Compounding:
Source of Exponential Growth and e

Year to Year

You work hard for your money.

And you want your money to work hard as well.

Simple interest doesn’t cut it.

The interest you earn, contributes nothing to future returns–it just sits there, idle.

The bank offers you a savings account.

No withdrawals for five years.
(Terms and conditions chosen for my convenience)

The word compounding comes from the Latin componere, meaning “to put together”. It means any interest earned is combined with your current balance, and this new amount is used to calculate future interest payments.

What does this look like?

You deposit $1,000, which becomes your “base” for interest calculations.

The base is just the money you have in the account at any time.

The bank calculates your interest at the end of the year and pays it into your account.

The interest you earn is added to your base—there is no distinction between “principal” and “interest” like with simple interest.

A simple formula in words: Base + Annual Interest = new Base

Let’s see how the bank applies this formula for the first two years:

1. End of Year 1 You started with $1,000. The bank calculates 10% of $1,000: $100 interest. Your account now has $1,100.

2. End of Year 2 Your starting base was $1,100. The bank calculates 10% of $1,100: $110 interest. Your account now has $1,210.

This creates a new pattern of growth.

Unlike the steady steps of simple interest, your balance growth now accelerates.

The growth itself grows, each step up is bigger than the last—which you can see in this chart:

Your final balance is $1,610.51, which is $110.51 more than the $1,500 you would have had with simple interest.

What happened here?

The interest you earned was added to your base, increasing your account balance.

And, unlike with simple interest, your entire base works to produce interest.

If your base gets bigger, you earn more interest.

It’s a financial snowball effect: the interest gets rolled back in, making the base bigger, generating more interest the next time.

This is the power of compounding.

The more you have, the more you get: this is the mathematics behind “the rich get richer”.

But it cuts both ways.

Your debt compounds the same way your savings do, which is why the poor fall deeper into debt.

The more they owe, the faster their debt grows.

You want compounding working fully for you.

You see that the base is only updated once a year—the year’s interest paid as a lump sum.

If the bank paid your interest out during the year, then your base would grow faster and you’d earn more interest.

Right now, your base just sits there all year—interest is only paid at the very end.

It’s a minute to midnight,
on New Year’s Eve, and your money has done nothing all year!

Totally understandable.

Okay, let’s make this more interesting.

Monthly Compounding

Your bank advertises 10% interest per year, called the annual percentage rate (APR).

But rather than pay you once per year, they offer to pay interest every month.

(Not daily like you wanted, but better than once a year)

Notice what’s happening here.

The bank is:

• stating a simple yearly rate,
• but compounding monthly.

A time gap now exists between the annual rate and the monthly change in your money.

If you are paid monthly, your money grows monthly, regardless if the rate is an annual one.

The MPR is the periodic rate that your money grows each month—and the bank needs to calculate it.

(The bar over the 3, called a vinculum, means it repeats forever in decimal format, i.e. 0.83333…)

The convention for calculating the Monthly Percentage Rate (MPR) is simple:

• the bank takes the 10% APR and
• divides it by the number of payment periods (12 months).

This gives a simple calculation: 10% APR / 12 months = 0.83 % MPR

This 0.83% rate is applied to your base every month. On your first $1,000, that’s an interest payment of $8.33 at end of January, rounding down.

I said this MPR calculation is a convention, a standard way of doing things—that doesn’t mean it’s the correct way to calculate the MPR.

The MPR calculation is correct—if you assume no compounding. If there were no compounding, after 12 months, you’d earn: $8.33 / month × 12 months = $100.00 And $100 interest is what you expect for a 10% APR on a $1,000 deposit: $1,000 × 10% = $100

But there is compounding—monthly compounding.

The bank’s MPR calculation is a simple rate applied to a compounding account!

It’s assuming the base doesn’t change, that:

• you can just apply the MPR of 0.83% each month, and
• end up with a 10% APR ($100) at the end.

For example, the first $8.33 payment is immediately rolled back into your base.

Now the February base is $1,008.33, so the next 0.83% is calculated on this new, larger amount.

