to compose functions. But intuitively, we've strapped a fast-forward device inside our function, speeding up the changes we experience.

Regular sin(x) keeps its derivative between -1.0 and +1.0, the limits imposed by cos(x). But now we have a fast-forward trick to make sin(x) move as fast as we want.

Extra: Imagine we weren't fast-forwarding at a constant rate of 2; what if sped up more, the further along we went? That function could be sin(x * x), where one x is our regular location in the cycle, and the other x is our fast-forward rate. For fun (yes!), you can take the derivative of sin(x * x) (answer).

We can run the exponential function (e^{x}) ahead of schedule with e^{x + b}.

But we can rewrite this to:

In other words, running the exponential function "ahead of schedule" can be seen as the regular exponential function with a bigger starting point.

Normal exponentials start at 1.0 and begin compounding continuously. Instead, we can see it as starting with a bigger starting value from 1.0. (For example, e^{x + 2} starts compounding from e^{2} = 7.389.)

Depending on the function, interpretations other than "ahead of schedule" might make more sense.

Happy math.

]]>If it's not (and it's not), you know you need to stretch more. The goal isn't the splits, just some self-determined level.

My math goals are similar. I don't need expert proficiency, just a "touch your toes" understanding for the topics I care about.

Here's an internal dialog I might have to verify my understand of exponents.

**Gutcheck: Roughly speaking, what's 2 ^{100}?**

It's a large, even, positive number. (This intuition should appear almost instantly. If it takes 10 seconds of thinking to realize it's large, even, or positive, exponents aren't natural.)

**Gutcheck: Roughly speaking, what's 2 ^{-100}?**

It's a tiny, almost undetectable positive decimal. Intuition: It's like going "back in time" by 100 doublings.

**Gutcheck: Roughly speaking, what's 2 ^{i}?**

Uh oh. Imaginary exponents! With enough intuition, you realize: "It's on the unit circle, at about ln(2) ~ .693 radians."

There's a few gutchecks here. The first is that an imaginary exponent puts you on the unit circle (no matter the base). The next level is a rough "important constant" gutcheck, where you remember ln(2) ~ .693. (Not as important, but good to remember. It helps with things like the Rule of 72)

**Gutcheck: Roughly speaking, what's i ^{i}?**

Oh, here's a tricky one. Remember how we blurted out that 2^{100} was large, even, and positive? How proud we were of our quick thinking? Well, what can you say about i^{i}, hotshot?

Yikes. Realizing I couldn't instantly rattle of any properties of i^{i} meant my intuition for exponents wasn't complete. After getting an intuition for imaginary exponents, the thought becomes:

i^{i} starts as growth pointing sideways, whose direction is rotated again. It's a positive real number less than 1.0.

Phew. If I truly understand exponents, the gutcheck for 2^{100} and i^{i} should be similar in speed and detail. A painful stretch means I need more understanding.

The gutcheck process doesn't quite translate to text. These internal back-and-forths happen pre-verbally: I think of a question and quickly feel/visualize/remember an analogy. (It's a gutcheck, not a think-aloud-for-minutes check.)

Here's a few examples I run through from time to time:

Imaginary numbers: What's the cube root of -1?

- Thought: Ok, i
^{2}= -1 means we go from 1 to -1 in two steps. Getting there in 3 steps means a 60-degree rotation (180/3). Oh, we can go the other way too (-60 degrees). Oh, we can flip 180 degrees (180 + 180 + 180 = 360 + 180 = net 180 degree rotation). So there's 3 cube roots of -1.

Fourier Transform: What's the transform of `[1 0 0 0]`

?

- Thought: We want 4 equally strong frequencies (0Hz, 1Hz, 2Hz, 3Hz). They split the strength "1" between them, so we have
`[.25 .25 .25 .25]`

(using the notation in the Fourier Transform article).

Trigonometry: What's the connection between the 6 major trig functions?

Thought: I think "dome, wall, ceiling" and visualize this trigonometry diagram:

Calculus: Explain the derivative of x^{3}

- Thought: x
^{3}is really x·x·x. We have 3 perspectives, each seeing a change of x·x. The result is x^{2}+ x^{2}+ x^{2}= 3x^{2}. I also visualize a cube with plates added to it.

Bayes Theorem: What's the plain-English description?

- Thought: chance evidence is real = true positive / (true positives + false positives)

Exponents: What does discrete vs. compound exponential growth look like?

- Thought: I see continuous exponential growth as "filling in the gaps" left by discrete growth.

The key element is speed: an intuitive response should bubble as you hear the question. Struggling for an hour to touch my toes, though admirable, still means I'm not flexible enough.

The goal isn't learning minutia, it's a working understanding of an idea, enough to solve a problem without tremendous effort. It's a diagnostic, not a value judgement. If I struggle, I simply need a better intuition.

