Linearity
Named by 10 essays across 5 fields — each of them below, with the objects they name alongside it.
Finding the error without reading the message
Three parity checks on a seven-bit word produce three bits. If they are all zero nothing is wrong; otherwise they are the number of the position that broke. The message is never consulted, because the answer does not depend on it.
How long until every one turns up
Draw at random from six equally likely kinds until all six have appeared. The wait is not six draws, and it is not sixty; it is fourteen point seven, and the number is a harmonic sum wearing a hat.
What a map throws away
A linear map redraws the grid, and the determinant measures how much it stretches area. When that measurement comes out zero the map has flattened the plane onto a line — and the question worth asking is not how much was lost but how much survived, because the two always add to what there was.
A rectangle grown on two sides
A product of two changing quantities is the area of a rectangle whose sides both move. The extra area is two strips and a corner, and the whole of the product rule is the observation that the corner is negligible and the strips are not.
The flat map that fits closest
A derivative is usually met as a number, which works because a line through a point is described by one. In more than one dimension the object that plays the same role is a linear map, and the number was always a one-by-one instance of it.
The one that hardly ever comes up
Make the kinds unequally likely and the tidy decomposition into stages fails, because a stage's rate now depends on which kinds turned up rather than on how many. What replaces it is an alternating sum over every subset — and the rarest kind turns out to be nearly the whole answer.
Two patterns, one chance, different waits
HTH and HTT are equally likely in any given window of three tosses. Waiting for HTH takes ten tosses on average and waiting for HTT takes eight, and the difference is not about probability at all — it is about what a failed attempt leaves behind.
Counted across and counted down
A rectangular array has a number of independent rows and a number of independent columns. The two are counted in different spaces, from different objects, by computations that share nothing — and they are always the same number, which is why 'rank' is one word.
A function that adds and is nowhere a line
Every continuous function with f(x + y) = f(x) + f(y) is a straight line through the origin. Drop continuity and, given the axiom of choice, there are others — functions that add perfectly and whose graphs are scattered densely over the whole plane. A finite piece of the construction can be drawn exactly; the whole of it needs a basis of the real numbers that no one can write down.
A slope for a curve that is no function
The folium x³ + y³ = 3xy loops back over itself, so no formula y = f(x) describes it, and yet at almost every point it has a perfectly good tangent. Differentiating the equation as it stands gives the slope, −(∂F/∂x) ÷ (∂F/∂y), and the only points where that fails are the ones where the curve turns vertical or crosses itself — which are exactly the points where it stops being a graph.
Named alongside it
The objects these essays reach for when they reach for this one.
ApproximationBasisDerivativeExpectationLimitMatrixSample spaceConvergence rateDeterminantDimensionHarmonic seriesKernel