Your money now grows in twelve distinct “jumps” or steps, each slightly larger than the last.

Okay! Look at the monthly interest payments in this table, and see the increase for yourself:

Visualizing this growth shows a “staircase” effect, as your money jumps, rests, then jumps again:

Note: There is no intra-period growth or “interest on interest” happening between payments: the base is at rest for the entire month between payments.

No, not magic—your money compounded, but the bank didn’t account for it.

After 12 months, you have $1,104.71, which is $4.71 more than expected!

The bank stated a 10% APR, which gives a nominal total of $1,100: $1,000 + 10% = $1,100 where nominal means it’s just stated for convenience.

But compounding monthly gave you an effective annual percentage yield (APY) of 10.47%, which means the bank paid out slightly more than they should, when you saved money with them.

How lovely of them!

Truly they have hearts of gold these bankers, paying out more than you agreed.

What a wonderful deal! Compounding working just for you!

But, yes—your suspicions are justified.

Banks are well aware of this discrepancy, and use it to their advantage when lending money.

They’ll quote you a loan using the annual percentage rate (APR), but you’ll pay interest based on the annual percentage yield (APY), which is larger than the APR.

Caveat debtor!

But now, let’s see what happens when we compound daily.

Daily Compounding

Then daily compounding you will have!

This frequency is common for both:

• high-yield savings accounts and
• credit card debt.

(Even if the interest is paid or charged monthly )

To make my calculations easier, the bank decides:

Once again, the 10% APR is a simple, nominal total for the year—it assumes no compounding.

But your “base” of money grows daily, based on this nominal 10% APR.

Same mechanism as always: base + interest = new base but done 365 times per year.

I showed you monthly compounding tables above: now, I’ll show you how to calculate these tables for any period of time.

The bank divides the nominal 10% APR by 365 days, to get the daily percentage rate (DPR): 10% ÷ 365 ≈ 0.0274% This means your base increases by 0.0274% each day, which is about $\frac{1}{40}$^th of 1%.

True, it seems puny—but maybe increasing 365 times in a row will give a great result?

We’ll see.

Note, while percentages are great for quoting rates like APR, they’re harder to calculate with than decimal numbers.

Calculating with percentages:

• adds more steps, and
• is more prone to error.

So, they’re ditched in favor of decimal numbers when actually calculating interest payments.

To increase by a DPR of 0.0274% is to increase by a decimal growth factor of 1.000274:

Not trying to pull anything!

I’ll explain this conversion, step by step, in “The Base Formula” below.

What matters now, for compounding, is the base is multiplied by this decimal growth factor: base × growth factor = new base and this is done every day, for a year.

So, starting with $1,000:

Compared with the other compounding periods, that’s:

• $5.16 more than annual ($1,100), but
• only 45¢ more than monthly ($1,104.71).

No, not a scam.

Something unexpected, for sure.

A disappointing discovery—but a discovery, nonetheless.

We compounded 30 times faster going from months to days, but all we gained was a measly 45¢.

What gives?

Notice the pattern:

• Annual: $1,100
• Monthly: $1,104.71
• Daily: $1,105.16

The gains from more frequent compounding are shrinking. We added $4.71 going from annual to monthly, but only 45¢ going from monthly to daily.

Still, shortening the time period did give you slightly more money.

I want the most money, the fastest growth possible!

Most understandable!

How fast can we go?

Hour-by-hour?

Minute-by-minute?

A split-second?

A nano-second?

There must be a limit.

What is it?

Diminishing Returns

The faster we compounded (from monthly to daily) the more money we made within the year: from $1,104.71 to $1,105.16

Can we make even more money if we compound faster?

What if we compound every hour instead of daily?

With 8,760 hours in a year:

• Hourly rate: 10% ÷ 8,760 = 0.001141%.
• After 1 year: $1,105.170287...

Every minute?

• Minute rate: 10% ÷ 525,600 = 0.00001903%.
• After 1 year: $1,105.170907...

Every second?

