- How BetterExplained got started (and staying motivated)
- Learning math through direct experience
- The ADEPT method (Analogy, Diagram, Example, Plain-English, Technical)
- How to get better at math and why some find it difficult
- Role of math in modern society
- Learning other skills from coding to snowboarding

It was a lot of fun and we shared a bunch of insights -- hope you enjoy it.

]]>The Golden Ratio (1.618...) is often presented with an air of mysticism as "the perfect proportion". Setting aside whether we can find the Golden Ratio in the leaves of a nearby houseplant, what makes it special from a math perspective?

Well, let's try to make a pattern that's as balanced (symmetric) as possible.

A quick guess is something like `1, 1, 1, 1, 1`

. Every item is identical, but it's not very interesting -- it's a song where every word is the same. Does it rhyme? Yeah. Do I care? Not really.

Ok, let's try `1, 2, 3, 4, 5`

. There's a symmetry in the relationship that every element is one more than the previous. But if we skip around beyond neighboring elements, there's no real connection: what do 3, 8, and 17 have in common?

Ok, fine. What about `1, 2, 4, 8, 16`

? Each element is twice the previous, and all the numbers are clean powers of two. But... what about addition? Can we build new elements from the previous ones?

- 1 + 2 = 3 [not quite 4...]
- 1 + 2 + 4 = 7 [not quite 8...]
- 1 + 2 + 4 + 8 = 15 [not quite 16...]

Argh. We can concoct a rule for addition ("Every element is the sum of all previous elements... plus an extra 1"), but that's not clean and you know it. Let's figure this out.

Our goal is a pattern that's connected with both addition and multiplication. No varying number of additions, no cute "+1" on the end. Just a clean, simple relationship.

First off, we need to make every item connected by multiplication. We need powers:

`1 x x^2 x^3 x^4`

Whatever number the pattern uses (`x`

), everything will be a power of it (just like 1, 2, 4, 8, although that sequence wasn't symmetric enough).

Next, we need an "addition symmetry" that connects the items. Every element should be buildable from the previous ones without extra rules. Throwing a few ideas against the wall:

- Can we have
`1 + x = x`

? If we subtract x from both sides, that leads to 1 = 0. Uh oh, we're breaking math. - How about
`1 + 1 = x`

? Maybe we can allow multiple copies of everything. But that's just another way of getting x=2, you sneak. - How about
`1 + x = 1`

? Ack, that just implies x = 0.`0, 0, 0, 0`

is symmetric but not interesting.

Hrm. The next addition pattern that might make sense is `1 + x = x^2`

. If we follow this idea, here's what happens:

A few notes:

- Starting from a single connection
`1 + x = x^2`

, we can multiply through by`x`

and get`x + x^2 = x^3`

. Every element is made from the previous two. - x
^{0}is another way to write 1. Makes the pattern easier to see, I think. - Everything in the pattern grid is just our original pattern, shifted a bit. Neat.

Ok. That's the goal for the pattern, let's try to solve it.

We can find the "x" that makes this relationship true using the quadratic formula:

Plugging in a=1, b=-1, and c=-1, we get:

We only want a positive solution (the new part can't be negative), so we have

We label the solution Phi (phi).

You can see it happening below. There are slight differences as the decimals go on -- computers have fixed precision.

Many descriptions of the Golden Ratio describe splitting a whole into parts, each of which is in the golden ratio:

I prefer the "growth factor" scenario, where we start with a single item (1.0) and evolve it, while keeping it linked to its ancestors with both arithmetic and multiplication. Just describing a ratio doesn't call out the symmetry we're able to achieve.

Visually, I see a growing blob, like this:

As we scale by Phi each time (1.618) we get:

- Addition symmetry: Each element is the sum of the previous two
- Multiplication symmetry: Each element is a scaled version of the previous
- Growth symmetry: Addition and multiplication change the pattern identically

Aha! That's a nice combo if I ever saw one. The "growing blob" can represent the length of a line, a 2d shape, an angle -- which can lead to interesting patterns:

The key relationship is we move from one blob to the next, such that multiplication and addition have the same effect:

We've described the Golden Ratio with different phrasing: the scaling factor (f) must equal the original (1) plus the previous item (1/f). We divide by our growth factor to find the previous element given the current one.

Solving f = 1 + frac(1)(f) yields the Golden Ratio as before.

