The Joy of X

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- Just as numbers are a shortcut for counting by ones, addition is a shortcut for counting by any amount. This is how mathematics grows. The right abstraction leads to new insight, and new power. (LocationĀ 132)
- math always involves both invention and discovery: we invent the concepts but discover their consequences. As weāll see in the coming chapters, in mathematics our freedom lies in the questions we askāand in how we pursue themābut not in the answers awaiting us. (LocationĀ 136)
New highlights added February 3, 2023 at 6:29 PM
- Notice something magical here: as the numbers inside the logarithms grew multiplicatively, increasing tenfold each time from 100 to 1,000 to 10,000, their logarithms grew additively, increasing from 2 to 3 to 4. Our brains perform a similar trick when we listen to music. The frequencies of the notes in a scaleādo, re, mi, fa, sol, la, ti, doāsound to us like theyāre rising in equal steps. But objectively their vibrational frequencies are rising by equal multiples. We perceive pitch logarithmically. (LocationĀ 852)
- Mathematicians and conspiracy theorists have this much in common: weāre suspicious of coincidencesāespecially convenient ones. There are no accidents. Things happen for a reason. While this mindset may be just a touch paranoid when applied to real life, itās a perfectly sane way to think about math. In the ideal world of numbers and shapes, strange coincidences usually are clues that weāre missing something. They suggest the presence of hidden forces at work. (LocationĀ 1059)
- Whenever a state of featureless equilibrium loses stabilityāfor whatever reason, and by whatever physical, biological, or chemical processāthe pattern that appears first is a sine wave, or a combination of them. Sine waves are the atoms of structure. Theyāre natureās building blocks. Without them thereād be nothing, giving new meaning to the phrase āsine qua non.ā (LocationĀ 1157)
- Calculus is the mathematics of change. It describes everything from the spread of epidemics to the zigs and zags of a well-thrown curveball. The subject is gargantuanāand so are its textbooks. Many exceed a thousand pages and work nicely as doorstops. (LocationĀ 1253)
- Roughly speaking, the derivative tells you how fast something is changing; the integral tells you how much itās accumulating. (LocationĀ 1257)
- So weāre long overdue to update our slogan for integralsāfrom āIt slices, it dicesā to āRecalculating. A better route is available.ā (LocationĀ 1393)