• After 1 year: $1,105.170918...

Huh.

Seems your gains are getting smaller and smaller, even though you are compounding faster and faster.

Sorry, but it doesn’t work that way.

The faster you compound your money within the year, the less additional money you get.

There seems to be a limit to the possible gains of compounding, which we discovered the hard way.

No, there really is a limit—a mathematical limit.

But it’s a limit that I cannot explain in words–it must be seen or calculated to understand.

Look at this animation and see for yourself, which shows 3 forms of compounding growth:

• Monthly: 12 jumps and rest.
• Daily: 365 jumps and rests.
• Continuous: No rests, just constant growth.

Periodic compounding (monthly and daily) means:

• Compounding happens at specific periods of time.
• Between the periods, your money sits at rest.

Continuous compounding means:

• Your money grows continually—no rest periods!

(Mathematicians say the number of periods, n, is “infinite”; I prefer to say there are no rest periods)

You can see that continuous compounding has the greatest growth—but the other two are not that far behind, are they?

This is why there are diminishing returns as you compound faster: even monthly compounding is close to the maximum possible growth, so increasing to daily doesn’t get you that much extra.

The “limit” shown on the graph is one that even continuous compounding cannot exceed.

It is set by the APR (10%), a rate that does not account for compounding, but does put a “ceiling” on what can be earned in a year—even with compounding.

Yes, your base grows during the year, but it does not grow infinitely.

At each payment period, the new base is set to grow by 10% in a full year, from that point in time.

Compounding growth is not infinite growth, merely “growth that grows” as time goes on.

And the absolute limit of compounding growth is continuous compounding at the given rate of interest.

This notion of a limit, a value that cannot be exceeded, is not an obvious concept.

Not obvious at all.

People have set interest rates for thousands of years—since the dawn of civilization—but this limit to compound growth was only first discovered in the 17^th century.

And it was discovered via a calculation on compounded interest payments.

A calculation, and I’ll show you the details next.

See for yourself.

The Base Formula

Not by hand, that’s for sure!

Adding interest paid by the second takes far too much time—even with a calculator.

There is a formula that gives us the total compounded yield.

(I’ve been using to calculate all the totals above)

No, I can show you how it is derived, no tricks.

We’ll use the “base” framework as our guide, the framework we have been using all along.

Recall: we focus only on the growing base of money, we do not consider “interest on interest” or anything like that.

Banks will say that your money, the “base”, grows when you add interest at each payment period: Base + Interest Payment = New Base And while this is correct in principal, we are not going to look at the overall growth of your money as repeated additions, i.e. adding a fixed percentage of interest at each payment period, e.g. $1,000 + 10% = $1,100

Rather, we are going to consider growth as repeated multiplications, i.e. multiplying the base by a fixed factor at each payment period, e.g. $1,000 × 1.1 = $1,100

While addition is easy,

Read why percentages are such a pain for calculations in the footnotes†.

Banks add interest at each payment period, yes, but calculating total growth is always done by multiplication——because we’re scaling the base repeatedly, not adding fixed amounts.

Actually, we can describe any repeated addition as a multiplication: 2 + 2 + 2 = 6 = 2 × 3

So, while the individual “steps” might be additions of interest, their total is found by multiplication, a mathematically legitimate approach.

Using multiplication at each step, and not just to find the total, fits the “base” framework better—since we are always growing in terms of the base at each payment period.

When multiplying, we say we are scaling the base, where “scaling” means “changing the size”.

• Scaling up: multiply by a number greater than 1 (get more than you started with), e.g. 10 × 1.5 = 15
• No change: multiply by 1 (stay the same), e.g. 10 × 1 = 10
• Scaling down: multiply by a number less than 1 (get less than you started with), e.g. 10 × 0.5 = 5

Okay, given:

• an APR of 10% and
• monthly compounding (n = 12)

we can calculate a monthly percentage rate (MPR) of: 10% / 12 = 0.83%

But rather than adding this to our base ($1,000), we can multiply by a decimal growth factor:

And this is how we calculate our new base from the current base, at each payment period: base × 1.0083 = new base

We just keep multiplying by this decimal growth factor, month after month, to find our compounded total in a year:

Given 12 months in 1 year, we have 12 multiplications. There is a shorthand for repeated multiplication, called exponential notation.

