— 1 of 10 —
PAGE 1 — INT. BACK ROOM OF A BARBERSHOP — EVENING
FADE IN:
A barbershop after hours. Clippers hung up, chairs empty. In the back room, EARL (60s, reading glasses on a chain, retired numbers man) sits at a folding table doing the books. His nephew TYRESE (20s, sharp, studying for his actuary exam) sits across from him with a calculator.
TYRESE
Uncle Earl, I've been staring at this practice exam for two hours. The mental math is killing me. How'd you run numbers for thirty years without a computer?
EARL
(not looking up from his ledger)
I didn't compute. I approximated. There's a difference.
TYRESE
But these problems want exact answers...
EARL
No they don't. They want you to pick the right option out of five. You don't need six decimals. You need to be in the right ballpark, fast. And for that, you need one trick: take out the big part.
TYRESE
Take out the big part?
EARL
(finally looking up)
Every number is made of a big part and a small correction. Handle the big part first — that's the structure. Then add the correction — that's the detail. Most of the time, the big part is enough.
— 2 of 10 —
PAGE 2 — MULTIPLICATION BY ONES AND FEWS
EARL
What's 31 times 29?
TYRESE
(reaching for calculator)
Let me...
EARL
(slapping the calculator away)
No. Think. What's 30 times 30?
TYRESE
Nine hundred.
EARL
Good. Now 31 is "30 plus a little" and 29 is "30 minus a little." The big part of both numbers is 30. The small part is plus or minus 1.
He writes on a receipt:
31 × 29 = (30 + 1)(30 - 1)
= 30² - 1²
= 900 - 1
= 899
EARL (CONT'D)
The big part gives you 900. The correction gives you minus 1. Total: 899. Three seconds, no calculator.
TYRESE
That's just difference of squares.
EARL
It IS difference of squares. But the PRINCIPLE is bigger than algebra. The principle is: decompose everything into "one" and "few." One is the big part — the round number, the easy part. Few is the correction — the small leftover. Multiply the ones first. Correct after.
TYRESE
What about 47 times 53?
EARL
Big part is 50. Correction is plus/minus 3.
50² - 3² = 2500 - 9 = 2491.
Done.
— 3 of 10 —
PAGE 3 — THE FRACTIONAL CHANGE
EARL
Now here's where it gets really useful. Fractional changes. Say a stock goes up five percent. What's 1.05 squared?
TYRESE
(thinking)
1.1025.
EARL
You computed that. Let me ESTIMATE it. The big part is 1. The small part is 0.05. When you square (1 + small):
(1 + x)² ≈ 1 + 2x (when x is small)
So (1.05)² ≈ 1 + 2(0.05) = 1.10.
TYRESE
You're off by 0.0025.
EARL
Quarter of a percent. On a hundred-dollar stock that's twenty-five cents. For a back-of-envelope calculation? Irrelevant. The big part — the 2x term — carries all the information.
He writes the general rule:
(1 + x)ⁿ ≈ 1 + nx (when x is small)
EARL (CONT'D)
This works for ANY power. Cube root of 1.06?
(1 + 0.06)^(1/3) ≈ 1 + (1/3)(0.06) = 1 + 0.02 = 1.02
Exact answer: 1.0196... Off by four ten-thousandths.
TYRESE
So any time something is "one plus a little bit" raised to a power...
EARL
You just multiply the little bit by the power and add it to one. The big part is always 1. The correction is always n times x. Everything else is noise.
— 4 of 10 —
PAGE 4 — LOW-ENTROPY EXPRESSIONS
TYRESE
Why does this work? Why is the approximation so good?
EARL
Because the full expansion is:
(1 + x)² = 1 + 2x + x²
When x is 0.05, the x² term is 0.0025. It's the SQUARE of something already small — so it's tiny. Each successive term in the expansion is smaller than the last by a factor of x.
TYRESE
So you're throwing away terms that are x-squared, x-cubed...
EARL
Which are 0.0025, 0.000125... Each one a hundred times smaller than the previous. The first correction — the nx term — captures almost everything.
He taps the table.
