In this Very Short Introduction, Peter M. Higgins, a renowned popular-science writer, unravels the world of numbers, demonstrating its richness and providing an overview of all the number types that feature in modern science and mathematics. Indeed, Higgins paints a crystal-clear picture of the number world, showing how the modern number system matured over many centuries, and introducing key concepts such as integers, fractions, real and imaginary numbers, and complex numbers. Higgins sheds light on such fascinating topics as the series of primes, describing how primes are now used to encrypt confidential data on the internet. He also explores the infinite nature of number collections and explains how the so-called real numbers knit together to form the continuum of the number line. Written in the fashion of Higgins' highly popular science paperbacks, Numbers accurately explains the nature of numbers and how so-called complex numbers and number systems are used in calculations that arise in real problems.

    In this small volume, Miranda Lundy presents a unique introduction to this most ancient and timeless of universal sciences.Sacred Geometry demonstrates what happens to space in two dimensions - a subject last flowering in the art, science and architecture of the Renaissance and seen in the designs of Stonehenge, mosque decorations and church windows. With exquisite hand-drawn images throughout showing the relationship between shapes, the patterns of coin circles, and the definition of the golden section, it will forever alter the way in which you look at a triangle, hexagon, arch, or spiral.

    Sousa discusses the cognitive mechanisms for learning mathematics and the environmental and developmental factors that contribute to mathematics difficulties. This award-winning text examines:Children's innate number sense and how the brain develops an understanding of number relationships Rationales for modifying lessons to meet the developmental learning stages of young children, preadolescents, and adolescents How to plan lessons in PreK-12 mathematics Implications of current research for planning mathematics lessons, including discoveries about memory systems and lesson timing Methods to help elementary and secondary school teachers detect mathematics difficulties Clear connections to the NCTM standards and curriculum focal points

    This is the amazing story of how the "map problem" was solved.The problem posed in the letter came from a former student: What is the least possible number of colors needed to fill in any map (real or invented) so that neighboring counties are always colored differently? This deceptively simple question was of minimal interest to cartographers, who saw little need to limit how many colors they used. But the problem set off a frenzy among professional mathematicians and amateur problem solvers, among them Lewis Carroll, an astronomer, a botanist, an obsessive golfer, the Bishop of London, a man who set his watch only once a year, a California traffic cop, and a bridegroom who spent his honeymoon coloring maps. In their pursuit of the solution, mathematicians painted maps on doughnuts and horseshoes and played with patterned soccer balls and the great rhombicuboctahedron. It would be more than one hundred years (and countless colored maps) later before the result was finally established. Even then, difficult questions remained, and the intricate solution--which involved no fewer than 1,200 hours of computer time--was greeted with as much dismay as enthusiasm.Providing a clear and elegant explanation of the problem and the proof, Robin Wilson tells how a seemingly innocuous question baffled great minds and stimulated exciting mathematics with far-flung applications. This is the entertaining story of those who failed to prove, and those who ultimately did prove, that four colors do indeed suffice to color any map.

    Behaviour that to our grandparents' generation might have seemed stupid, harmful or simply wicked now seems rational, natural, woven into the very logic of things. And, asserts Jonathan Aldred in this revelatory new book, it's economics that's to blame.Licence to be Bad tells the story of how a group of economics theorists changed our world, and how a handful of key ideas, from free-riding to Nudge, seeped into our decision-making and, indeed, almost all aspects of our lives. Aldred reveals the extraordinary hold of economics on our morals and values. Economics has corrupted us. But if this hidden transformation is so recent, it can be reversed. Licence to be Bad shows us where to begin.Over the past fifty years, the way we value what is 'good' and 'right' has changed dramatically. Behaviour that to our grandparents' generation might have seemed stupid, harmful or simply wicked now seems rational, natural, woven into the very logic of things. And, asserts Jonathan Aldred in this revelatory new book, it's economics that's to blame.Licence to be Bad tells the story of how a group of economics theorists changed our world, and how a handful of key ideas, from free-riding to Nudge, seeped into our decision-making and, indeed, almost all aspects of our lives. Aldred reveals the extraordinary hold of economics on our morals and values. Economics has corrupted us. But if this hidden transformation is so recent, it can be reversed. Licence to be Bad shows us where to begin.

    More than two centuries after Euler's death, it is still regarded as a conceptual diamond of unsurpassed beauty. Called Euler's identity or God's equation, it includes just five numbers but represents an astonishing revelation of hidden connections. It ties together everything from basic arithmetic to compound interest, the circumference of a circle, trigonometry, calculus, and even infinity. In David Stipp's hands, Euler's identity formula becomes a contemplative stroll through the glories of mathematics. The result is an ode to this magical field.

    Just about everyone has at least heard of Albert Einstein's formulation of 1905, which came into the world as something of an afterthought. But far fewer can explain his insightful linkage of energy to mass. David Bodanis offers an easily grasped gloss on the equation. Mass, he writes, "is simply the ultimate type of condensed or concentrated energy," whereas energy "is what billows out as an alternate form of mass under the right circumstances." Just what those circumstances are occupies much of Bodanis's book, which pays homage to Einstein and, just as important, to predecessors such as Maxwell, Faraday, and Lavoisier, who are not as well known as Einstein today. Balancing writerly energy and scholarly weight, Bodanis offers a primer in modern physics and cosmology, explaining that the universe today is an expression of mass that will, in some vastly distant future, one day slide back to the energy side of the equation, replacing the "dominion of matter" with "a great stillness"--a vision that is at once lovely and profoundly frightening. Without sliding into easy psychobiography, Bodanis explores other circumstances as well; namely, Einstein's background and character, which combined with a sterling intelligence to afford him an idiosyncratic view of the way things work--a view that would change the world. --Gregory McNamee

    ENTANGLEMENT. REALITY. Books containing these three words are typically fluff or incomprehensible; this one is not. "Q is for Quantum" teaches a theory at the forefront of modern physics to an audience presumed to already know only basic arithmetic. Topics covered range from the practical (new technologies we can expect soon) to the foundational (old ideas that attempt to make sense of the theory). The theory is built up precisely and quantitatively. Deceptively vague jargon and analogies are avoided, and mysterious features of the theory are made explicit and not skirted. The tenacious reader will emerge with a better technical understanding of why we are troubled by this theory than that possessed by many professional physicists.

    Our senses are not equipped to see the universe as it is, and routinely let us down in that quest—the world appears flat, the stars seem fixed in the heavens. That is why we invented math and science—the ultimate toolkit to explain how the world really works.In this book, mathematician Hannah Fry and geneticist Adam Rutherford investigate everyday mysteries (does your dog love you?) alongside the burning questions of time, space, and the origin of the universe. What will the end of the world look like, and when will it happen? What is time, and where does it come from? Approaching these questions and more with science and math rigorous and playful, Rutherford and Fry celebrate the weirdness of the cosmos, and reveal its secrets along the way.