We put a superscript “12” above one instance of the decimal growth factor to indicate “we do this 12 times”:

Yes, that’s all there is to it.

This type of multiplicative growth is called exponential growth, and it means the rate of growth depends on how much you currently have: the more you have, the faster you grow etc.

We need to abstract this formula: it’s specific to one case, and we want to apply it to any case of periodic compounding.

In banking terminology, our initial base is called the “Principal”: Principal × (1.0083)¹²  ­= 1-year Amount and we can replace $1,104.71 with “1-year Amount” for now, since we haven’t extended this formula to handle multiple years—yet.

We can substitute P for “Principal” and save some space: P × (1.0083)¹²  ­= 1-year Amount

Now, we can swap n for 12, where n means “the number of payment periods”: P × (1.0083)ⁿ  ­= 1-year Amount

The growth factor is more complicated, but can be broken down into:

• 1 (the original amount) and
• 0.0083 (the gain)

giving us: P × (1 + 0.0083)ⁿ  ­= 1-year Amount

The gain is given by the APR divided by the number of periods, n: P × (1 + $\frac{\text{APR}}{\mathrm{n}}$  )ⁿ  ­= 1-year Amount where n is the number of times that interest is paid within the year.

We can visualize this grouping of 12 payment periods into one year as follows:

We group these payment periods into years, as our total rate of growth is specified as an “annual” rate (APR). We could use any time period we want, but the “year” is a basic unit of time for financial affairs.

To emphasize, our periodic compounded growth gives an annual multiplier: P × (1 + $\frac{\text{APR}}{\mathrm{n}}$ ) ⁿ  ­= 1-year Amount written as “annual m.”, which is the amount our initial principal is multiplied by after one year: (1 + $\frac{\text{APR}}{\mathrm{n}}$ ) ⁿ Rather than an annual percentage rate (APR), let’s just call it a “rate”, r, to make it general: P × (1 + $\frac{\mathrm{r}}{\mathrm{n}}$ ) ⁿ = 1-year Amount

Now we have a general formula to calculate total compounded growth after one year.

Agreed! This formula must work across years—not just for one year.

Say the bank were (very generously) doubling our principal in a year, but only paying out once per year. P × 2 = 1-year Amount

Our money is compounding year-to-year, but not within the year.

How would we calculate our total after two years? We’d just double again, wouldn’t we? P × 2 × 2 = 2-Year Amount giving us 4 times our principal after 2 years, which is 2².

You can see the pattern here, one of repeated multiplication from year to year: P × 2 × 2 × 2 = P × 2³ which we can write using exponential notation.

Now, we are talking about years, but this could be any time period—years just make sense for the world of human finances.

We’ll stick with years for practical purposes, and say we want to grow for a time period of t years.

Our annual multiplier here is “2”, so we swap in “annual m.” to get a general, multi-year formula:

We can write this in exponential format: P × (annual m.)^t = Total Amount and write A for “Total Amount”: P × (annual m.)^t = A

Yes, this is all very detailed and tedious, but we’re nearly there.

We have two formulae:

• Compounded total after 1 year, for n payment periods within the year: P × (1 + $\frac{\mathrm{r}}{\mathrm{n}}$ )ⁿ = 1-year Amount

• Compounded total for t years, no payment periods within the years:

  P × (annual m.)^t = A

We must stitch them together somehow, and you can probably see where the “stitch” will happen.

Just swap in the actual annual multiplier (“annual m.”) value from the first equation: P × (1 + $\frac{\mathrm{r}}{\mathrm{n}}$ )ⁿ = 1-year Amount into the “annual m.” placeholder in the second equation: P × (annual m.)^t = A to get the following: P × ((1 + $\frac{\mathrm{r}}{\mathrm{n}}$ )ⁿ)^t = A

This formula represents:

• n periods of compounding,
• grouped into years, and
• applied for t years.