EARL (CONT'D)
Here's the philosophy. I call it "low-entropy expressions." An expression like 1 + 2x is LOW entropy — simple, easy to read, easy to use. An expression like 1.1025 is HIGH entropy — just a pile of digits. No structure.
TYRESE
So 1 + 2x is better than 1.1025 even though it's less precise?
EARL
Because it tells you the STORY. The "2" tells you it's squared. The "x" tells you what drives the change. The digits 1-0-2-5 tell you nothing — they're just a dead number.
TYRESE
Structure over precision.
EARL
Every time. A street fighter keeps the structure and lets the decimals go.
— 5 of 10 —
PAGE 5 — GENERAL EXPONENTS
EARL
Let me show you the power of this. What's the volume of a sphere if the radius increases by two percent?
TYRESE
Volume is (4/3)πr³. So if r goes up by two percent...
EARL
New volume is (4/3)π(1.02r)³. Factor out the old volume:
V_new = V_old × (1.02)³
Now apply the rule: (1 + x)ⁿ ≈ 1 + nx.
(1.02)³ ≈ 1 + 3(0.02) = 1.06
Volume goes up by six percent. A two percent increase in radius gives a six percent increase in volume — because of the cube.
TYRESE
The exponent 3 multiplies the fractional change.
EARL
THAT'S the insight. It works for any exponent. Negative, fractional, whatever.
Area of a circle (r²): 2% radius increase → 4% area increase
Period of pendulum (√L): 2% length increase → 1% period increase
Gravitational force (1/r²): 2% distance increase → 4% force DECREASE
EARL (CONT'D)
The rule (1+x)ⁿ ≈ 1 + nx is the single most useful approximation in all of applied math. It's how I set odds. It's how engineers check designs. It's how physicists estimate everything.
TYRESE
And you're saying the "big part" is 1 — the original — and the correction is nx.
EARL
Take out the big part. The rest is manageable.
— 6 of 10 —
PAGE 6 — THE COMPOUND INTEREST TRICK
EARL
Here's a classic from my betting days. A guy wants to know what happens to a hundred dollars at five percent interest compounded annually for ten years. What's (1.05)¹⁰?
TYRESE
The rule of 72 says it doubles in about 14 years... so in 10 years it's maybe 1.6 something?
EARL
Good instinct. But let me show you the "take out the big part" approach. Rewrite:
(1.05)¹⁰ = e^(10 × ln(1.05))
Now ln(1.05) ≈ 0.05 — because ln(1+x) ≈ x for small x. So:
e^(10 × 0.05) = e^0.5
TYRESE
e to the 0.5 is about 1.65.
EARL
Exact answer is 1.6289. We got 1.6487. Off by one percent. Not bad for mental math.
TYRESE
You turned a compound interest problem into... e to the half?
EARL
That's the move. Take the log, approximate the log with x, multiply by the exponent, exponentiate. Each step uses "take out the big part."
ln(1 + small) ≈ small ← big part of log
e^(small) ≈ 1 + small ← big part of exponential
EARL (CONT'D)
For bigger exponents or bigger rates, you keep more terms. But for five percent over ten years? One term is enough to pick the right answer on a multiple-choice exam.
— 7 of 10 —
PAGE 7 — SUCCESSIVE APPROXIMATION
EARL
Now let me show you the advanced version. Sometimes one correction isn't enough. So you iterate — use the first approximation to get a better one.
TYRESE
Like Newton's method?
EARL
Same spirit. Here's a physical problem. You drop a rock into a well. You hear the splash after T seconds. How deep is the well?
TYRESE
The rock falls for time t₁, the sound comes back up in time t₂. T = t₁ + t₂.
EARL
Right. If the well is depth d:
t₁ = √(2d/g) [free fall time]
t₂ = d/v_sound [sound travel time]
T = √(2d/g) + d/v_sound
Solving that exactly requires solving a quadratic in √d. Messy. But here's the street-fighting approach.
EARL (CONT'D)
First approximation: ignore the sound travel time. It's small compared to the fall time for a shallow well. So T ≈ √(2d/g). Solve for d:
d₀ ≈ gT²/2
That's the big part.
