“the unreasonable effectiveness of mathematics in the natural sciences.”

We can decide what we mean by things like 6 and +, but once we do, the results of expressions like 6 + 6 are beyond our control. Logic leaves us no choice. In that sense, math always involves both invention and discovery: we invent the concepts but discover their consequences.

book A Mathematician’s Lament, Paul Lockhart

a composite number—

Ogawa’s charming novel The Housekeeper and the Professor,

To enjoy working with numbers you don’t have to be Einstein (German for “one stone”), but it might help to have rocks in your head.

Andy Ruina has pointed out, people have concocted all sorts of funny little mental strategies to sidestep the dreaded negative sign.

conceive of multiplication visually.

in many real- world situations, especially where money is concerned, people seem to forget the commutative law,

we’re wired to doubt the commutative law because in daily life, it usually matters what you do first.

It’s the story of the quest for ever more versatile numbers.

Fractions always yield decimals that terminate or eventually repeat periodically—

it can’t be equal to the ratio of any whole numbers. It’s irrational.

Whole numbers and fractions, so beloved and familiar, now appear scarce and exotic.

When we divide an hour into 60 minutes, or a minute into 60 seconds, or a full circle into 360 degrees, we’re channeling the sages of ancient Babylon.

The Roman approach was to elevate a few favored numbers, give them their own symbols, and express all the other, second- class numbers as combinations of those.

the automation of arithmetic was made possible by the beautiful idea of place value.

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.

concentrating on their most beautiful, important, and far- reaching ideas.

solving for x and working with formulas.

Checking that the units cancel properly helps avoid this kind of blunder.

physicist Richard Feynman,

straightedge and compass, the only tools allowed in Greek geometry.

They occur whenever an existing operation is pushed too far, into a domain where it no longer seems sensible.

Here “complex” doesn’t mean “complicated”; it means that two types of numbers, real and imaginary, have bonded together to form a complex, a hybrid number like 2 +

roots of any polynomial are always complex numbers. In

Electrical engineers love complex numbers for exactly this reason.

fractal—an intricate shape whose inner structure repeated at finer and finer scales.

In much harder problems where it may be impossible to find an exact answer— not just in math but in other domains as well— this sort of partial information can be very valuable.

The silver lining is that even wrong answers can be educational . . . as long as you realize they’re wrong.

The undistracted reasoning that this problem requires is one of the most valuable things about word problems. They force us to pause and think, often in unfamiliar ways. They give us practice in being mindful.

relationships are much more abstract than numbers. But they’re also much more powerful. They express the inner logic of the world around us. Cause and effect, supply and demand, input and output,

We should be up front about the fact that word problems force us to make simplifying assumptions. That’s a valuable skill— it’s called mathematical modeling. Scientists do it all the time

negative solution was ignored in ancient times, since a square with a side of negative length is geometrically meaningless. Today, algebra is less beholden to geometry and we regard the positive and negative solutions as equally valid.

instead of wood and steel, the materials that functions pound away on are numbers and shapes and, sometimes, even other functions.

Our brains perform a similar trick when we listen to music. The frequencies of the notes in a scale— do, re, mi, fa, sol, la, ti, do— sound to us like they’re rising in equal steps. But objectively their vibrational frequencies are rising by equal multiples. We perceive pitch logarithmically.

from the Richter scale for earthquake magnitudes to pH measures of acidity, logarithms make wonderful compressors.

What’s important is the axiomatic method, the process of building a rigorous argument, step by step, until a desired conclusion has been established.

Euclid had begun with the definitions, postulates, and self- evident truths of geometry— the axioms— and from them erected an edifice of propositions and demonstrations, one truth linked to the next by unassailable logic.

Having allowed intuition to guide us this far, now and only now is it time for logic to take over and finish the proof.

Once you begin to appreciate the focusing abilities of parabolas and ellipses, you can’t help but wonder if something deeper is at work here. Are these curves related in some more fundamental way?

In the ideal world of numbers and shapes, strange coincidences usually are clues that we’re missing something. They suggest the presence of hidden forces at work.

according to the definition, a parabola consists of all the points that lie just as far from F as they do from

They’re collectively known as conic sections— curves obtained by cutting the surface of a cone with a plane.