This formula is a composite of the two formulae we discussed above, and can be represented visually like this:

Right now, the formula is a little ugly, with the double set of round brackets cluttering things up: P × ((1 + $\frac{\mathrm{r}}{\mathrm{n}}$ )ⁿ )^t = A

But we can use a basic rule of exponents to tidy it up.

Given: (2²)² we can multiply the exponents to simplify: (2²)² = 2^{2 × 2}
= 2⁴
= 16

Leaving us with this simple formula, which calculates the total amount of money you’ll earn after compounding for t years, with n payment periods per year: P × (1 + $\frac{\mathrm{r}}{\mathrm{n}}$ ) ^nt = A

Note: n is the number of payment periods within each time period, but t is the number of time periods.

No need for blind trust—I can show you, directly.

We’ll use our earlier case of monthly compounding:

• Principal, P = $1,000
• An APR of 10%, so r = 0.1 (in decimal)
• Number of payment periods, n = 12 (monthly)
• For one year, so t = 1.

Given our formula: P × (1 + $\frac{\mathrm{r}}{\mathrm{n}}$ ) ^nt = A We just swap in the relevant values: $1,000 × (1 + $\frac{0.1}{12}$ )^{12 × 1} = $1,000 × (1 + 0.0083)¹² = $1,104.71 which is exactly what you get if you multiply $1,000 by 1.0083, 12 times in a row, since the exponent 12 means “repeat this multiplication 12 times”.

This formula is a mathematical shorthand for the repeated multiplications of periodic compounding.

And it will lead us to the limit of this process, the source of our diminishing returns: e.

An Interesting Discovery

We have a way to calculate the total amount from periodic compounding: P × (1 + $\frac{\mathrm{r}}{\mathrm{n}}$ ) ^nt = A

Where:

• P, the principal, the initial “base” of money.
• r, the total rate of growth in a year (the APR).
• n, the number of payment periods in a year.
• t, the number of full years you want to calculate for.
• A, the final amount of money.

For our starting base of $1,000 and annual growth rate of 0.1 (10%), the compounded totals as the number of payment periods, n, increase:

We’re approaching a limit of $1,105.17—the value that periodic compounding converges to, as we move towards increasingly shorter payment periods.

And by “converges” I means “approaches a definite value”.

A great question!

While his private notes reveal his early investigations into compound interest, his first work on the topic was published in 1690, titled “Certain Questions on Usury” (Latin: Quæstiones nonnullæ de usuris). ^[1]

And he found something very interesting as n becomes extremely large—as it “tends towards infinity”, as the mathematicians say.

Now, mathematicians always want their work to be as generalizable as possible, so he didn’t use an initial deposit of $1,000 and an APR of 10%, like I’ve been using.

Instead, Bernoulli used an initial “unit of principal” as his starting base and applied “unit growth” to it.

That is, he set:

• P = 1
• r = 1 (i.e. 100%)
• t = 1

which means the initial base grows by 100% in one year, i.e it doubles.

In other words, he looked at a base of $1 that doubled in a year, using simple interest, and then used the formula to calculate the compounded total amount, A:  1  × (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ = A Since multiplying by 1 doesn’t change anything, we can drop the ‘1’ that is P from our formula: (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ = A Next, Bernoulli used this simplified formula to see what happened to the total amount, A, as n became increasingly large.

Note: You can read in the footnotes* why “unit growth” means: P = 1 = r

His results followed the same pattern of diminishing returns we saw, soon hitting an upper limit to periodic compound growth:

Where the upper limit on his total was ≈2.718 times the original unit of base.

