Greene, B. (Brian), 1963-

New York : WW Norton & Co., ©2003.

xv, 448 pages : illustrations ; 21 cm

Pt. 1. Edge of knowledge -- ch. 1. Tied up with string -- pt. 2. Dilemma of space, time, and the quanta -- ch. 2. Space, time, and the eye of the beholder -- ch. 3. Of warps and ripples -- ch. 4. Microscopic weirdness -- ch. 5. Need for a new theory : general relativity vs. quantum mechanics -- pt. 3. Cosmic symphony -- ch. 6. Nothing but music : the essentials of superstring theory -- ch. 7. "Super" in superstrings -- ch. 8. More dimensions than meet the eye -- ch. 9. Smoking gun : experimental signatures -- pt. 4. String theory and the fabric of spacetime -- ch. 10. Quantum geometry -- ch. 11. Tearing the fabric of space -- ch. 12. Beyond strings : in search of m-theory -- ch. 13. Black holes : a string/m-theory perspective -- ch. 14. Reflections on cosmology -- pt. 5. Unification in the twenty-first century -- ch. 15. Prospects.

Relates the scientific story and the human struggle behind the search for the string theory--the ultimate theory which scientists believe is capable of describing all physical phenomena, large and small; and discusses how the theory is impacting human understanding of space and time.

New York, NY : Penguin Random House, Inc.

volumes : illustrations ; 28 cm

At head of title: The Princeton Review.

New York, NY : Penguin Random House, Inc.

volumes : illustrations ; 28 cm

At head of title: The Princeton Review.

New York, NY : Penguin Random House, c2015-

At head of title: The Princeton Review.

New York, NY : Penguin Random House, c2014-

At head of title: The Princeton Review.

Ananthaswamy, Anil, author.

©2018

x, 290 pages : illustrations ; 24 cm

Prologue: The story of nature taunting us -- The case of the experiment with two holes : Richard Feynman explains the central mystery -- What does it mean "to be"? : the road to reality, from Copenhagen to Brussels -- Between reality and perception : doing the double slit, one photon at a time -- From sacred texts : revelations about spooky action at a distance -- To erase or not to erase : mountaintop experiments take us to the edge -- Bohmian rhapsody : obvious ontology evolving the obvious way -- Gravity kills the quantum cat? : the case for bringing spacetime into the mix -- Healing an ugly scar : the many worlds medicine -- Epilogue: Ways of looking at the same thing?

"It's the story of quantum mechanics told through the lens of the 'double-slit' experiment, showing how light passing through two slits cut into a cardboard sheet first challenged our understanding of light and the nature of reality almost two hundred years ago--and continues to do so"-- Provided by publisher.

New York, NY : Random House,

v. : ill. ; 28 cm.

At head of title: The Princeton Review.

New York, NY : Random House

At head of title: The Princeton Review.

Zager, Tracy, 1972-

Portland, Maine : Stenhouse Publishers, [2017]

xvi, 376 pages : illustrations (chiefly color) ; 26 cm

Acknowledgments -- Breaking the cycle -- What do mathematicians do? -- Mathematicians take risks -- Mathematicians make mistakes -- Mathematicians are precise -- Mathematicians rise to a challenge -- Mathematicians ask questions -- Mathematicians connect ideas -- Mathematicians use intuition -- Mathematicians reason -- Mathematicians prove -- Mathematicians work together and alone -- "Favorable conditions" for all math students.

"While mathematicians describe mathematics as playful, beautiful, creative, and captivating, many students describe math class as boring, stressful, useless, and humiliating. In Becoming the Math Teacher You Wish You'd Had, Tracy Zager helps teachers close this gap by making math class more like mathematics. Tracy spent years with highly skilled math teachers in a diverse range of settings and grades. You'll find this book jam-packed with new thinking from these vibrant classrooms. You'll grapple with big ideas: How is taking risks inherent to mathematics? How do mathematicians balance intuition and proof? How can teachers value both productive mistakes and precision? You'll also find dozens of practical teaching techniques you can try in your classroom right away--strategies to stimulate students to connect ideas; rich tasks that encourage students to wonder, generalize, conjecture, and persevere; routines to teach students how to collaborate. All teachers can move toward increasingly authentic, delightful, robust mathematics teaching and learning for themselves and their students. This important book helps us develop instructional techniques that will make the math classes we teach so much better than the math classes we took." -- Provided by publisher.

Becker, Adam, 1984- author.