Strangely enough, not everyone wants to keep math insights top-of-mind. But pick something that's important to you and occasionally try a 5 second gutcheck on the essentials.

Happy math.

]]>- Imaginary multiplication directly rotates our position
- Imaginary exponents rotate the direction of our exponential growth; we compute our position after the sideways growth is complete

I think of imaginary multiplication as turning your map 90 degrees. East becomes North; no matter how long you drove East, now you're going North.

An imaginary exponent is like turning just the *steering wheel*. Where you will end up? Depends how long you drive!

That's the intuition, let's work through the details.

The math is straightforward when multiplying by i. We perform a 90-degree rotation around the origin, so 1 becomes 1i, 2 becomes 2i, and so on:

This is Multiplication World, where numbers are plopped on the number line then slid around (added), stretched (multiplied), shrunk (divided), and so on. Rotating a number is new, but not overly strange.

Unfortunately, the Multiplication World perspective isn't great for exponents. If we see exponents as "repeated multiplication" we're stuck when we try to count i times. It's the wrong model.

Nope -- to use i in the exponent, we need to enter Exponent World.

Here, numbers are *grown*, not simply plopped down on the number line. Every number starts at 1.0, then we run an exponential growth engine at 100% for some period of time:

- e^0 = 1.0 (no fuel, you stay the same)
- e^1 = 2.71828... (1 unit of fuel, with continuously compounding interest)
- e^2 = 7.3189... (2 units of fuel, with continuously compounding interest)

In Exponent World, a familiar number like "2" is just 1.0, grown for .693 seconds at a 100% continuous interest rate. In other words:

And in general:

e^{x} is an rocketship that pushes our numbers ever further from our starting point of 1. At t=3 we're around 20, and at t=10 we're over 20,000.

So, what happens if we drop an imaginary number into the exponent (e^{1} rightarrow e^{1i})? We keep the same amount of fuel, but rotate our engine sideways:

With regular exponential growth, we expect to speed along the real dimension. With our sideways engine, we'll need to compute what will happen.

Euler's Formula gives us the answer: constant force in a perpendicular direction creates an orbit:

and in general:

A few notes/gotchas:

**We always start at 1.0.**When seeing e = 2.718..., it's tempting to think growth starts from 2.718. But no -- when we write e^{x}we're still begin the growth process at 1.0 (e^{0}= 1 implies no change from 1.0).I try to remember we can swap e for its official definition at any time:

The only starting point we see in the definition is 1.

**Every number orbits at a radius of 1.0.**In exponent world, every number is grown from 1.0, just with varying amounts of fuel. When we put the engine sideways, the orbits are at 1.0, for varying distances around the circle.**The orbit doesn't get faster.**Regular exponential growth has runaway compounding because our changes accumulate in the same direction. With sideways growth, changes don't accumulate (always in a new direction) and we spin at a constant speed. In other words, e^{10}is thousands of times larger than e^{1}, but e^{10i}is only ten times around the circle compared to e^{i}.

So, a setup like 2^{i} tells us to use .693 units of fuel in a sideways direction:

To get the coordinates for our final position, we see how far ln(2) = .693 units of fuel takes us around the unit circle:

Phew! Working out 2^{i} (rotated exponential growth) is much trickier than 2i (a simple rotation).

Now let's get tricky. What is i^{i}?

Remember, we're in Exponent World, and even i is something we had to grow to! In other words, we start at 1.0 and orbit a quarter of the way around the circle (90 degrees, or frac(π)(2) radians).

Whoa. Don't like how i appears in its own exponential definition? It must also bother you that every word in the dictionary is defined by other words.

Coming back to i^{i}, we have two operations.

- The bottom i (in the base) is shorthand for running our engine sideways, with frac(π)(2) units of fuel
- The top i (in the exponent) says "Nah, spin that engine one more time"
- The engine is facing 180 degrees (backwards on the real axis) with frac(π)(2) units of fuel

Ah. The result is

And we end up with a real number. Intuitively, we can roughly predict this because we start at 1.0 and point the engine backwards on the real axis. Once you can mentally estimate the direction i^{i} goes, in seconds, you've got a sold intuition on imaginary exponents.

Happy math.

Just for fun, what about i · i^{i}?

We know that i^{i} is a purely real number smaller than 1.0. The first i (doing the multiplication) will just rotate us, so now we have a purely imaginary number smaller than 1i.

Not bad! Work through the exponents, then rotate the final position.

From a physics perspective, if f(x) = e^{ix} is our position, then f'(x) is our velocity. Working this out we get:

In other words, our velocity is perpendicular to our position. Taking the derivative of e^{ix} might seem weird, but treat i like any other constant: frac(d)(dx) e^{ax} = ae^{ax}.