Bernoulli didn’t calculate the limit precisely, but he proved it existed and that it lay between 2$\frac{1}{40}$ and 3: 2$\frac{1}{40}$ < (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ < 3

We can describe the results of his process as follows:

• The limit: $lim$

• As n tends to infinity (∞): $\underset{\mathrm{n}\to \mathrm{\infty }}{lim}$

• Of unit periodic compounding: $\underset{\mathrm{n}\to \mathrm{\infty }}{lim}$(1 + $\frac{1}{\mathrm{n}}$ ) ⁿ

• Is e: $\underset{\mathrm{n}\to \mathrm{\infty }}{lim}$(1 + $\frac{1}{\mathrm{n}}$ ) ⁿ = e

It’s the limit that  (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ  approaches as n grows arbitrarily large (as large as you like).

This is how it was first discovered, and it is how it is defined: one of the most important constants in science and mathematics, first revealed in a financial problem.

Bernoulli spent several years investigating compound interest, but never gave his limit a name. A later mathematician, Leonhard Euler, did: e.

Writing in the 1730s, Euler didn’t name it after himself—“e for Euler”—but because a, b, c and d were already in use.

a, b… c, d e me… exponential.

He also didn’t pick ‘e’ because of the word “exponential”: the association between the number e and exponential growth came after Euler.

Note: Euler wrote ‘e’ in italics (slanted), e, rather than in roman (upright) type, e, as I have been doing. Why?
The traditional convention in mathematics is to write both variables (placeholders) and constants (fixed numbers) in italic. The modern international standard ISO 80000-2 specifies that ‘e’ be considered a constant (since its value is fixed) and be written in roman type as ‘e’.

In his 1748 work Introductio in analysin infinitorum, Euler used an infinite sum to calculate e to 23 decimal places—a stunning achievement at the time! ^[3]

2.71828182845904523536028…

Euler studied e in depth, revealing a range of fascinating properties. By the late 19^th century, mathematicians honored these contributions by naming the number after him: “Euler’s number”.

The specific numerical value of e (≈ 2.718...) comes from choosing a unit principal (P = 1), and a unit rate (r = 1, i.e. 100%).

But this value is much less important than the conceptual limit it represents: the maximum possible growth of our principal, in a year, if our interest rate is 100% and we compound continuously.

Periodic compounding, at rate r = 1, is described by: P × (1 + $\frac{1}{\mathrm{n}}$ ) ^nt = A which we can group together like this, with square brackets ([ ]): P × [ (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ ]^t = A This makes it easier to see what happens when we take n to the limit (∞), making the process continuous, rather then periodic: P × [ (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ ]^t = A

Continuous compounding over one year, at rate r = 1, is defined by: P × e^t = A since e is the limit of the expression: (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ when n approaches infinity.

And by “expression” I mean “something that produces a number”, e.g. ‘2 + 2’ is an expression that produces ‘4’, once you calculate it.

No faster sequence of payments can increase the total over one year—the process has become perfectly continuous, increasing at every instant.

But how can we describe our original APR of 10%, r = 0.1, in these terms?

That’s what we’ll explore next.

Origins of e^rt

There is a limit to compounded growth.

We’ve physically seen it.

Over a single year, with an APR of 100% (r = 1), that limit is e (≈ 2.718) times the original amount.

This relation was discovered by Jacob Bernoulli, ^[1] who published his initial findings in 1690: P × e^t = A Where:

• P, the principal, the initial “base” of money.
• e, Euler’s number, a constant ≈ 2.718.
• t, the number of full years you want to calculate for.
• A, the final amount of money.

This relation between variables is called a formula, and a formula allows us to calculate one variable (A) in terms of other things (P, e^t )

Again, this formula assumes “unit growth”, r = 1, i.e. doubling (APR of 100%) in a year.

What about other growth rates?

Like an APR of 10% (used throughout this article), rather than 100%?

Do we have to define a new limit number for every case?

That doesn’t sound practical at all.

Yes, this is the way!

How could we do this?

We have a formula for continuous compounded growth, at rate r = 1, defined by: P × e^t = A where e is the limit of the expression: (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ when n approaches infinity.

For an APR of 10%, we need: r = 0.1 We have $\frac{1}{\mathrm{n}}$ in the expression that defines e: (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ which means we are close except for that rate of ‘1’, rather than 0.1.

