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Classical physics uses complex numbers constantly and can always discard them at the end. Quantum physics cannot, and the reason is exact: take the real part of a classical solution and you still have a solution — take the real part of a wavefunction and you have nothing. A lesson for philosophers and theologians on what F = ma actually claims, and what quantum mechanics does and does not license. … readThe i Is in the Equation
Turning is not a special kind of motion. It is the only thing that can happen when a thing’s rate of change is perpendicular to the thing itself — one line of algebra, not a mystery. Why i is the quarter turn, where the minus signs in the cosine series come from, and exactly how much circle each Taylor term buys you. … readRotation Is What Perpendicular Does
Your clock and mine disagree for the same reason two people at different angles disagree about a door: we are reading components off different axes. What a frame of reference actually is, why your time differs from mine, and which quantities were never yours to disagree about. … readA Frame Is an Angle
Momentum is mass times velocity, and a real number cannot rotate a complex one — so momentum inherits velocity’s imaginary axis whole. Why the quantum momentum operator has an i in it, what the Born rule actually deletes, and exactly where the metaphor stops being true. … readMass Cannot Turn You
Confined to a single line, straightedge and compass reach exactly the integers — not one-half, not √2. Every other number on the number line is a souvenir brought back from a dimension the line does not have. … readThe Line Cannot Do Its Own Arithmetic
Position, velocity, acceleration, jerk, snap, crackle, pop. Your body reads exactly half that ladder — the real half. A Tegmark-style lesson for philosophy majors: you cannot feel the Earth’s thousand miles an hour because velocity is imaginary. The mathematics is not the description. It is the situation. … readYou Only Feel the Real Part
Students meet e twice — at the bank and as its own derivative. Here is a third story: run the circle lesson’s engine without the turn, and e is where the walk lands — and the exchange rate for how many steps any walk needs. … reade Is Where You Land If You Don’t Turn
Every term of the Taylor series of e^iθ is the previous term turned 90° and shrunk. Chain those perpendicular steps and a circle appears — an interactive lesson that builds rotation without assuming the unit circle. … readHow Straight Lines Make a Circle
A logarithm is the bridge between a world that multiplies and a mind that adds. Once you see that, my equation for felt experience stops being a modelling choice and becomes the only option available. … readWhat Is a Logarithm
Access in 2030: Education Without Borders By 2030, a child