Friends,
I usually have a concrete plan in advance of writing, but today’s letter is totally spontaneous, other than knowing that something would be published. I wrote it all, but found myself stuck on a title. I hope it will make sense by the time you’re done.
So…one of my oldest friends, someone I consider family really, is enjoying a sabbatical year. He and his wife crashed with us for 10 days. It’s one of those slices of time that you know before it’s even over that you will have nostalgia for.
We didn’t do anything overly special, although it was a great catalyst to convene with the rest of our Bay Area college crew over the weekend. The week was something in between a staycation and just a far more elevated (ie joyful) routine. I would work during the day while they went on an excursion, but we’d make sure to go to the gym or at very least a walk together daily, and the evenings were filled with good food (his baked ziti is in contention for my electric chair meal which I don’t anticipate needing unless they start rounding up the dorks) and games (Scrabble and Decrypto mostly) with the whole family.
One of my favorite parts of it was to have them around in such an informal way, just like family crashing. The kids would hang around and be part of the discussion like these were the aunties and uncles they normally see. Everyone actually gets to know each other. My kids get to see that their dad’s friends are weird just like their dad is. It’s funny for them to see them process it, but I hope it models to them what friendship is, even if they do think we’re old aliens. I’m also happy to see my friends know my kids. To see their different personalities and for them to be more than just names.
Personally, the week has been so much fun, especially because my friend and I have always had a nerd bond. He’s much more educated and technical than I (this sabbatical very likely ends with him working for Jane Street or Waymo), so I also just have permission to geek out without worrying that I’ll be talking to myself soon. I also convinced him, although it didn’t require much, to show me the game Factorio. This confirmed what I expected…I can’t introduce THAT into my life. Diet soda is addicting enough.
A nice bonus feature of the week was getting to chat about various math topics and education broadly because we were both trying to help Zak with his Math Academy lessons by trying to break down the concepts in intuitive ways. He’s currently jamming on logs and exponentials, which I’ll come back to in a moment. But I want to share a nice analogy first. When Zak hits a wall, he’ll do that thing all kids do when they feel frustrated. “Why do I need to learn this, I’m never going to need it.”
A reflexive, true, and entirely uncompelling response to such pleas is “Actually, you might. It depends what you end up doing for a living.” But kids think the future is as distant as the afterlife, so the argument for doing homework is as convincing as telling them they’ll burn in hell for punching their brother. My friend used an analogy that meets Zak on his terms. “Why do you do pushups or lift weights? You’re never going to do a pushup on the court.”
My mother shared this exact point of view when I was growing up. It’s training. Let’s be honest, when it comes to actual application, the most useful classes you take in school are home ec and typing. And while I certainly have many gripes with the non-useful stuff they teach, there’s stuff that you will not use but counts as training like math and critical reading. Numeracy and literacy. Even if their utility were diminished, they make for a richer interior existence, allowing you to be amused and intrigued by the world for free. Anyway, a nerd writing on the internet comes off as one-note at best and self-flattering at worst when carrying a flag for hokey ideas like doing your times tables, so I’ll stop there on all that.
Back to the log stuff real quick. I sometimes wonder if it’s such a challenging topic because our minds struggle naturally with non-linearity or if we should actually just learn it earlier. It seemed helpful for Zak to realize that all logs are is another rung on the ladder of basic math operations.
We start with addition.
Subtraction is the inverse of addition. It “undoes” addition.
We then move up to multiplication. Multiplication is repeated addition. 4 tires, 40 times is the same as 4+4+4…(repeated 37 more times) which is the same as a Nascar race.
Division undoes multiplication by repeating subtraction. 40 divided by 8 is how many times can I take away 8 from 40.
Then we move up to exponents. Exponents are repeated multiplication of the same number.
Logs undo exponents by repeatedly dividing by the same number (ie the base).
In infographic form:
Zak was struggling with understanding natural log. I took a stab at it from the compounding angle during a car ride last week.
“If you invest $100 and receive back $110 in a year what interest rate did you receive?”
10%
“What if I told you you compounded semi-annually…do you think the interest rate that got you to $110 is greater than or less than 10%?”
Less than.
“Good. Forget the calculator, let’s just guess and say the semiannual compounding at 9.8% got us to $110. What if we compound daily, is the rate greater than or less than 9.8%?”
Less than.
“Now imagine we keep shortening the interval from daily compounding to minute-compouding to seconds to nanoseconds. We can shorten the interval until it gets close to zero without touching zero. Later in calculus you’re learn that this is a useful trick where you approach zero but don’t touch it. It’s called a limit. If you shorten the interval until the limit, almost zero, we call that continuous compounding. The natural log gives you the rate if we assume continuous compounding. So in the case of our investment, we can compute the continuous rate by taking the natural log of 1.1 because our return was $110/$100.
LN(1.1) is about 9.5% going off memory and represents the continuously compouded rate that would give a total return of 10%.”
Then I did that thing he hates which is try to give him more than he asked for.
“You know how if you double your money that’s a 100% return. Well, the continuous compounded rate comes from taking LN(2). Before we compute that, do you think the continuous compound rate is going to be less or greater than 100% if we doubled our money?”
Less than 100%
“Exactly. LN(2) is about 72%. The cool thing about logs is you can simply divide the already compounded return by the number of years to get an annual compounding rate. So if you double your money in 10 years, the annual rate is 7.2%
That’s where the rule of 72 comes from!
It’s just inverting the logic. If you continuously compound at 10% per year it takes ~7 years to have a continuously compounded return of 70%, which corresponds to doubling your money.”
e (Euler’s constant)
We talked just a little about e.
