Foundation lessons
Start with the idea that is missing.
45 short lessons on the mathematics and physics the four papers take for granted. Each asks one question, works one example, and says where it stops, so you can go back to the passage you came from.
Written for this edition.
No algebra needed
Arithmetic, a ruler and a little patience are enough.
Adding and averaging
What does an average keep, and what does it lose?
Used in Brownian motion
A sign records direction
How can movement add up to zero?
Used in Brownian motion
Fractions and ratios
What does dividing one quantity by another tell you?
Used in Light quanta · Special relativity
An equals sign states a balance
Is an equals sign an instruction to calculate, or a statement?
Used in Mass and energy
A letter stands for a quantity
What does a letter such as t mean in a formula?
Used in Brownian motion
Squares and square roots
If the mean square grows four times, why does the typical distance only double?
Used in Brownian motion · Special relativity
Powers of ten and physical units
How do you write a very small length without counting zeros?
Used in Light quanta · Mass and energy
Reading a graph
How does a curve show the relationship between two physical quantities?
Used in Light quanta
Marks that look alike and do different jobs
What do Δ, d, a prime and a small raised number each ask of the reader?
Used in Special relativity
A probability, a frequency and a single trial
What does a probability of 1/4 say about one try, and about a hundred?
Used in Light quanta
Units and sizes
What a number means once it carries a unit, and how big the quantities of 1905 are.
Quantities and units
Can a number of metres be compared with a number of seconds?
Used in Light quanta
Ratios and scaling
If one quantity doubles, what happens to the others?
Used in Brownian motion
Orders of magnitude
How big, how fast and how energetic are the things in the 1905 papers, measured against each other?
Used in Light quanta · Brownian motion
The units of 1905
How do Einstein's numbers, printed in centimetres, grams, ergs and the old electrical units, become SI units?
Used in Light quanta · Brownian motion · Special relativity
Rates, curves and sums
The calculus the papers use, one tool at a time.
Functions and graphs
How does a graph record the way one quantity depends on another?
Used in Light quanta
Rates of change and derivatives
How can a body have a speed at a single instant?
Used in Special relativity
Partial derivatives and held-fixed quantities
When a quantity depends on two things, what does its rate of change mean?
Used in Light quanta · Brownian motion · Special relativity
Adding continuously
How do you add up something that varies continuously?
Used in Light quanta · Brownian motion · Special relativity
Approximating a curve near a point
Why can a curve be replaced, close to one point, by a few simple terms?
Used in Brownian motion · Mass and energy
Exponential change
What kind of change does an exponential describe?
Used in Brownian motion
Logarithms: turning products into sums
Why does entropy need a logarithm?
Used in Light quanta · Brownian motion
Vectors, matrices and changing frame
The geometry of arrows and of changing coordinates that the relativity paper works in.
Vectors and their components
How can one arrow have different numbers in different coordinate systems?
Used in Special relativity
Dot and cross products
How much of one arrow lies along another, and which way does a magnetic force push?
Used in Special relativity
Matrices and linear maps
How can a table of four numbers turn one frame's coordinates into another's?
Used in Special relativity
Hyperbolic functions and rapidity
Two speeds along one line do not simply add. Is there a number that does?
Used in Special relativity
Chance and spread
What an average hides, and how a random walk spreads.
Probability and independence
When you square a sum of random steps, why do the mixed terms average to zero?
Used in Light quanta · Brownian motion
Density is not probability
What does the height of a probability curve mean?
Used in Brownian motion
Mean, variance and RMS
If particles wander both ways, which average shows how far they got?
Used in Brownian motion
From steps to spread
Why does typical displacement grow as the square root of the step count?
Used in Brownian motion
The Gaussian and its width
After a time t, where is a spreading particle likely to be?
Used in Brownian motion
Uncertainty, evidence and inference
What can a set of measurements tell you, and what can it not?
Used in Brownian motion
Two measurements, two unknowns
Can a measured spreading rate tell you how big the particle is?
Used in Brownian motion
Physics the papers take for granted
Ideas a reader of the Annalen in 1905 was expected to know.
Energy of motion and inertia
If a body moves at the same speed but carries less energy of motion, what has changed?
Used in Light quanta · Special relativity · Mass and energy
Conservation and symmetry
When an experiment is turned, moved or watched from a moving frame, what should stay the same?
Used in Mass and energy
Temperature and thermal energy
What does a temperature say about the jostling of molecules and grains?
Used in Light quanta · Brownian motion
Entropy and temperature
What does temperature have to do with entropy?
Used in Light quanta
Entropy and the number of ways
Why is entropy the logarithm of a probability?
Used in Light quanta · Brownian motion
Counting what crosses a boundary
If particles keep crossing the edges of a region, what decides whether it fills up or empties?
Used in Brownian motion · Special relativity
What the diffusion coefficient means
What changes when D changes?
Used in Brownian motion
Viscosity and Stokes drag
How hard does a liquid hold back a small sphere drifting through it?
Used in Brownian motion
Osmotic pressure and free energy
Why should grains floating in water push on a wall the way dissolved molecules do?
Used in Brownian motion
Events and distant clocks
What does it mean to say that two distant events happen at the same time?
Used in Special relativity · Mass and energy
Fields and waves
What is a wave in a field, and what does a light detector measure?
Used in Light quanta · Special relativity
Charges, fields and light
What does an electric or a magnetic field do to a charge?
Used in Light quanta · Special relativity
The push of light
How hard does a beam of light push on what it hits?
Used in Special relativity