While researching why car insurance rates are so extremely high in Las Vegas, I started thinking about the three-second rule and its validity. As I’ve always heard, the three-second rule refers to how far you should be behind a car in traffic. The idea is that you pick out a fixed roadside marker and you are supposed to pass that marker at least three seconds after the car in front of you. That rule is simple enough, yet deceptively deep once you unpack the physics.

“Three seconds is a rule of thumb. Physics reveals the truth.”

Every summer, it feels like a small miracle when the pool finally warms up enough to swim. In Nevada, where the air temperature can sit above 100°F (38°C) for weeks, you’d expect the water to keep pace. Yet, somehow, it takes forever to warm, and only a few cool nights can undo all that progress.

The same phenomenon shows up in a stick of butter. Butter melts quickly, while margarine stays stubbornly firm even under the same heat. That’s not coincidence; it’s thermodynamics.

The butter versus margarine comparison is a staple example in nutrition science. It shows how the proportions of fat, water, and solids affect how much energy it takes to change temperature. Butter, with more fat and less water, heats up and melts quickly. Margarine, full of water and unsaturated oils, absorbs more energy before softening because water’s specific heat is much higher.

“A pool in the desert and a stick of margarine in the kitchen both tell the same story: water resists change.”

When you interact with a large language model (LLM) such as ChatGPT or Claude , the model seems to respond instantly relative to the question’s degree of difficulty. What’s easy to forget is that every word it predicts comes from a long history of learning where billions of gradient steps have slowly sculpted its understanding of language.

Large language models don’t memorize text. They optimize it. Behind that optimization lies calculus. I’m not referring to the calculus you did with pencil and paper. I’m talking about a sprawling, automated version that computes millions of derivatives per second.

At its heart, every LLM is a feedback system. It starts with random guesses, measures how wrong it was, and then adjusts itself to be slightly less wrong. The word “slightly” in this context is the essence of calculus.

“Each gradient step represents a measurable reduction in error, guiding the model toward a more stable understanding of language.”

When you enter a sentence into a Large Language Model (LLM) such as ChatGPT or Claude , the model does not process words as language. It represents them as numbers.

Each word, phrase, and code token becomes a vector — a list of real-valued coordinates within a high-dimensional space. Relationships between meanings are captured not by grammar or logic but by geometry. The closer two vectors lie, the more similar their semantic roles appear to the model.

This is the mathematical foundation of large language models: linear algebra. Matrix multiplication, vector projection, cosine similarity, and normalization define how the model navigates this vast space of meaning. What feels like understanding is actually the alignment of high-dimensional vectors governed by probability and geometry.

“Linear algebra and geometry do more than support AI; they create its language of meaning.”

Developers are accustomed to thinking about code in terms of syntax and semantics, the how and the why. Syntax defines what is legal; semantics defines what it means. A compiler enforces syntax with ruthless precision and interprets semantics through symbol tables and execution logic. But a Large Language Model (LLM), reads code the way a seasoned engineer reads poetry, recognizing rhythm, pattern, and context more than explicit rules.

“When an AI system ‘understands’ code, it is not executing logic; it is modeling probability.”

When you need to parse a number, the first instinct is often to convert it to a string and slice it. That works well for data that comes from people — like phone numbers, credit cards, or postal codes — where formatting and leading zeros matter. But when you are working with raw numeric data that is guaranteed to be fixed-width and free of formatting, numeric parsing with integer division (//) and modulus (%) is the better option.

String parsing is flexible, but numeric parsing is faster and cleaner when the data is truly numeric.

Most of us reach for NumPy whenever math shows up in a project. But sometimes, you don’t want approximate answers, you want exact math. That’s when you pull SymPy out of your programmer’s toolkit and get to work.

It’s easy to think of SymPy only in academic terms, like running physics simulations where small rounding errors can snowball into nonsense, or checking algebraic identities where a value such as 0.0000001 should really be treated as exactly 0. Those are valid use cases, but they barely scratch the surface.

In real-world business applications, imprecision can be just as costly. Financial software is the most obvious example, where a few pennies lost to rounding errors can add up to millions at scale. Supply chain and logistics systems can also suffer when tolerances or unit conversions drift slightly off, leading to incorrect shipments or mismatched inventory. Even common scenarios such as pricing models or tax calculations can go sideways if the math behind them is not exact.

“Floats guess. SymPy knows.”

This is where SymPy shines. To see the difference between floating-point approximations (Python or NumPy) and symbolic precision (SymPy), let’s look at a simple but very real example from finance.

You know the story: drop a cookie on the kitchen floor, swoop in before five seconds are up, and declare it safe. It is comforting. It is also wrong.

“Germs don’t wait five seconds. They start the party the instant your food hits the floor.”

The truth is much more interesting than the myth. Germs do transfer gradually, but they are especially fast at the beginning. That means if you want to know whether your floor-cookie is still edible, you need to think in curves, not in timers. And curves are something we can model.

Every week your calendar fills with more meeting invites than you can reasonably handle. Which ones are worth the time and energy, and which should you politely decline? What if there was a way to quantify that choice?

“Your calendar is a knapsack. Every meeting takes space, but only some add enough value to justify carrying them.”

The good news: math can help. By modeling your schedule as a 0/1 knapsack problem with two constraints , you can treat meetings like items with value, time cost, and energy cost. Classic optimization techniques then help decide which meetings to attend. In this post, we’ll walk through framing the problem, prompting AI to scaffold the code, and running a simulation to visualize your optimal “meeting diet.”

Let’s be honest: argument validation code is rarely the proudest part of anyone’s repo.

Most of us start with the usual suspects:

❌ The dreaded inverted-V tower of if/else statements
❌ A graveyard of guard clauses scattered line after line

Using a dispatch table for validation rules means: one dictionary, one loop, infinite sanity.

Both work fine… until they don’t. Then you’re left maintaining a wall of conditionals that feels like it was designed by a committee of goblins.

There’s a better way: dispatch tables!