TYRESE
And the correction?
EARL
The sound takes time d₀/v_sound to travel up. So the actual fall time is shorter — not T, but T minus that correction:
t₁ ≈ T - d₀/v_sound
Recalculate d using the corrected fall time. That's your second approximation. Closer. You can iterate again if you want — each round peels off a smaller correction.
— 8 of 10 —
PAGE 8 — HOW DEEP IS THE WELL
EARL
Let's put numbers to it. Say T is 3 seconds. g is 10 m/s². Speed of sound is 340 m/s.
First approximation — ignore sound:
d₀ = (10)(3²)/2 = 45 meters
Sound travel time at that depth:
t₂ = 45/340 ≈ 0.13 seconds
Second approximation — correct the fall time:
t₁ ≈ 3 - 0.13 = 2.87 seconds
d₁ = (10)(2.87²)/2 ≈ 41.2 meters
TYRESE
So the first guess was 45, the refined guess is 41.
EARL
And if you solve the exact quadratic? About 40.5 meters. Our second approximation is off by less than two percent.
TYRESE
Two iterations and you're within two percent.
EARL
That's successive approximation. Take out the big part — get d₀. Use d₀ to compute the correction. Apply the correction to get d₁. Each round is better. Each round is easy.
He leans back.
EARL (CONT'D)
This is how GPS works, by the way. First estimate of your position, then corrections based on satellite signals. Iterate until you converge. Same principle — take out the big part, then refine.
TYRESE
I thought GPS was high-tech...
EARL
The tech is in the satellites. The MATH is from the barbershop.
— 9 of 10 —
PAGE 9 — STACKING CORRECTIONS
TYRESE
So the pattern is always the same? Big part, then correction, then correction to the correction?
EARL
Always. And each correction is smaller than the last — usually by a fixed ratio. Like a geometric series of improvements.
He writes on the receipt:
Answer = Big Part × (1 + ε + ε² + ε³ + ...)
where ε is the small parameter — the ratio of
correction to big part.
EARL (CONT'D)
In the well problem, ε was about 0.13/3 ≈ 0.04. Each correction is four percent of the previous one. Two iterations is overkill — one was almost enough.
TYRESE
How do you know when to stop?
EARL
When the correction is smaller than your uncertainty. If you don't know gravity to better than two percent — no point computing a one percent correction. Match the precision of your answer to the precision of your inputs.
TYRESE
My professors always want exact answers.
EARL
Your professors live in a world where the inputs are exact. In the real world — where I set odds, where engineers build bridges, where doctors dose medicine — the inputs are never exact. So the answer doesn't need to be either.
TYRESE
Take out the big part. Stop when the correction doesn't matter.
EARL
Now you're thinking like a numbers man.
— 10 of 10 —
PAGE 10 — THE BIG PART PHILOSOPHY
Earl closes his ledger. Tyrese closes his practice exam book.
EARL
(leaning forward)
Let me tell you what "taking out the big part" really is. It's a way of seeing the world.
Most people look at 1.05 to the tenth power and see a wall. A computation they can't do in their heads. So they freeze. Or grab a calculator.
A street fighter sees: 1 is the big part. The 0.05 raised to the tenth — that's a journey from 1 to somewhere. And (1+x)ⁿ ≈ 1 + nx tells me the destination is about 1.5. That's enough to ACT.
TYRESE
Act before you compute.
EARL
Estimate before you calculate. Approximate before you solve. Take out the big part before you drown in the details.
He stands, stretches.
EARL (CONT'D)
Three tools in one:
Multiplication — split into round numbers, compute the big product, add the cross term.
Fractional changes — (1+x)ⁿ ≈ 1 + nx. The exponent multiplies the fractional change. Period.
Successive approximation — solve the simple version first, then use that answer to correct itself.
All three are the same idea: separate the important from the unimportant. Handle the important first. If you need more, come back for the rest.
TYRESE
(standing, grabbing his books)
I think I can pass this exam now.
EARL
(turning off the back-room light)
You could've passed it an hour ago. You just didn't know what to ignore.
FADE OUT.
