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mathematics, in pictures
Concept

Linearity

The property that a quantity applied to a sum is the sum of the quantities, and scales with a multiple.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

received1011010H111010001110101101001010synsyndromethe bit it names101bit 1111bit 2110bit 3011bit 4100bit 5010bit 6001bit 7000no error1011010 gives syndrome 010, which is column 6 — so bit 6 is wrong and 1011000 is thecodewordthe syndrome is three bits and the message is four: the check finds the error without everrecovering what was sent

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.

computation · error correcting codes 10 → 11.201 → 21.502 → 323 → 434 → 565 → 614.70 draws expected in all6 kinds, and the last is the expensive onethe waits are 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 14.70 drawsthe last one alone costs 6 draws on average, which is why the total grows faster than the number of kinds

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.

probability · expectation

Named alongside it

The objects these essays reach for when they reach for this one.

ApproximationCollisionConvergence rateCosetDecision procedureError correcting codeThe Euler–Mascheroni constantExpectationHamming codeHarmonic seriesNatural logarithmParity

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