Understanding Discrete vs. Continuous Growth

There are two types of exponential growth, and it's easy to mix them up:

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 isalwayshappening. We can't point to an event and say "It changedhere". The pattern is always in motion (radioactive decay, a bacteria colony, or perfectly compounded interest).

(Brush up onthe number eand the2022世界杯预选赛 .)

Insight: Convert between discrete and continuous

I visualize change as events along a timeline:

离散的变化以明显的绿色斑点的形式发生。我们可以把它们分成更小，更频繁的变化，并把它们分散开来。有了足够的分裂，我们就可以进行平稳、持续的变化。

因此，离散变化可以用一些等价的平滑曲线来表示。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: $\frac{\text{new}}{\text{old}}$.

Mathematically,

$\displaystyle{ 2^x = e^{\ln(2)x} = e^{.693 x} }$

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.)

那么我们用哪个版本呢?

现在的问题是:我们应该如何谈论增长?It depends on the scenario:

If growth happens in a man-made system, discrete growth works better ($2^x$, $3^x$)
If growth occurs a natural system, continuous growth is better ($e^x$)

让我们一起来看看吧。

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 $2^n$ wherenwas the number of flips.

I'm using "n" (not x) by convention: x could mean any value on the x-axis (-3, 1.234, $\sqrt{14}$), while n represents an integer (1, 2, 3, 4).

Couldwe say the number of outcomes was $e^{\ln(2)x}$, where x was the number of coin flips? Yes. But it's confusing: in a man-made system, where we have changeevents，我会用离散版本来描述可能性。

Example: Binary Numbers

Binary numbers follow the same pattern: if we havenbits, we get $2^n$ 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

当放射性物质衰变时，我们经常谈论它的半衰期:多长时间一半的物质会消失?

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

$\displaystyle{ \text{percent of carbon left} = (1/2)^{\text{years}/5700} }$

If we wait 5700 years, we expect $(1/2)^1= .50$ of the carbon remaining. If we double that and wait 11,400 years, we'd expect $(1/2)^2 = .25$ 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 becausewe humanswant 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 thecontinuous decay rateby converting the discrete growth into a continuous pattern:

$\displaystyle{ (1/2)^{\text{years}/5700} = (e^{\ln(1/2)})^{\text{years} / 5700} = e^{-.693 \cdot \text{years} / 5700} = e^{-0.00012 \cdot \text{years}} }$

This helps me understand why the natural log isnatural-- 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%:

$\begin{aligned} e^{{-.479} \cdot \text{years}} &= 0.5 \\ \ln(e^{{-.479} \cdot \text{years}}) &= \ln(0.5) \\ {-.479} \cdot \text{years} &= {-.693} \\ \text{years} &= {-.693} / {-.479} = 1.44 \end{aligned}$

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:

$\displaystyle{ \text{expected value} = \text{investment} \cdot (1 + 8\%)^{\text{years}} }$

连续速率的形式是$e^x$对系统没有意义。我们并没有试图以每一瞬间为基础来为投资组合的价值建模:我们想知道30年后会发生什么。

Example: Population Growth

人口是棘手的:根据动物的不同，离散或连续模型都有意义。

A bacteria colony is made of billions of organisms. Althougheach bacteria cellgrows discretely (it has to wait until it splits before splitting again), the entirecolony因为很多细菌都处于不同的生长阶段，所以生长很顺利。

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 ($e^x$) that expresses the colony's instant rate, and a discrete rate ($2^x$) 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 justhappensto 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.)

把咆哮放一边，模拟老虎的数量怎么样?老虎有繁殖季节。他们全年都没有孩子，所以人口在一个离散事件中变化。

$\displaystyle{ \text{new population} = \text{current population} * (1 + \text{growth rate})^{\text{years}} }$

(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 myvideo on e关于2^x$如何代表“阶梯增长”的问题。这不也是一条平滑的曲线吗?

Sure, but most of the time we use 2 as a base to model discrete patterns. $2^n$ (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 $e^{\ln(2) x}$?

这里没有对错之分，只有我们传达的信息。整数基数($2^x$， $3^x$)意味着你希望人们考虑x的整数值(半衰期就是一个很好的例子)。使用$e$作为基数($e^{\text{rate} \cdot \text{time}}$)意味着您希望人们考虑每时每刻发生的变化。

无论哪种方式，都要熟练掌握这两种模式，并学会在这两种模式之间切换。

快乐数学。