Where do the numbers in a formula come from?

What Is That Number Recording?

The speed of light and 137. This book separates the numbers that change when you change units from those that do not, and asks whether a number in a formula records nature or the scale we chose.

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Where Did That Number Come From?

Who chose the units? How does dimensional analysis break down? Why does the equals sign have six meanings? A textbook, in sixteen chapters and forty-three exercises, on how to check where the numbers in a formula come from.

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Numbers That Move with the Dimension, and Numbers That Do Not

The volume of a ball of radius 1 is largest in five dimensions; with radius 2, it is twenty-four. In twenty-one stories the book asks whether a number moves when the dimension changes. It does not claim that higher-dimensional worlds exist.

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Where Did That π Come From?

The 4π of a sphere’s surface, the 6π of a ball sinking in syrup, the 8π of the equations of gravity. Across twenty-three formulas the book checks whether each π is a spread in every direction, a full turn, or a number put there by the notation. The same number need not have the same reason.

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There Were Four Kinds of Time

The one word “time” points to four different things: the marks on a clock, the length of a path, the direction of spreading disorder, and the scale of a flow. When someone says time does not exist, this book asks which time they mean.

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Without Assuming the Speed of Light

This book does not begin relativity with “the speed of light is the same for everyone”. From Galileo’s ship alone, a formula with a single dial appears. Twenty-three stories sort assumption from measurement and convention. It is not a book that doubts relativity.

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There Are Three Edges of the Universe

How many light years to the edge of the universe? Counted on the same model, there are three edges. Twenty-four stories, from the distance ladder to black-hole shadows and rotating galaxies, ask what each number actually measures.

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How Many Meanings Do + − × ÷ = Have?

The + − × ÷ = learned in primary school, counted again scene by scene. The same symbol does different work in different situations, and this book checks each difference one at a time.

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The Words of Mathematics Have More Than One Meaning

“Converges” has four meanings. Distance, completeness, independence, unprovable: in nineteen stories about the words of mathematics, the book asks how many things a single word points to.

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The Words of Physics Have More Than One Meaning

There are seven kinds of “mass”: one can be negative, and one changes with the direction of the push. In eighteen stories, from pressure and stability to conservation and symmetry, the book separates what a single word of physics points to.

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The Angle of a Rainbow Does Not Contain the Size of the Drops

The size of the drops does not enter the angle of a rainbow. Rainbows, halos, mirages, the blue sky, soap bubbles: in twenty-two stories of sky, light and colour, the book counts what really decides each one.

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The Boiling Point Is Not a Property of the Substance

A boiling point is not set by the substance alone. Pans, kettles, condensation, salt, rust, hourglasses: twenty-eight stories from kitchen and home show that a number is fixed by the substance together with its surroundings. It is not a book of cooking or health advice.

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A Bicycle Is Not Held Up by Its Gyroscope

A bicycle does not stay up because its wheels spin; counted out, the numbers are orders of magnitude short. Thirty-three stories of vehicles and motion separate the explanation that feels right from the one that adds up.

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Measure Two Shadows, and the Earth Comes Out

Measure two shadows and the size of the Earth appears. Twenty-five stories, from the distance to the Moon to tides and leap years, count how each number was measured. From the same observation, one choice of assumption can shift the answer by forty per cent.

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Strength Is Not a Constant of the Material

Strength is not set by the material alone, but by shape, size, direction, how it was made and what it is paired with. Twenty-five stories, from columns and glass to screws and springs, count what decides it. It is not a design manual.

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A Lifetime of Heartbeats Does Not Depend on Body Weight

The number of heartbeats in a lifetime does not depend on body weight. Thirty-two stories of living things, from brains, bones and sleep to immunity and striped patterns, ask what each number is counting. It is not a book of medical advice.

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Why Does a Drum Have No Pitch?

Why does a drum have no pitch? Seventeen stories, from strings, pipes and octaves to echoes, beats and voices, show that what we took for properties of sound are fixed by shape, ratio and the listener.

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Solubility Is Not a Constant of the Substance

Solubility is not set by the substance alone. Twenty-one stories, from colloids and half-lives to osmotic pressure, batteries, rust and catalysts, ask whether the words of chemistry really point to properties of a substance.