For any algebraic equation, you can substitute a variable with another variable—since a variable is just a placeholder for a value.

(A “value” just means “a number”)

No, by substituting variables, we are suiting ourselves—by extending our continuous compounding formula to handle any rate of growth, not just unit growth (r = 1).

Consider converting between dollars ($) and cents (¢): $1 = 100¢ So, whenever you have an equation with dollars 3 × $1 = $3 You can swap in 100¢ and preserve the established relations: 3 × 100¢ = 300¢ giving the same result, but written in a different unit (cents versus dollars). It’s a like-for-like swap.

Starting with our original definition for e, which assumes 100% growth (r = 1): (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ We want a formula for 10% growth (r = 0.1): (1 + $\frac{0.1}{\mathrm{n}}$ ) ⁿ

How to get this in terms of our original definition of e?

We’re going to substitute n with 10m, where m is just another variable we created, one that is 10 times smaller than n: n = 10m

Why ‘m’?
Next letter after ‘n’.

Why 10 times smaller?
Because 0.1 (10%) is 10 times smaller than 1 (100%).

We swap in 10m for every instance of n in our original expression: (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ resulting in the following modified expression: (1 + $\frac{1}{10\mathrm{m}}$ ) ^10m

We divide the ‘1’ by the ‘10’: (1 + $\frac{\sout{1}0.1}{\sout{10}\mathrm{m}}$ ) ^10m

We now have: (1 + $\frac{0.1}{\mathrm{m}}$ ) ^10m

What it means is:

• periodic compounding for m periods,
• at an APR of 10% (r = 0.1),
• for 10 time periods (exponent of ¹⁰ )

And to be clear, this modified expression (with m) is equal to our original one (with n): (1 + $\frac{0.1}{\mathrm{m}}$ ) ^10m = (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ since 10m is equal to n.

Continuing, we re-arrange the exponent 10m as ‘m × 10’: (1 + $\frac{0.1}{\mathrm{m}}$ ) ^10m since: 10 × m = 10m = m × 10 Which will take us closer to having the formula we want. (1 + $\frac{0.1}{\mathrm{m}}$ ) ^{m × 10}

Leaving us with this expression, once we pull the exponent ¹⁰ outside the square brackets: [ (1 + $\frac{0.1}{\mathrm{m}}$ ) ^m ] ¹⁰ which is a valid rule of exponents that we saw earlier.

Note: both round ( (…) ) and square brackets ( […] ) do the same thing: group terms together. I use them here to draw attention to different groups of things.

Yes, we are close to being done.

We define e as the limit that  (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ  approaches as n grows arbitrarily large: $\underset{\mathrm{n}\to \mathrm{\infty }}{lim}$(1 + $\frac{1}{\mathrm{n}}$ ) ⁿ = e

But what about our modified expression, which has  (1 + $\frac{0.1}{\mathrm{m}}$ ) ^m instead? [ (1 + $\frac{0.1}{\mathrm{m}}$ ) ^m ] ¹⁰ What is its limit as m tends to infinity? $\underset{\mathrm{m}\to \mathrm{\infty }}{lim}$(1 + $\frac{0.1}{\mathrm{m}}$ ) ^m = ?

I don’t know.
But we do know the modified expression is equal to our original one: [ (1 + $\frac{0.1}{\mathrm{m}}$ ) ^m ] ¹⁰ = (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ since we arrived at the modified expression by swapping like-for-like into the original expression.

Let’s take both of these expressions to their limits:

We know, by definition, that (1 + $\frac{1}{\mathrm{n}}$ ) ⁿ taken to its limit is equal to e: [$\underset{\mathrm{m}\to \mathrm{\infty }}{lim}$(1 + $\frac{0.1}{\mathrm{m}}$ ) ^m ] ¹⁰ = e

I don’t know what the limit for this expression is—yet.

For clarity, I’ll drop the limit notation, and focus on the highlighted part of the expression [( 1 + $\frac{0.1}{\mathrm{m}}$ )^m ] ¹⁰ = e

What is the value this highlighted part?