These four types of curves appear even more intimately related when viewed from other mathematical perspectives. In algebra, they arise as the graphs of second- degree equations where the constants A, B, C, . . . determine whether the graph is a circle, ellipse, parabola, or hyperbola.

planets move in elliptical orbits with the sun at one focus; or that comets sail through the solar system on elliptic, parabolic, or hyperbolic

trigonometry,

It’s the key to the mathematics of cycles.

Finding the areas of triangles and squares is easy. But working with curved shapes like circles is hard.

This approach is known as the method of exhaustion because of the way it traps the unknown number pi between two known numbers that squeeze it from either side. The bounds tighten with each doubling, thus exhausting the wiggle room for pi.

the strategy is to find a series of approximations that converge to the correct answer as a limit.

Their confusion is understandable. It’s caused by our reliance on graphs to express quantitative relationships.

all scientists translate their problems into the common language of mathematics.

These four steps require a command of geometry, algebra, and various derivative formulas from calculus— skills equivalent to fluency in a foreign language and, therefore, stumbling blocks for many students.

That was a typical use of integrals. They’re all about taking something complicated and slicing and dicing it to make it easier to add up. In a 3-

there’s no surer way to hone the facility with integrals needed for advanced work in every quantitative discipline from physics to finance.

James Gregory, Isaac Barrow, Isaac Newton, and Gottfried Leibniz established what’s now known as the fundamental theorem of calculus.

has a thing or two to say about how many people you should date before settling down.

clue to the ubiquity of e. It often arises when something changes through the cumulative effect of many tiny events.

the heartbreaking uncertainties of romance.

differential equations, which describe how interlinked variables change from moment to moment, depending on their current values.

the nearly 350 years since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations.

“made his head ache, and kept him awake so often, that he would think of it no more.”

greatest achievements of vector calculus lie in that twilight realm where math meets reality.

dragonfly as it hovered in place.

aloft. In this way, vector calculus is helping to explain how dragonflies, bumblebees, and hummingbirds can fly— something that had long been a mystery to conventional fixed- wing aerodynamics.

light. Although experimenters like Faraday and Ampère had previously found key pieces of this puzzle, it was only Maxwell, armed with his mathematics, who put them all together.

Everybody wants to connect the dots, to find the needle of meaning in the haystack of data.

For, as I’ll try to explain, much of modern life makes a lot more sense when you understand these distributions. They’re the new normal.

because fluctuations in stock prices don’t follow normal distributions. They’re better described by heavy- tailed distributions.

more benign things like word frequencies in novels and the number of sexual partners people have.

Although reformulating the data in terms of natural frequencies is a huge help, conditional- probability problems can still be perplexing for other reasons. It’s easy to ask the wrong question or to calculate a probability that’s correct but misleading.

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.

Worrying about content turned out to be an impractical way to rank webpages.

Popularity means nothing on its own. What matters is having links from good pages.

A page is good if good pages link to it, but who decides which pages are good in the first place?

how important that page is relative to the others by computing the proportion of time that a hypothetical Web surfer would spend there.

fluid, a watery substance that flows through the network, draining away from bad pages and pooling at good ones.

algorithm seeks to determine how this fluid distributes itself across the network in the long run.

egalitarian stance.

Those limiting values are what Google’s algorithm would define as the PageRanks for the network.

Linear equations, as opposed to those containing nonlinear terms like x ² or yz or sin x, are comparatively easy to solve.

One of the central tasks of linear algebra, therefore, is the development of faster and faster algorithms for solving such huge sets of equations.

These issues are central to number theory, the subject that concerns itself with the study of whole numbers and their properties and that is often described as the purest part of mathematics.

In other words, we’ve rigged the definition of prime numbers to give us the theorem we want.

The naive view is that we make our definitions, set them in stone, then deduce whatever theorems happen to follow from them.

That would be much too passive. We’re in charge and can alter the definitions as we please— especially if a slight tweak leads to a tidier theorem, as it does here.

How many primes are less than or equal to 10? Or 100? Or an arbitrary number N? This construction is a direct parallel

Mathematicians call it the counting function for the primes.

They never die out completely— we’ve known since Euclid they go on forever—

The validity of the lnN formula as N tends to infinity is now known as the prime number theorem.

