Mathematics & Probability
Section 1
Core Idea
Dimensional analysis checks whether an equation or estimate makes sense by verifying that the units on both sides match. If you're calculating cost per customer and your answer comes out in "dollars per square metre," something is wrong — regardless of whether the arithmetic looks right. The method originated in physics and engineering, where mixing up units has caused real disasters (the Mars Climate Orbiter was lost because one team used metric and another used imperial). As a thinking tool, dimensional analysis is a fast, low-cost sanity check that catches errors before they compound into bad decisions.
Free playbook
Get The Business Model Playbook
58 business models, one visual page each: how the money flows, the metrics that matter, and who runs it. Free when you join the Faster Than Normal email.
Free. No spam. Unsubscribe anytime.
Section 2
How to See It
Finance
You're seeing it when someone presents a financial model and the output units don't match the question — projecting "revenue per employee" but accidentally calculating "revenue per department."
Operations
You're seeing it when a supply chain estimate predicts delivery in "tons per hour" when the question requires "units per day" — the number may be mathematically correct but dimensionally wrong.
Strategy
You're seeing it when a back-of-envelope calculation produces an answer that's off by orders of magnitude, and checking the units reveals that a conversion step was missed or a ratio was inverted.
Section 3
How to Use It
Before trusting any calculation, check the units. Write out what each variable represents and trace the units through each step. If the output units don't match the question's units, there's an error in the logic — even if the arithmetic is correct. This is especially valuable for Fermi estimates and financial models, where a single unit mistake can produce answers that are off by factors of 10 or 100.
Decision filter
"Do the units of my answer match the question I'm trying to answer — or have I introduced a dimensional error somewhere in the calculation?"
As a founder
Before presenting any financial model or metric to investors or your board, check the units. Is your CAC in dollars per customer or per lead? Is your runway in months or weeks? Unit errors in high-stakes presentations destroy credibility and lead to wrong decisions. Make dimensional checking a standard step.
Section 5
Founders & Leaders
Andy GroveCo-founder & CEO of Intel; author of High Output Management
Grove's management philosophy centred on measurable outputs — but he insisted that measurements be dimensionally consistent. In High Output Management, he stresses that a manager's output must be measured in the same units as their team's output, not in activity metrics that don't convert. Tracking "meetings per week" when the goal is "chips shipped per quarter" is a dimensional mismatch that obscures reality. For founders, Grove's lesson is that metrics only drive performance when their units map directly to the outcome you're optimizing. Mismatched dimensions create the illusion of progress.
Section 7
Connected Models
Pairs-with
Algebra
Algebra sets up relationships between variables. Dimensional analysis validates those relationships — if the units don't balance, the algebraic expression is wrong regardless of its mathematical form.
Supports
Fermi Problem
Fermi problems estimate unknown quantities through decomposition. Dimensional analysis is the error-checking step — verifying that each sub-estimate's units combine correctly to produce an answer in the right dimensions.
Pairs-with
Order of Magnitude
Order-of-magnitude thinking asks whether an answer is in the right ballpark. Dimensional analysis operates at a more fundamental level — checking whether the answer is even in the right units before assessing whether the magnitude is reasonable.
Section 8
One Key Quote
— Lord Kelvin"When you can measure what you are speaking about, and express it in numbers, you know something about it; when you cannot, your knowledge is of a meagre and unsatisfactory kind."
Section 11
Summary & Further Reading
Dimensional analysis checks that the units of an answer match the question. It's a fast, low-cost sanity check that catches logical errors before they compound into bad decisions. If the units don't match, the answer is wrong.
01
Book
On measuring managerial output in the right units and why dimensional consistency drives effective management.
02
Book
On dimensional analysis, estimation, and the art of checking whether an answer is in the right ballpark.
03
Book
On understanding stocks, flows, and the dimensional relationships between system variables.