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Half of the 15 Puzzle Cannot Be Solved

Half of all arrangements of the 15 puzzle can never be solved. Twenty-one stories, from Lights Out and Nim to noughts and crosses and Othello, check known theorems by counting every case on small boards. It is not a book of winning strategies.

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In a 99 % Accurate Test, 9 in 10 Positives Are Wrong

A positive result on a 99% accurate test may be right 9% of the time or 99%, depending on who is tested. Twenty-eight stories, from the three doors to regression to the mean and significance levels, ask what each number is a number about.

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The ABC Conjecture Is a Theorem for Polynomials

The abc conjecture is unsolved for whole numbers, yet for polynomials it is a theorem; the difference is whether you can differentiate. Seventeen stories, from primes and Collatz to the Riemann hypothesis, separate theorem, conjecture and mere counting. No conjecture is proved here.

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“Unsolvable” Comes in Three Kinds

The quintic cannot be solved, yet its roots can be written to a hundred digits. “Unsolvable” is about a number and the tools you allow. Twenty-six kinds of “cannot” are checked by actually counting small cases.

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Six Brakes

When the same number turns up in different formulas, it is tempting to write that they are the same thing. This is a book of six questions for stopping right there. They came from the author’s own drafts, and in the second half they are turned on themselves.

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How Outsiders Think

Someone who did not assume the speed of light; someone who read the space of velocities as curved. Stories of people who stood a little off the main road, and of four working habits a self-taught author has relied on.

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Zero Tension

The masters had learned, in their bodies, how the rod tip moves when nothing is biting. Stories of the anglers the author met on a breakwater as a schoolboy, and of how that stance has the same shape as the mathematics of reading causes from results. It is not a fishing manual.

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What Do Chart Tools Measure?

What do the numbers from chart indicators actually measure? The book checks them against four markets from 2019 to 2024. It is neither a winning method nor investment advice.

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What Does a Backtest Count?

Try forty-nine strategies with no real skill and keep the best one, and it will still look good. Twenty-four stories, from maximum drawdown to win rate and expected value, ask what each number is counting. It is not investment advice.

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A Bar Is a Clock

A one-minute candle lasts 57.5 seconds at the median, not 60. Time candles, tick candles, renko: in thirteen stories the book asks what each kind of “one candle” holds constant. It is not investment advice.

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What Are We Exchanging for Money?

Behind the Price Tag / What We Eat

Why did rice balls get dearer? The book divides prices by the hourly wage of each year and, using national statistics, splits where 100 yen goes into six lines. Ramen, kerosene, rent, rice, eggs, beer: a look at what lies behind the price tag.

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What Are We Exchanging for Money?

What We Burn / What We Ride

Kerosene, petrol, electricity and gas, buses and taxis, new cars, monthly payments, and the money that leaves just for owning a car. The book uses national statistics to count what lies behind the price of what we burn and what we ride. It is not a book about making money.

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How Big Must a Town Be for That Trade to Appear?

“There is nothing in this town.” Using national statistics, the book counted how many establishments of 526 kinds of business exist in every municipality. The number of kinds of business in a town turned out to be set almost entirely by its population. It is not a guide to starting a business.

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Reading the Kushiro Region Through Records

What has been caught in the sea off Kushiro? How many days a year is there fog? Why is Kushiro’s soba green? Twenty-seven stories of the Kushiro region in eastern Hokkaido, written from public records. It is not a travel guide.

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A Field Guide to Patterns Born from Formulas

Roll a gear and a flower appears; three lines of equations give a butterfly. A field guide of sixty-six patterns computed and drawn from short formulas and rules. Each comes with a picture, its formula and what was checked by counting.

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Twenty Questions Seen in 3D

Why can’t you comb away a crown? Why must every world map distort? The captions and pictures of videos that read twenty questions with moving 3D images, gathered in one book. Each chapter also says what it does not claim.

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Building the Universe out of Light

In the formulas, the speed of light appears raised to the second power, the fourth, and more. Could those exponents be counting dimensions? A book of imagination that keeps verified physics and the dream clearly apart.

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