The Golden Ratio tends to be oversold in its occurrences. While it may appear occasionally in nature, buildings, and portraits, if you draw lines thick enough many things have a ratio of about 1.5 to 1.

I think the deeper intuition comes from realizing we've made addition and multiplication symmetric.

The Fibonacci sequence is built from having every piece built from the two before:

Using a certain formula, we can jump to a Fibonacci number by repeated scaling (exponents) instead of laboriously adding the parts. And maybe we'd come to *expect* the Golden Ratio here, since it's the scaling factor that allows two parts to add to the next item in the sequence. (The specifics of dividing by √(5) are because we need to adjust based on the starting pieces.)

Rather than hunting for examples of the Golden Ratio in the produce aisle, let's soak in the beauty of balancing the forces of multiplication and addition.

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.)

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 (frac(text(new))(text(old))).)

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.)

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 (2^x, 3^x)
- If growth occurs a natural system, continuous growth is better (e
^{x})

Let's take a look.

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} 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, √(14)), while n represents an integer (1, 2, 3, 4).

*Could* we 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 change *events*, I'd use the discrete version to describe the possibilities.

Binary numbers follow the same pattern: if we have *n* bits, 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.)

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 (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 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.

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 e^{x} 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.

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 (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 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.)

I wrote this post because my video on e had questions about how 2^x 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. 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)?

There's no right and wrong here, just the message we convey. A whole-number base (2^{x}, 3^{x}) implies you want people to think about whole-number values of x (and half-life is a good example). Using e as a base (e^{rate · time}) 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.

]]>And the formula we found was:

It seems that regular arithmetic, algebra, geometry, or even statistics could help work out the equation.

But how about Calculus? Is this bringing a nuclear missile to a gun fight?

Let's find out.

The sequence to add (1 2 3 4 5 6 7 8 9 10...) looks a lot like f(x) = x. At every position on the x-axis, we put in a number and get the same one out.

Intuitively, the integral is "repeatedly adding a bunch of stuff" -- it seems like we could put it to work. From the rules of Calculus (or using Wolfram Alpha) we get this:

Intuitively: Add up things following the f(x) = x pattern and you end up with frac(1)(2) x^{2}.

Well, let's see: the actual sum from 1 to 100 is 5050. But using the Calculus equation we get:

Uh oh: there's a difference. What's going on?

**Calculus works with continuous patterns, and we used a discrete one**.

Here's what's happening:

Calculus was built to measure *smoothly changing functions*, like a line, parabola, circle, etc. The pattern we have is a jumpy staircase (going from 1 to 2 without ever passing through 1.5, or 1.1, or 1.0001). In math class, books harp on analyzing whether a function is "continuous", aka changes smoothly enough for Calculus to work.

So when a pattern changes smoothly, Calculus works great. If a pattern changes suddenly, Calculus can only give an approximate answer. So what's the plan?

Use Calculus where possible, on the smooth part, and adjust for errors in the jumpy part.

The area under the line is the integral. We a bunch of triangles above the line we need to include.

- How many of them? 1 for each item (x)
- How big are they? They're half a of a 1x1 square, so they have area 1/2.
- What's the total area to add back in? x frac(1)(2) = frac(x)(2)

So our final formula should be

Aha! Learning Calculus doesn't mean we hunt around for Official Calculus Problems.

Nope. Take your scenario (adding 1 to 100) and realize what Calculus brings to the table: finding patterns in smoothly changing functions. Use Calculus on the smooth parts and adjust (or ignore) the other parts.

(Ironically, Calculus works by making jumpy approximations for smooth functions, and is in fact "jumpy" under the hood. If you are planning on working with jumpy patterns, use Discrete Calculus.)

Let's take this further: what's your guess for the sum of the first 100 square numbers?

Hrm. Getting the exact formula is tricky. But maybe we don't need the exact count, just an estimate.

With Calculus, we'd say: The pattern isn't continuous, but it looks like f(x) = x^{2}. Let's integrate x^2 from 0 to 100.

The indefinite integral is frac(1)(3) x^{3} , the running total for how much we have. From 0 to 100 it would be

That's our guess, without a calculator. And the actual answer? 338350.

How close were we? 99.9%. Not bad for something we worked out by hand in a minute!

Truly internalizing Calculus means it helps other elements of your math understanding, even regular addition problems.

Happy math.

PS. To keep building your intuition, check out the Calculus Guide.

]]>