©2018

ix, 370 pages : illustrations ; 24 cm

Introduction -- Prologue : The impossible done -- Part I. A tranquilizing philosophy. The measure of all things -- Something rotten in the eigenstate of Denmark -- Street brawl -- Copenhagen in Manhattan -- Part II. Quantum dissidents. Physics in exile -- It came from another world! -- The most profound discovery of science -- More things in heaven and earth -- Part III. The great enterprise. Reality underground -- Quantum spring -- Copenhagen versus the universe -- Outrageous fortune -- Appendix. Four views of the strangest experiment.

"Quantum mechanics is humanity's finest scientific achievement. It explains why the sun shines and how your eyes can see. It's the theory behind the LEDs in your phone and the nuclear hearts of space probes. Every physicist agrees quantum physics is spectacularly successful. But ask them what quantum physics means, and the result will be a brawl. At stake is the nature of the Universe itself. What does it mean for something to be real? What is the role of consciousness in the Universe? And do quantum rules apply to very small objects like electrons and protons, but not us? In What is Real?, Adam Becker brings to vivid life the brave researchers whose quest for the truth led them to challenge Bohr: David Bohm, who picked up Einstein's mantle and sought to make quantum mechanics deterministic, all while being hounded by the forces of McCarthyism; Hugh Everett, who argued that everything, big and small, must be governed by the same rules; and John Bell, who went to great lengths to eradicate the power of the god-like observer from the core of quantum physics. And they paid dearly, their reputations, careers, and sometimes lives ruined completely. But history has been kinder to them than their contemporaries were. As Becker shows, the brave intellectual giants have inspired a growing army of physicists and philosophers intent both on making a philosophically more satisfying theory of the universe and a more useful one as well. A gripping story of some of humanity's greatest ideas and the high cost with which many have pursued them, What is Real? is intellectual history at its passionate best"-- Provided by publisher.

Gray, Theodore W., author.

New York, NY : Black Dog & Leventhal Publishers, Hachette Book Group, 2017.

vii, 216 pages : illustrations (chiefly color) ; 26 cm

Includes index.

Chemistry is magic. Is that really magic? ; Ancient magic was mostly chemistry ; The reactions I remember from when I was but a wee lad -- Atoms, elements, molecules, reactions. What is a molecule? ; What force holds molecules together? ; What is a chemical reaction? ; Energy ; Follow the energy ; The arrow of time ; Entropy -- Fantastic reactions and where to find them. In the classroom ; In the kitchen ; In the lab ; In a factory ; On the street ; In you -- On the origin of light and color. Absorbing light ; Emitting light ; The ancient Chinese art of chemical arranging -- The boring chapter. Watching paint dry ; Watching grass grow ; Watching water boil -- The need for speed. Weathering ; Fire ; Fast fire ; Really fast fire ; The fastest reaction of all.

The book begins with a brief recap of elements and molecules and then goes on to explain important concepts the characterize a chemical reaction, including Energy, Entropy, and Time. It is then organized by type of reaction including Combustion, Photosynthesis, Respiration, Oxidation, and Fermentation. A special section dedicated to chemical cycles includes The Carbon Cycle, The Iron Cycle, and The Lime Cycle.

Rovelli, Carlo, 1956- author.

New York : Riverhead Books, 2018.

240 pages : illustrations ; 20 cm

Originally published in Italian: L'ordine del tempo (Milan : Adelphi Edizioni, 2017).

Preface : perhaps time is the greatest remaining mystery -- The crumbling of time -- Loss of unity -- Loss of direction -- The end of the present -- Loss of independence -- Quanta of time -- The world without time -- The world is made of events, not things -- The inadequacy of grammar -- Dynamics as relation -- The sources of time -- Time is ignorance -- Perspective -- What emerges from a particularity -- The scent of the madeleine -- The source of time.

Why do we remember the past and not the future? What does it mean for time to "flow"? Do we exist in time or does time exist in us? In lyric, accessible prose, Carlo Rovelli invites us to consider questions about the nature of time that continue to puzzle physicists and philosophers alike.-- Provided by publisher.

McManaman, Yelena, author.

Cary, NC : Delta Stream Media, [2013]

90 pages : color illustrations ; 24 cm

Why play this book? -- Questions & answers -- Symmetry -- Number -- Function -- Grid.

"How do you want your child to feel about math? Confident, curious and deeply connected? Then Moebius Noodles is for you. It offers advanced math activities to fit your child's personality, interests, and needs. The book shows you how to go beyond your own math limits and anxieties to do so. It opens the door to a supportive online community that will answer your questions and give you ideas along the way. Learn how you can create an immersive rich math environment for your baby. Find out ways to help your toddler discover deep math in everyday experiences. Play games that will develop your child's sense of happy familiarity with mathematics."--Provided by publisher.

