Driver reaction time (notice → decide → act) is ~1 second.

So if an obstacle appears closer than 16 meters (at 60 km/h), you probably cannot do anything in time. If the speed is three times higher, what distance is critical?

So if you spend 1 hour per day commuting (half an hour each way), you spend 250 hours per year on the road.

Want 10% annual growth on your investments? They double in 7.2 years.

Want your investments to double in 5 years? You need a rate of 72/5, or about 15%.

You can use this rule for any time period, not just years.

Inflation is 4%? That cuts your money in half in 72/4, or 18 years.

10,000 = hundred hundred

million = thousand thousand

billion = thousand million

trillion = million million

1% of 10k is 100. The Dow is roughly 10k (it’s about 12k now). So if the Dow drops 100, it’s about a 1% loss.

What’s 5k × 50k? That’s 250 × thousand × thousand, or 250 million.

12 days = 1 million seconds

1 year = 31 million seconds (about π × 10 million)

30 years = 1 billion seconds

30,000 years = 1 trillion seconds

One “part per million” means an accuracy of 1 second every 12 days. One “part per trillion” means an accuracy of 1 second every 30,000 years.

It’s not immediately clear, but it’s true: a% of b = 0.01 × a × b, which is the same as b% of a (0.01 × b × a).

What’s 16% of 25? The same as 25% of 16: 4.

What’s 43% of 200? The same as 200% of 43: 86.

But to build intuition and really understand it, you need to derive the formula yourself.

Below are four ways to derive it.

 1   2   3   4   5
10   9   8   7   6

Each vertical pair has the same sum: \(1 + 10 = 2 + 9 = \ldots = n + 1\). There are \(\frac{n}{2}\) pairs.

 1   2   3   4   5   6   7   8   9  10
10   9   8   7   6   5   4   3   2   1

Add both rows: every column sums to \(n + 1\), and there are \(n\) columns:

We only want one row, so divide by 2:

x
x x
x x x
x x x x
x x x x x

\(n\) rows, row \(k\) has \(k\) cells — \(1 + 2 + \ldots + n\) in total. Imagine fitting two such triangles together tooth-to-tooth to form a rectangle. Derive the formula yourself from that picture.

We all know that

which we can rewrite as

For \(1, 2, \ldots, n\) the average is easy to eyeball from the middle of the range: \(\frac{n+1}{2}\). There are \(n\) items, so:

For \(1 \ldots 100\): average \(\approx 50.5\), sum \(50.5 \times 100 = 5050\).

Every operation is a transformation. In the real world there are things we want to slide, smoosh and stretch — arithmetic is exactly what gives us the tools to model that.

1. accumulation — counting quantities (total at checkout)
2. slide — move a mark along a scale (temperature)
3. combination — a new quantity from two different ones (sound wave)

1. repetition — several additions in a row
2. scaling — a number grows or shrinks all at once

1. multiply by 1/2: turn a profit of 1 into a profit of 1/2 (“unscale”)
2. multiply by −2: turn a profit of 1 into a loss of 2 (“invert”)

1. share (\(a/b\)) — how much of the whole: ate \(3/8\) of the pizza, tank is \(2/5\) full
2. split (\(a \div b\)) — how many groups of \(b\) in \(a\): \(12 \div 3\) → 4 piles of 3
3. undo scaling — multiplied by 3 → divide by 3 to get back

A fraction looks at part of a whole; division looks at how many times it fits.

1. repeated multiplication — \(2^3 = 2 \times 2 \times 2\)
2. growth across dimensions — length \(\times 2\) → area \(\times 4\) (\(n=2\)), volume \(\times 8\) (\(n=3\))
3. root — inverse power: \(2^3 = 8 \Leftrightarrow \sqrt[3]{8} = 2\); \(a^{1/n}\) links \(\sqrt[n]{a}\) and \(a^n\)

1. \(a^{-n}\) — “invert” growth: not 8, but \(1/8\)
2. \(a^{1/n}\) — “unscale” the exponent: from 8 back to 2 without flipping sign