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.

Meaning 1: Repeated Multiplication

We first learn that exponents like or are repeated multiplication: multiply , times.

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

Common usage of : Counting problems. If you flip a coin times, you have possible outcomes.

Meaning 2: Growth Microwave

Let's say I have an exponent like . I mentally convert it to , and then .

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

What values can have?

• If t is positive, we go forward in time and get larger (assuming ). 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 : Man-made systems. If I agree to pay you 15% at a certain discrete interval (yearly), we can model the outcome as . If I decide to cut the payments short (2.5 years) we can exponentially interpolate between the two intervals ( is the square root). We often set , so we could write .

Aside: Let's prove isn't flexible.

However, this shows the special case of does work.

Meaning 3: Continuous Growth Engine

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 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 gets the same final result.

Common usage of : 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. )

Meaning 4: A Power Series Calculation

We can treat as a fancy mathematical function:

You may see written , treated like any other function . Here, 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 () and our "repeated multiplication" interpretation shows up again! The power of the exponent, , switches from the number of multiplications to the base being multiplied. (The ciiiircle of life.)

Common usage of : When we see 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 ).
• Exponential approximations become easy: for small values of , since the higher-order powers become negligible.
• Other math patterns click. Sine and cosine have expansions similar to , hinting that trig functions and exponents are connected (Euler's Formula).
• looks like a polynomial of infinite degree, and will eventually surpass any finite polynomial. (While in the beginning, will eventually exceed it.)

So which version of exponents is best?

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.

First off, e was discovered, not chosen. Think of the speed of light, c. It wasn't originally decided to be 299,792,458 m/s -- we did experiments and realized under ideal, universal conditions (a vacuum), this was the fastest light could move¹.

Let's look at growth and ask under ideal, universal conditions, what's the fastest something can possibly grow? Ideal, universal assumptions would be:

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

Turning these assumptions into a formula, we get:

If we actually use the formula (using large values of n for more accuracy) we get e = 2.718281828459...

Objection: But can model exponential growth just like 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, 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 (in discrete systems) or (in continuous systems) are "universal".

Objection: But things can grow faster than , which is just -- what about ?

What is it with you and 13.74? Yes, you can beat in an exponential footrace, if you use a rate more than 100%. is really . Because ln(13.74) ~ 2.6, you are assuming a 260% continuous interest rate, more than the 100% uses. (Alternatively, you can grow for 260% of the unit time period that uses.)

Related:

• Intuitive guide to e
• Common definitions of e (colorized)

The key intuition: e represents 100% continuous growth.

Today let's revisit each definition with a colorization viewpoint, describing continuous growth from a few different perspectives.

Definition 1: Compound Interest Perfectly

This definition of e was my starting point on understanding the concept. We start with 1 growing to 2 (100% interest), and then compound that growth more and more frequently.

Eventually, we see that 1 grows to 2.71828..., hitting a speed limit of e.

The trick is distinguishing the role of each "1" in the definition. One is the base quantity, one is the interest, and another is the implicit single unit of time we plan to grow for. Math is so abstract that we don't have these separations labeled individually, they are just quantities interacting.

Definition 2: Track Each Interest Contribution

This definition splits up the compounding process into chunks we can see separately. I like to see each component like a "factory" that is earning money. We start with our initial amount, which builds interest. That interest builds its own interest, which builds its own interest, and so on (read more).

From a calculus perspective, here's what's happening:

• Our initial quantity is 1 (for all time)
• This principal earns 100% continuous interest, and after time has earned . (After 1 unit of time, this is 1)
• After time x, that interest () has earned . (After 1 unit of time, this is )
• After time x, that interest () has earned (After 1 unit of time, this is )

And so on. Every instant, the entire chain of interest is growing. When learning calculus, you might have repeatedly tried to integrate just for fun (whatever gets you going). That game is how we end up with .

Instead of plugging in to compute , we can leave unspecified to handle any amount of time:

I like seeing how each piece of interest contributes to the whole. Later terms have larger powers, but fight a larger factorial. For very small values of (like 1%), we can approximate as:

For example, earning 1% continuous interest for a single year is still around 1.01, because there isn't much growth from compounding. (And yep, .)

