The Joy of X by Steven Strogatz

• 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.
• Roman numerals are only slightly more sophisticated than tallies. You can spot the vestige of tallies in the way Romans wrote 2 and 3, as II and III.
• All place-value systems are based on some number called, appropriately enough, the base. Our system is base 10, or decimale. After the ones place, the subsequent consecutive places represent tens, hundreds, thousands, and so on, each of which is a power of 10.
• Our world has been changed by the power of 2. In the past few decades we’ve come to realize that all information--not just numbers, but also language, images, and sound--can be encoded in streams of zeros and ones.
• Algebra is the language in which such patterns are most naturally phrased.
• Complex numbers are magnificent, the pinnacle of number systems. They enjoy all the same properties as real numbers--you can add and subtract them, multiply and divide them--but they are better than real numbers because they always have roots. You can take the square root or cube root or any root of a complex number, and the result will still be a complex number.
• Complex numbers are the culmination of the journey that began with 1.
• Perhaps even more important, word problems give us practice thinking not just about numbers, but about relationships between numbers.
• Relationships are much more abstract than numbers. But they’re also much more powerful. They express the inner logic of the world around us.
• A mathematician needs functions for the same reason that a builder needs hammers and drills. Tools transform things. So do functions.
• Exponential growth is almost unimaginably rapid.
• Parabolic curves and surfaces have an impressive focusing power of their own: each can take parallel incoming waves and focus them at a single point. This feature of their geometry has been very useful in settings where light waves, sound waves, or other signals need to be amplified.
• Sine waves are the atoms of structure. They’re nature’s building blocks. Without them there’d be nothing, giving new meaning to the phrase “sine qua non”.
• The key to thinking mathematically about curved shapes is to pretend they’re made up of lots of little straight pieces. That’s not really true, but it works...as long as you take it to the limit and imagine infinitely many pieces, each infinitesimally small. That’s the crucial idea behind all of calculus.
• Calculus is the mathematics of change. It describes everything from the spread of epidemics to the zigs and zags of a well-throw curveball.
• Roughly speaking, the derivative tells you how fast something is changing; the integral tells you how much it’s accumulating.
• Mathematical signs and symbols are often cryptic, but the bet of them offer visual clues to their own meaning. The symbols for zero, one, and infinity aptly resemble an empty hole, a single mark, and an endless loop.
• How does individual randomness turn into collective regularity? Easy--the odds demand it.
• An idealized version of these bell curves is what mathematicians call the normal distribution. It’s one of the most important concepts in statistics. Part of its appeal is theoretical. The normal distribution can be proven to arise whenever a large number of mildly random effects of similar size, all acting independently, are added together. And many things are like that.
• The normal distribution is not nearly as ubiquitous as it once seemed.
• Power Law distributions have counterintuitive properties from the standpoint of conventional statistics.
• Whether you want to detect patterns in large data sets or perform gigantic computations involving millions of variables, linear algebra has the tools you need.
• One of the central tasks of linear algebra, therefore, is the development of faster and faster algorithms for solving such huge sets of equations. Even slight improvements have ramifications for everything from airline scheduling to image compression.