(See what a kid can do when not distracted by an Xbox?)

According to the prime number theorem, any particular prime near N has no right to expect a potential mate much closer than lnN away, a gulf much larger than 2 when N is large.

The twin prime conjecture says couples like this will turn up forever.

Brian Hayes explores this problem in the title essay of his book Group Theory in the Bedroom.

As these examples suggest, group theory bridges the arts and sciences. It addresses something the two cultures share— an abiding fascination with symmetry.

(Images like this abound in a terrific book called Visual Group Theory, by Nathan Carter. It’s one of the best introductions to group theory— or to any branch of higher math— I’ve ever read.)

The mattress group is special. Many other groups violate the commutative law. Those fortunate enough to obey it are particularly clean and simple.

That’s one of the charms of group theory. It exposes the hidden unity of things that would otherwise seem unrelated

In topology, two shapes are regarded as the same if you can bend, twist, stretch, or otherwise deform one into the other continuously— that is, without any ripping or puncturing.

But the one thing such a deformation can’t get rid of is the intrinsic loopiness of a circle and a square. They’re both closed curves. That’s their shared topological essence.

You can stretch and twist a Möbius strip all you want, but nothing can change its half- twistedness, its one- sidedness, and its one- edgedness.

Vi Hart

“Möbius music box”

In an era of globalization, Google Earth, and intercontinental air travel, all of us should try to learn a little about spherical geometry and its modern generalization, differential geometry.

If it’s part of a great circle, you can keep the front wheel pointed straight ahead at all times.

this is why the subject is called differential geometry: it studies the effects of small local differences on various kinds of shapes, such as the difference in length between the helical path and its neighbors.)

Likewise, surfaces with holes and handles permit many locally shortest paths, distinguished by their pattern of weaving around various parts of the surface.

They bend to conform to the surface but don’t bend within it. To make this clear, Polthier has produced another illuminating video.

geodesics, like great circles, are the natural generalization of straight lines.

what’s the most efficient way to find the shortest path through a network?

The same is true of all the math that’s been created to help you find the shortest way from here to there when you can’t take the easy way out.

But in private, math is occasionally insecure. It has doubts. It questions itself and isn’t always sure it’s right. Especially where infinity is concerned. Infinity can keep math up at night, worrying, fidgeting, feeling existential dread.

And the therapy that eventually helped it through this crisis came to be known, coincidentally, as analysis.

The debate over the series 1– 1 + 1– 1 + ∙ ∙ ∙ raged for about 150 years, until a new breed of analysts put all of calculus and its infinite processes (limits, derivatives, integrals, infinite series) on a firm foundation, once and for all.

Two of their key notions are partial sums and convergence.

So suppose we keep to the straight and narrow— no dallying with the dark side— and restrict our attention to only those series that converge. Does that get rid of the earlier paradoxes?

If you add up its terms in a different order, you can make it sum to anything.

It’s as if the series had utter contempt for the commutative law of addition. Merely by adding its terms in a different order, you can change the answer— something that could never happen for a finite sum.

And surprisingly enough, it matters in real life too. As we’ve seen throughout this book, even the most abstruse and far- fetched concepts of math often find application to practical things.

with Georg Cantor’s groundbreaking work on set theory.

He defined a rigorous way to compare different infinite sets and discovered, shockingly, that some infinities are bigger than others.

The argument I’ve just presented is a famous one in the theory of infinite sets. Cantor used it to prove that there are exactly as many positive fractions (ratios p/ q of positive whole numbers p and q) as there are natural numbers (1, 2, 3, 4, . . .). That’s a much stronger statement than saying both sets are infinite. It says they are infinite to precisely the same extent, in the sense that a one- to- one correspondence can be established between them.

For it implies we could make an exhaustive list of all positive fractions, even though there’s no smallest one!

We’ve already found it. The fraction p/ q corresponds to passenger p on bus q, and the argument above shows that each of these fractions can be paired off with a certain natural number 1, 2, 3, . . . , given by the passenger’s room number at the Hilbert Hotel.

The conclusion is that the Hilbert Hotel can’t accommodate all the real numbers. There are simply too many of them, an infinity beyond infinity.