Definition 3: Maintain 100% Continuous Growth

This is the shortest definition, but relies on calculus machinery. In short, we're saying that always changes by exactly the amount that we have.

The derivative () measures instantaneous change:

And we say is that input that makes this machine return the original value. (In a similar way, "0" is the input that makes addition return the original; 1.0 is the input that makes multiplication return the original value.)

We can check this works. We saw earlier that is really this equation:

If you take the derivative each term in the right-hand side we get:

which simplifies to

In other words, every term gets "pulled over" to the left, with the constant 1.0 disappearing (it doesn't change). We have our original pattern, therefore is its own derivative. (There's no +C chicanery here, because ).

While other functions like or may momentarily equal their derivative at certain instants, they don't keep it up for all values of . is that Disneyland ride that keeps the magic going forever.

Definition 4: Define e using the natural log

This definition is the most gnarly: instead of talking about e directly, we work backwards.

Define the natural logarithm as the time needed to grow from 1 to (our desired number), assuming 100% continuous interest. What does that look like?

Let's say we've grown to 4.0. How long to grow to 5.0? Well, assuming 100% interest, we grow 4 units per time period, so it takes us 1/4 of a unit to grow to 5.

And when we're at 5, it'll take us 1/5 of a unit to get to 6.

And so on. The time to grow from to 1 to a is the time from 1 to 2 (1 unit), plus 2 to 3 (1/2 unit), plus 3 to 4 (1/3 unit), until we reach .

(In reality, we need to break time down microscopically, growing from 1, to 1.1, to 1.2, etc. That's what the integral really does.)

Phew! Once we have the notion of "time to grow from 1 to a" defined, we say e is the number that takes 1 unit of time to reach. In other words:

Here, e is the "base of the natural logarithm".

Trace the similarities

I feel comfortable with an idea when I can hop between definitions and notice similarities. For example, look at the items that show up in each colorization: do you see where "interest" shows up in each definition? The unit quantity? The idea of perfection or infinitely precise change?

It feels great when e becomes a flexible tool on your bat-belt and not an incantation to memorize.

Happy math.

• Discrete growth: change happens at specific intervals
• Continuous growth: change happens at every instant

Here's the difference:

The key question: When does growth happen?

With discrete growth, we can see change happening after a specific event. We flip a coin and get new possibilities. We have a yearly interest payment. A mating season finishes and offspring are born.

With continuous growth, change is always happening. We can't point to an event and say "It changed here". The pattern is always in motion (radioactive decay, a bacteria colony, or perfectly compounded interest).

(Brush up on the number e and the natural logarithm.)

Insight: Convert between discrete and continuous

I visualize change as events along a timeline:

Discrete changes happen as distinct green blobs. We can take them, split them into smaller, more frequent changes, and spread them out. With enough splits, we could have smooth, continuous change.

So, discrete changes can be modeled by some equivalent, smooth curve. What does it look like?

The natural log finds the continuous rate behind a result. In our case, we grew from 1 to 2, which means our continuous growth rate was ln(2/1) = .693 = 69.3%. The natural log works on the ratio between the new and old value: .

Mathematically,

In other words: 100% discrete growth (doubling every period) has the same effect as 69.3% continuous growth. (Continuous growth requires a smaller rate because of compounding.)

So which version do we use?

Now here's the question: how should we talk about growth? It depends on the scenario:

• If growth happens in a man-made system, discrete growth works better (, )
• If growth occurs a natural system, continuous growth is better ()

Let's take a look.

Example: Flipping Coins

Let's say we flip a coin. What are the possible outcomes?

• 1 flip: 2 outcomes (H or T)
• 2 flips: 4 outcomes (HH, HT, TH, TT)
• 3 flips: 8 outcomes (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT)

You see where this is going. I'd describe the number of possibilities as where n was the number of flips.

I'm using "n" (not x) by convention: x could mean any value on the x-axis (-3, 1.234, ), while n represents an integer (1, 2, 3, 4).

Could we say the number of outcomes was , where x was the number of coin flips? Yes. But it's confusing: in a man-made system, where we have change events, I'd use the discrete version to describe the possibilities.

