THE JOY OF X by Steven Strogatz

• numbers are wonderful shortcuts.
• Sure, they are great timesavers, but at a serious cost in abstraction.
• Just as numbers are a shortcut for counting by ones, addition is a shortcut for counting by any amount.
• The right abstraction leads to new insight, and new power.
• Subtraction can generate negative numbers.
• in many real-world situations, especially where money is concerned, people seem to forget the commutative law, or don’t realize it applies.
• Roman numerals may look impressive, but they’re hard to read and cumbersome to use.
• 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.
• Ten is marked by a position—the tens place—instead of a symbol. The same is true for 100, or 1,000, or any other power of 10. Their distinguished status is signified not by a symbol but by a parking spot, a reserved piece of real estate.
• All numbers can be expressed with the same ten digits, merely by slotting them into the right places.
• Any calculation involving any pair of numbers, no matter how big, can be performed by applying the same sets of facts, over and over again, recursively.
• With place-value systems, you can program a machine to do arithmetic.
• But the unsung hero in this story is 0. Without 0, the whole approach would collapse. It’s the placeholder that allows us to tell 1, 10, and 100 apart.
• All place-value systems are based on some number called, appropriately enough, the base.
• 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, for example, may have struck you as a dizzying mix of symbols, definitions, and procedures, but in the end they all boil down to just two activities—solving for x and working with formulas.
• Solving for x is detective work.
• Working with formulas, by contrast, is a blend of art and science.
• In fact, you never get back to even when you lose and gain by the same percentage in consecutive years.
• 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.
• a grand statement called the fundamental theorem of algebra says that the roots of any polynomial are always complex numbers.
• So multiplying by i produces a rotation counterclockwise by a quarter turn.
• Electrical engineers love complex numbers for exactly this reason. Having such a compact way to represent 90-degree rotations is very useful when working with alternating currents and voltages, or with electric and magnetic fields, because these often involve oscillations or waves that are a quarter cycle (i.e., 90 degrees) out of phase.
• By looking at an extreme, or limiting, case, we can see that that answer can’t possibly be right.
• relationships are much more abstract than numbers. But they’re also much more powerful.
• Today, algebra is less beholden to geometry and we regard the positive and negative solutions as equally valid.
• A mathematician needs functions for the same reason that a builder needs hammers and drills. Tools transform things. So do functions.
• Power functions like these are the building blocks that scientists and engineers use to describe growth and decay in their mildest forms.
• Exponential growth is almost unimaginably rapid.
• it’s useful to have tools that can undo the actions of other tools.
• We perceive pitch logarithmically.
• In every place where they arise, from the Richter scale for earthquake magnitudes to pH measures of acidity, logarithms make wonderful compressors. They’re ideal for taking quantities that vary over a wide range and squeezing them together so they become more manageable.
• The Pythagorean theorem tells you how long the diagonal is compared to the sides of the rectangle.
• EVERY MATH COURSE contains at least one notoriously difficult topic. In arithmetic, it’s long division. In algebra, it’s word problems. And in geometry, it’s proofs.
• What’s important is the axiomatic method, the process of building a rigorous argument, step by step, until a desired conclusion has been established.
• 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 focusing property of parabolas is just as useful when deployed in reverse.
• Mathematicians and conspiracy theorists have this much in common: we’re suspicious of coincidences—especially convenient ones.
• Sine waves are the atoms of structure. They’re nature’s building blocks.
• The key to thinking mathematically about curved shapes is to pretend they’re made up of lots of little straight pieces.
• Calculus is the mathematics of change.
• Roughly speaking, the derivative tells you how fast something is changing; the integral tells you how much it’s accumulating.
• Change is most sluggish at the extremes precisely because the derivative is zero there. Things stand still, momentarily.
• The optimal strategy, however, is to stop playing the field a little sooner, after only 1/e, or about 37 percent, of your potential dating lifetime. That gives you a 1/e chance of ending up with Dreamboat. As long as Dreamboat isn’t playing the e game too.
• In all cases, the business of theoretical physics boils down to finding the right differential equations and solving them.
• things that seem hopelessly random and unpredictable when viewed in isolation often turn out to be lawful and predictable when viewed in aggregate.
• 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.
• Power-law distributions have counterintuitive properties from the standpoint of conventional statistics.
• Events don’t have to follow their probabilities;
• And just as everything is composed of atoms, every number is composed of primes.
• Like straight lines in ordinary space, great circles on a sphere contain the shortest paths between any two points.