Something to the power of 10 is equal to e.

What could it be?

Yes, exactly! The only way something raised to a power is equal to e, is that the something is a root of e itself.

Like how the square root of 2, √2, when squared (multiplied by itself), equals 2: (√2)² = √2 × √2 = 2

This means our something is the 10^th root of e (¹⁰√e): [( 1 + $\frac{0.1}{\mathrm{m}}$ )^m ] ¹⁰ = e¹ [ ¹⁰√e ] ¹⁰ = e¹ We can re-write ¹⁰√e with a decimal exponent: [ e^0.1 ] ¹⁰ = e¹

(You can read more about these exponent rules in the footnotes‡ )

e^{0.1 × 10} = e¹

We can now express a 10% APR in terms of e: $\underset{\mathrm{m}\to \mathrm{\infty }}{lim}$(1 + $\frac{0.1}{\mathrm{m}}$ ) ^m = e^0.1

The point is this:

• We have a way to express any growth rate in terms of e.

Look at our equation showing continuous compounding at 10% APR: $\underset{\mathrm{m}\to \mathrm{\infty }}{lim}$(1 + $\frac{0.1}{\mathrm{m}}$ ) ^m = e^0.1 where the rate, r, is 0.1.

We’ve shown that continuous compounding at 10% gives a total of e^0.1 times the original principal, P: P × e^0.1 = A over the course of a single year (t = 1).

We can use our specific equation, with r = 0.1: $\underset{\mathrm{m}\to \mathrm{\infty }}{lim}$(1 + $\frac{0.1}{\mathrm{m}}$ ) ^m = e^0.1 to generalize for any rate we want: $\underset{\mathrm{m}\to \mathrm{\infty }}{lim}$(1 + $\frac{\mathrm{r}}{\mathrm{m}}$ ) ^m = e^r And since we know we can find the total after t years by raising e to t : (e^r )^t = e^r t We can now calculate the total amount (A) after continuous compounding of our principal, P :

• At any rate, r.
• For any period of time, t.

with the following formula: P × e^r t = A

The proof is the derivation, but we can verify the formula with a specific example.

From our earlier table, a princiale of $1,000 has a limit of $1,105.17, at an APR of 10% after one year of compounded growth:

We can plug these values into our formula, and check the result: $1,000 × e^{0.1 × 1} = A Combining the exponents: $1,000 × e^0.1 = A Evaluate e ≈ 2.7182: $1,000 × 2.7182^0.1 = A Then calculate to total amount, A, in dollars $1,000 × 1.105167591 = A $1,105.16167591 = A Rounding to nearest cent, we get: $1,105.17 = A which is the same as the total we calculated via periodic compounding in the table above.

Yes, well spotted!

This difference is due to rounding.

We set e = 2.7182, but the number itself is irrational, so it has an infinite decimal expansion—with no repeated pattern.

If you want a more accurate estimate of a limit, you’ll need more decimal places—of which has an infinite amount.

e^rt is Everywhere

Because e appears wherever the rate of change depends on the current amount.

Consider:

All these are discrete events, i.e. individually separate and distinct.

(Some of them also being discreet, hopefully!)

But if we aggregate many such events together, we can view them as a population changing continuously over time.

• From radioactive atoms to some radium decaying into radon gas.

• From a single bacterium to a growing bacterial colony.

• From a couple to the changing figures of a census.

We “zoom out” and take longer slices of time—and find a measure of change in every moment.

This continuous view is the limit of countless discrete changes.

The pattern is universal:

• the more you have, the more you get (or lose).

All else being equal, more radium means more radon gas.

Same for the other cases:

• the amount of change depends on the size of the population.

You’re in a world of e^r t and the only constant is change.

Compounding means: “the more you have, the more you get.”

Push compounding to the limit and you’ll find e.

The principle of “have more, get more” is all around us:

• debt and decay,
• banks and bacteria,
• contagion and collapse.

And change is happening all the time.

That’s why e is everywhere—because compounding is everywhere.

It’s not just a number—e is the mathematics of change itself.