Example: Binary Numbers

Binary numbers follow the same pattern: if we have n bits, we get possibilities. For example, 8 bits have 256 possible values, and 16 bits have 65536.

(There may be some cases where intermediate values make sense, like representing the number of bits required, even though we need a whole number of bits in practice. This is similar to saying the average family has 2.3 kids.)

Example: Radioactive Decay

When radioactive material decays, we often talk about its half-life: how long until half the material is gone?

For example, the half-life of Carbon-14 is 5700 years. We could write it like this:

If we wait 5700 years, we expect of the carbon remaining. If we double that and wait 11,400 years, we'd expect of the carbon left.

However, this equation is written for our convenience. Carbon doesn't decay in jumps, politely waiting around 5700 years and suddenly decaying by half. We use (1/2) as the base because we humans want to count the number of halvings (decaying into half, decaying into a quarter, decaying into an eighth...).

The radioactive material is changing every instant. From a physics perspective, a continuous rate is more telling. We can find the continuous decay rate by converting the discrete growth into a continuous pattern:

This helps me understand why the natural log is natural -- it's describing what nature is doing on an instant-by-instant basis. None of this "wait until we decay by 50% so humans can count it easier" nonsense.

In practice, you don't discover the half-life by waiting for carbon to decay 50%. You'd wait a reasonable about of time (a year?), use the natural log to find the continuous rate over that period, and work out the half life.

Example: Material X decayed from 53kg to 37kg over 9 months. What's the continuous decay rate and half life (in years)?

The ratio between new and old was 37/53, so ln(37/53) = -.359 = -35.9% continuous growth over our time period. This happened over 9 months, so the monthly continuous rate is -35.9/9 = -3.98%. Scaling this up, the yearly continuous rate is -3.98% * 12 = -47.9%. (Notice how the rate must be scaled to match the time period. Earning "12% interest" isn't helpful without a time period. "12% interest per day" is different than "12% interest per year".)

Now that we know the continuous rate is -47.9% per year, we can work out how long until we're at 50%:

The half-life is 1.44 years.

Example: Stock Market Growth

This is a tricky one: the stock market changes every day, so it seems like it'd continuous, but there isn't an underlying predictable rate. We see a lot of jumpy changes, and sample them at yearly intervals to see how we're doing. The market is usually described with an annual average growth rate:

A continuous rate of the form doesn't really make sense for the system. We aren't trying to model our portfolio's value on a per-instant basis: we want to know what to expect in 30 years.

Example: Population Growth

Population is tricky: depending on the animal, discrete or continuous model can make sense.

A bacteria colony is made of billions of organisms. Although each bacteria cell grows discretely (it has to wait until it splits before splitting again), the entire colony grows smoothly because so many bacteria are in different stages of growth.

Like the radioactive decay example, we can sample the colony at different time periods and work out how long it takes to double. We might have a continuous rate () that expresses the colony's instant rate, and a discrete rate () that helps us humans count the doublings.

One of my pet peeves were problems like "A bacteria colony doubles after 24 hours...". Argh! Are you telling me the bacteria colony just happens to have a continuous rate of precisely ln(2) over the course of a day?

I'd prefer you told me the colony doubled while a grad student stared at a petri dish for 24 hours straight. (1.98kg... 1.99kg... 2.00kg. I found the doubling time, I can go home! What's that Professor? I...ok, I'll work out the continuous rate after an hour next time.)

Rant aside, how about modeling a tiger population? Tigers have breeding seasons. They aren't having kids throughout the year, so the population changes in a discrete event.

(The model gets more complex as you account for how long it takes for cubs to have children of their own.)

Onward and Upward

I wrote this post because my video on e had questions about how represents "staircase growth". Isn't that a smooth curve too?

Sure, but most of the time we use 2 as a base to model discrete patterns. (where n is an integer) models discrete scenarios like coin flips or binary digits. If your system does change continuously, why not provide the continuous rate and write ?

There's no right and wrong here, just the message we convey. A whole-number base (, ) implies you want people to think about whole-number values of x (and half-life is a good example). Using as a base () implies you want people to think about change that happens at every moment.

Either way, be fluent in both models and learn to hop between the two.

Happy math.