If you continuously compound at 100% for 1 year, you end up with e, or about 2.718x what you started with.
e1 = 2.718
Undo it:
ln(2.7128) = ln(e) = 1 = 100%
If it takes 10 years for your money to grow to 2.718, then you are continuously compounding at 10% or 100% / 10
Contrast this with solving for the annual compounded return where you compute:
2.7181/10 - 1 = 10.5% annual compouding
We just did ln(2.7128)/10 = 10% continuous compounding
Continuous return in finance
2 properties make log returns convenient for financial math.
A) Logreturns are linearly proportional to time making them easier to manipulate.
The wealth equation is:
W = ert where r is the logreturn. Note its proportionality to t.
but for annual compounding:
W = (1+r)t where r is a discretely compounded return.
In discrete land, we need the t root to back out the rate.
2.7181/10 - 1 = 10.5% annual compouding
In continuous terms,
ln(2.718)/10 = 10% continuous compouding
Logs linearize the time component of returns so you can aggregate them as time-weighted sums or take simple averages to get the mean logreturn. Discrete compounding requires multiplying returns then taking the geometric mean (t root)
By example, suppose you grow by 10% in year 1, 26 % in year 2, and 0% in year 3. What’s average total growth?
If those are log returns, the average annual log return is simply 12%
If they are annual returns, we must compute:
(1.1 * 1.26 * 1.00 )1/3 - 1 = 11.49%
B) Log returns are normally distributed, making them a convenient measure of return for statistical modeling.
Learning Math in the AI Era
A fun thing my friend told me was that the natural log function was discovered and the term coined before e, which is surprising because I think of the natural log as just a logarithm with e as a base. But it had relevance before e!
I needed Claude to spell it out:
The natural log arrived through the hyperbola. Saint-Vincent’s equal-area observation was around 1647, Sarasa spelled out “this area behaves like a logarithm” in 1649, and Mercator published the series and the name logarithmus naturalis in 1668. So by the late 1660s people had the natural log as a working object — they could compute it, they had its series, they knew it was the log that came out of the 1/x area with no choice of base.
What they did not have was e as a recognized number. Nobody in that group pointed at 2.718… and said “there’s a special constant here, and it’s the base of this thing.” Napier’s 1614 construction secretly ran on 1/e, and the hyperbola log secretly had base e, but “secretly” is the operative word.
e gets isolated as a number of interest later and from the side door — Jacob Bernoulli, around 1683, poking at compound interest and the limit of (1 + 1/n)ⁿ. He found the value sat somewhere between 2 and 3 but didn’t tie it cleanly to the logarithm or give it a name. The unification is Euler’s, in the early-to-mid 1700s: he named it e, nailed it as the base of the natural log, established eˣ and ln as inverses, and made the whole structure look inevitable rather than accidental.
Charlie Munger was on a podcast shortly before his death and he spoke of how a solid understanding of grade-school and high school math basics was fundamental to thinking. He had a highly utilitarian perspective rather than an academic one.
If we combine the natural log story with Munger’s perspective, I think we land at an interesting idea. A math history approach to the basics.
In elementary school, the focus should certainly be operations. There’s a grammar to math that complements the many abstractions of counting, which is what I think you’re ultimately learning. But by late middle school, we should include an appeal to stories, history, mystery, and pragmatism by personalizing the context of the math we learn. To put a student in the shoes of someone trying to solve a problem for the first time in history with the tools that were available at the time. Obviously, asking students to do what geniuses did is not the goal. But AI would be an amazing tool for placing the student in an RPG where you drip as much information as they need to get to the next step within an appropriate level of difficulty for the individual.
It wouldn’t be a substitute for instrumental math education but a way to deepen our relationship with the fundamental concepts Munger thinks we could benefit from deepening. And it’s not limited to math. It’s more like STEM History 101. Science ed seems to have a bit more focus on the individual scientists and stories of discovery, but many of these figures are fascinating eccentrics if not crazies. It’s a colorful way to captivate.
I admit it’s less than a half-baked idea, so it’s really more of a “here’s a side-project that could be cool, feel free to run with it” but I do keep coming back to it as something I’d like to see.
I also want to take a moment to repeat myself — AI is a tireless tutor. A gift to the curious.
This investor has been live-tweeting his own learning arc:
It’s a great example of the similar projects I’ve been doing for self-help.
I’ve unpaywalled the below article Socrates 2026: How to Use Highlights.
It’s stuffed to the gills with things I think are fun and can hopefully help you help yourself.
From the Options Trench
📺Erik and I tackle the question…”Should you start trading options?”
From My Actual Life
Both my friend and I are Diet Coke addicts. My addiction actually started when I lived with him sophomore year of college and we did the Atkins diet. Between eating sticks of butter and bacon (oh to be 19 again), I’d crave anything remotely sweet. It started with diet Sunkist sodas and turned into a lifelong DC dependency. You’re welcome Warren Buffett.
One of the activities we did during this visit was a blind taste test of Diet Coke, Coke Zero, Pepsi Zero, and Diet Pepsi. All but the DC were served from a bottle, which would be insultingly sloppy to a serious researcher, but we procured what we could from 7-11. Regardless, the results were consistent…we all preferred Diet Pepsi or Pepsi Zero! My friend had recently converted his habit to Pepsi, so he wasn’t surprised, but I was. Another steady finding was that Coke Zero was the most easily identifiable by all 6 of us and the least preferred. Made it a bit more tragic that I was stuck drinking CZ in Europe the past couple summers since it’s almost impossible to find DC or Pepsi products. Woe to the American tourist.
We haven’t done a taste test night since our first batch of Japanese exchange students visited several years ago. That was a pizza night. In any case, taste tests are engaging activities to bolt on to or make central to your next get-together. Let me know if you’ve done any fun ones!
Stay groovy
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You have some really amazing ideas for the future of education!