Basics on Blog

BetterExplained - Part 2 - Arithmetic

Mon, 01 Jun 2026 00:00:00 +0000

Mental tricks

60 km/h is 1 km per minute

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?

1 year = 250 work days = 2000 work hours

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

Rule of 72: years to double = 72 / interest rate

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.

Mental arithmetic

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.

Visualizing numbers

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.

a% of b = b% of a

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.

Sum from 1 to n

We’re usually just given the formula and told to memorize it

We're usually just given the formula and told to memorize it

\[ \text{Sum from 1 to } n = \frac{n(n+1)}{2} \] \[ \text{Sum from 1 to 100} = \frac{100(100+1)}{2} = (50)(101) = 5050 \]

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

Below are four ways to derive it.

1) Split the sequence in half and add in pairs

 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.

\[ \text{Number of pairs} \times \text{sum of each pair} = \frac{n}{2}(n+1) = \frac{n(n+1)}{2} \]

2) Two rows of numbers

 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:

\[ \text{Sum of both rows} = n(n+1) \]

We only want one row, so divide by 2:

\[ \frac{n(n+1)}{2} \]

3) Picture a triangle

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.

4) Via the average — the most interesting one, in my view

We all know that

\[ \text{average} = \frac{\text{sum}}{\text{number of items}} \]

which we can rewrite as

\[ \text{sum} = \text{average} \times \text{number of items} \]

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:

\[ \text{sum} = \frac{n+1}{2} \cdot n = \frac{n(n+1)}{2} \]

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

Visual arithmetic

Every operation is a transformation

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.

Addition gives tools for:

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

Multiplication gives tools for:

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

Negatives and inverses

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

Division gives tools for:

share (\(a/b\)) — how much of the whole: ate \(3/8\) of the pizza, tank is \(2/5\) full
split (\(a \div b\)) — how many groups of \(b\) in \(a\): \(12 \div 3\) → 4 piles of 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.

Powers and roots give tools for:

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

\(a^n\) and \(\sqrt[n]{a}\) — don’t mix up “reverse”

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

Math with Bad Drawings — overview of Ben Orlin's books

Fri, 13 Mar 2026 00:00:00 +0000

Another way to build math intuition before the formulas — through stories and sketches. Ben Orlin writes as if explaining to a friend over tea: with humor, no pretense, and a focus on why any of this matters.

Author’s site: Math with Bad Drawings.

Math with Bad Drawings

The flagship book. Orlin covers school and early college topics — probability, geometry, statistics, infinity — via comics and everyday analogies.

The strength: math as a way of thinking, not a memorization exam. Pairs well with BetterExplained: more pure intuition there, more narrative and humor here.

Change Is the Only Constant

The second book — calculus and the history of ideas: how people arrived at derivatives, integrals, and limits. A good bridge to derivatives and the circle area example: story and meaning first, symbols second.

Math Games with Bad Drawings

A collection of games and puzzles — logic, combinatorics, strategy. Less directly about ML, but it trains mathematical thinking: spot structure, test hypotheses, don’t fear being wrong.

When to read in this series

After BetterExplained — same “understand, don’t memorize” vibe, in book form.
Alongside the interview math overview — so the checklist topics aren’t just a dry list.

You don’t need all three in order: start with Math with Bad Drawings for a broad tour; with Change Is the Only Constant if calculus is the sticking point.

BetterExplained - Part 1 - Intuitive Math

Tue, 10 Mar 2026 00:00:00 +0000

First article in the BetterExplained series on this blog — not about formulas, but about how to understand them.

Learn Right, Not Rote.

Math is no more about equations than poetry is about spelling.

Formulas and rules are tools. Without a sense of why something works, they become symbols that are easy to forget and hard to apply. In machine learning this shows up everywhere: gradient, matrix, integral — the meaning slips away if there is no intuition behind them.

BetterExplained

BetterExplained is Kalid Azad’s site of intuitive math explanations. The material is built from images and meaning toward notation, not the other way around.

The usual pattern: first what is going on (geometry, analogy, story), then symbols. That makes it easier to connect school math with linear algebra, calculus, and probability for AI.

What is useful for this series

Derivatives and integrals — rate of change and accumulation before the formulas; good prep for derivatives and the circle area example.
Exponents and e — why e^x behaves as it does; useful for softmax, loss, and growth in models.
Vectors and dot products — geometric meaning before matrix algebra.
Euler’s formula — one intuition linking exponentials, sin/cos, and complex numbers.

You do not need to read everything: pick the topic that still feels opaque in a textbook or an ML article.

How to use it with this blog

Here — why the math matters for AI and short ML-oriented write-ups.
On BetterExplained — deeper intuition on classic topics.
Return to the formulas with a mental picture — articles, code, and lectures become easier.

Next in the series: BetterExplained — Part 2 — Arithmetic.