We can’t all be mathematicians, but we certainly ought to try. So says math scholar Jordan Ellenberg in this demanding, fascinating book.

Ellenberg, a professor of mathematics at the University of Wisconsin, preaches to the unconverted every chance he gets. He writes a column on the subject for the popular website Slate and served as a consultant to the CBS TV series “Numb3rs,” about a team of mathematical crime fighters. So Ellenberg knows how to tell a story. There’s far more zest and wit in these pages than you’ll find in your typical algebra textbook. At times, a bit too much. I occasionally wished he’d slow things down a bit for those of us who’ve fallen out of practice. But never mind; this is a book worth taking your time over, preferably with pencil and paper nearby.

But don’t get the wrong idea. Mathematics is not about calculation. It’s about thinking. “Math,” writes Ellenberg, “is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.” The complex, tedious calculations that made no sense to most of us are like the exercises that the New England Patriots perform in training camp. They’re mainly useful in preparing our brains to use mathematical principles as a guide to accurate thinking about the world.

In one of his most magical illustrations, Ellenberg tells of Abraham Wald, who used mathematical reasoning to help design more robust warplanes during World War II. Military officers had noticed that planes that made it back to base tended to have more bullet holes in the wings and fuselage than in the engine. So they figured to make the aircraft more survivable the key was to add armor to parts that were taking more frequent hits.

Wrong answer, said Wald. He began from the common-sense assumption that bullet hits would occur randomly all over the plane. So why were there fewer holes in the engines? Because those planes that had taken more hits there did not make it back. Wald’s solution: Don’t worry so much about the wings and fuselage. Armor up the engines instead.

The book is divided into five sections, each devoted to a math concept with frequent applications to real life — linearity, inference, expectation, regression, and existence. In linearity, for instance, Ellenberg shows how math overturns our instinctive belief that life moves in straight lines. For instance, will raising tax rates lead to higher or lower tax revenues? Correct answer: It depends, because the relation between rates and revenue is not a straight line. Set rates too low and you won’t collect enough money. But raise rates too high and at some point revenue starts to fall as people decide to work less or make fewer investments to avoid higher taxes.

Or consider inference — the art of applying mathematical analysis to real-world experience to tease out unexpected insights. For instance, when people are asked to pick a random number between zero and seven, they most frequently choose seven. Apparently, people think it sounds more random than the other digits. So when two Columbia University grad students wanted to analyze the controversial 2009 election in Iran, they looked at the vote counts from each of Iran’s 29 provinces. Sure enough, the number seven showed up almost twice as often as would be expected in a fair vote — as if the figures were recorded by individuals wanting the results to appear random. Not absolute proof of vote fraud, but a significant hint.

Ellenberg shows time and again how a mathematician’s knack for asking the right question can avoid all manner of mistakes. For instance, during the 2012 presidential campaign, Republican candidate Mitt Romney proclaimed that under President Obama, 92.3 percent of those who had lost jobs were women.

The claim was based on accurate numbers, but lousy logic. The Romney campaign had calculated net job losses among women from January of 2009 to March 2012, compared with net job losses for both men and women for the same period. But men had lost twice as many jobs as women during this period, only to gain most of them back by 2012. So the Romney campaign’s calculation exaggerated the recession’s impact on women and downplayed a sizable recovery in male employment. Romney was arithmetically correct, but mathematically wrong.

Ellenberg also has some fun with the idea that the orderliness of the universe proves it was created. Maybe, he replies. But with a bit of math, he shows how a universe run by multiple gods is an even more plausible explanation. For that matter, so is the idea that we’re all software running inside a computer created by more intelligent creatures than ourselves. Ellenberg’s not taking sides on the existence of God; he’s merely noting that it’s one of those rare questions about which mathematics can tell us nothing.

There are plenty more practical insights here. For instance, Ellenberg warns that many scientific research studies may be worthless, because of the method used to calculate the reliability of the results. He explains why red-hot, home-run hitters inevitably cool off, and spectacularly successful businesses always lose their edge over time. (I’m looking at you, Microsoft, and Apple, too.) And in a dazzling confirmation of what we already know, Ellenberg proves that it makes no sense to play Powerball — unless the jackpot is really, really big.

If the feel of sand between your toes gets you thinking about Zeno’s Paradox or Pascal’s Wager, Ellenberg’s book is ideal beach reading. But even if your interests lie elsewhere, you may find it a challenging but welcome companion.

*Hiawatha Bray can be reached at hiawatha.bray@globe.com.*