Mathematics for Fermi Calculation

The Toolkit

Fermi calculation doesn’t need calculus or algebra. It needs a small, sharp set of tools: logarithms for magnitude reasoning, a handful of approximations for common operations, and discipline about when to round and when not to.

Logarithms Base 10

The log₁₀ of a number is the power of 10 you’d need to produce it. It converts multiplication into addition — which is why it’s the natural language of order-of-magnitude reasoning.

log₁₀(1,000) = 3
log₁₀(1,000,000) = 6
log₁₀(ab) = log₁₀(a) + log₁₀(b)
log₁₀(a/b) = log₁₀(a) − log₁₀(b)

Key values to memorise:

Number	log₁₀
2	0.301 ≈ 0.3
3	0.477 ≈ 0.5
5	0.699 ≈ 0.7
7	0.845 ≈ 0.85
π	0.497 ≈ 0.5
e	0.434 ≈ 0.43

From these, you can derive any log mentally:

log(6) = log(2) + log(3) = 0.3 + 0.5 = 0.8
log(50) = log(100/2) = 2 − 0.3 = 1.7
log(300) = log(3) + log(100) = 0.5 + 2 = 2.5

In Fermi work, this means: instead of multiplying a chain of numbers, add their logs, then convert back.

Converting Between Log and Number

Given a log value, recover the number:

• Integer part → power of 10
• Fractional part → mantissa

log⁻¹(2.3) = 10^2.3 = 10² × 10^0.3 = 100 × 2 = 200
log⁻¹(6.7) = 10^6.7 = 10⁶ × 10^0.7 = 10⁶ × 5 = 5,000,000

The fractional-part lookup:

Fractional part	Mantissa
0	1
0.3	2
0.5	3
0.6	4
0.7	5
0.78	6
0.85	7
0.9	8
0.95	9

With this table you can read any log back to a number in seconds.

Useful Approximations

The Rule of 72

Time to double at a given growth rate: 72 / rate%

At 6% annual growth → doubles in 72/6 = 12 years
At 10% → doubles in 7.2 years

Works because ln(2) ≈ 0.693, and the rule approximates the continuous compounding formula.

Small percentage changes

For small x: (1 + x)ⁿ ≈ 1 + nx

1.05³ ≈ 1 + 3×0.05 = 1.15   (exact: 1.157)
0.97² ≈ 1 − 2×0.03 = 0.94   (exact: 0.941)

Good for growth rates, discount factors, compounding — anything under ~10% per step.

Square roots of powers of 10

√10 ≈ 3.16 ≈ π
√(10³) = √1000 ≈ 31.6
√(10⁷) ≈ 3,162

Useful for geometric means: √(10² × 10⁶) = √10⁸ = 10⁴.

Stirling’s approximation (for large factorials)

log(n!) ≈ n·log(n) − n·log(e) = n·log(n) − 0.43n

Comes up in combinatorics estimates — how many ways can something be arranged?

Rounding Strategy

The goal is to control errors deliberately, not eliminate them.

Round to one significant figure. 6.4 billion → 6 × 10⁹. 3.14 → 3. This is not laziness — it’s a statement that your uncertainty is already at the order-of-magnitude level.

Round in alternating directions. If one factor rounds up, round the next one down. Errors partially cancel across the chain.

Keep the mantissa when it matters. The difference between 1 × 10⁶ and 5 × 10⁶ is a factor of 5 — half an order of magnitude. If that half-order matters to your conclusion, keep it.

Never add false precision. Writing 7,432 when you mean “about 7,000” makes the estimate look more reliable than it is. Round visibly: “~7 × 10³.”

Combining Uncertainties

Each factor in a Fermi chain carries uncertainty. If you’re uncertain about a factor by a ratio r (i.e., could be r× higher or lower), and you have n independent factors, the combined uncertainty is roughly rⁿ.

• 3 factors, each uncertain by 3×: combined uncertainty = 3³ = 27×, about 1.5 orders of magnitude
• 5 factors, each uncertain by 2×: combined uncertainty = 2⁵ = 32×, about 1.5 orders of magnitude

This is why Fermi answers are usually stated as “within an order of magnitude” — that’s what the math supports. Claiming tighter precision requires either better data or fewer steps.

A Full Example in Log Space

How many red blood cells are produced in India per second?

Factor	Value	log₁₀
India population	1.4 × 10⁹	9.15
RBCs produced per person per second	~2.4 × 10⁶	6.38
Product		15.53

log⁻¹(15.53) = 10¹⁵ × 10^0.53 ≈ 10¹⁵ × 3.4 ≈ 3.4 × 10¹⁵

About 3 quadrillion red blood cells per second, across India. The log addition took seconds; the magnitude is unambiguous.