Still, Ben, author.

©2017

176 pages : illustrations (chiefly color) ; 24 cm

Includes index.

Building blocks and construction rules -- Building a universe -- Electromagnetism and QED -- The strong force and QCD -- The weak force and breaking symmetries -- Broken symmetry and mass -- Problems with ghosts -- Violated symmetry -- Future.

"A unique and fascinating exploration of the building blocks that make up our Universe, Particle physics brick by brick illustrates and illuminates the twelve core building block particles and the forces that act upon them to create the world as we know it. Starting with the Big Bang and ending with the Higgs boson particle and the future beyond, this is a comprehensive and uniquely visual guide to quantum physics"--Back cover.

Dyson, Freeman J., author.

New York : Liveright Publishing Corporation, [2018]

xvi, 400 pages ; 25 cm

Great minds around the billiard table -- War and peace -- Truth and reconciliation -- Cornell student -- Go west, young man -- Demigods on stilts -- Nolo contendere -- Well, doc, you're in -- The physicist in love -- Cornell professor -- Mycenean tablets and spin waves -- Moscow and La Jolla -- The forsaken merman -- A spaceship and a wedding -- Homecoming -- Working for peace -- Marching for justice -- Sitting in judgment -- Two deaths and two departures -- Adventures of a psychiatric nurse -- Whale worshippers and moonchildren.

"Both recalling his life story and recounting many of the major advances in twentieth-century science, a renowned physicist shares his autobiography through letters. While recognizing that quantum mechanics "demands serious attention," Albert Einstein in 1926 admonished fellow physicist Max Born that the theory "does not bring us closer to the secrets of the Old One." Aware that "there are deep mysteries that Nature intends to keep for herself," Freeman Dyson, the 94-year-old theoretical physicist, has nonetheless chronicled the stories of those who were engaged in solving some of the most challenging quandaries of twentieth-century physics. Written between 1940 and the early 1980s, these letters to relatives form an historic account of modern science and its greatest players, including J. Robert Oppenheimer, Richard Feynman, Stephen Hawking, and Hans Bethe. Whether reflecting on the horrors of World War II, the moral dilemmas of nuclear development, the challenges of the space program, or the considerable demands of raising six children, Dyson offers a firsthand account of one of the greatest periods of scientific discovery of our modern age"-- Provided by publisher.

Havil, Julian, 1952- author.

©2014

xv, 279 pages : illustrations ; 24 cm

Life and lineage -- Revelation and recognition -- A new tool for calculation -- Constructing the canon -- Analogue and digital computers -- Logistics : the art of computing well -- Legacy.

John Napier (1550-1617) is celebrated today as the man who invented logarithms--an enormous intellectual achievement that would soon lead to the development of their mechanical equivalent in the slide rule: the two would serve humanity as the principal means of calculation until the mid-1970s. Yet, despite Napier's pioneering efforts, his life and work have not attracted detailed modern scrutiny. John Napier is the first contemporary biography to take an in-depth look at the multiple facets of Napier's story: his privileged position as the seventh Laird of Merchiston and the son of influential Scottish landowners; his reputation as a magician who dabbled in alchemy; his interest in agriculture; his involvement with a notorious outlaw; his staunch anti-Catholic beliefs; his interactions with such peers as Henry Briggs, Johannes Kepler, and Tycho Brahe; and, most notably, his estimable mathematical legacy. Julian Havil explores Napier's original development of logarithms, the motivations for his approach, and the reasons behind certain adjustments to them. Napier's inventive mathematical ideas also include formulas for solving spherical triangles, "Napier's Bones" (a more basic but extremely popular alternative device for calculation), and the use of decimal notation for fractions and binary arithmetic. Havil also considers Napier's study of the Book of Revelation, which led to his prediction of the Apocalypse in his first book, A Plaine Discovery of the Whole Revelation of St. John--the work for which Napier believed he would be most remembered. John Napier assesses one man's life and the lasting influence of his advancements on the mathematical sciences and beyond. -- Inside jacket flaps.

Butterworth, Jon, author.

©2018

xvi, 284 pages : maps ; 23 cm

"Originally published in the UK as A Map of the Invisible by Jon Butterworth in 2017"--Title page verso.

Prologue: The journey begins -- Expedition I: Sea legs -- Setting sail -- The ocean wave... -- ...Or particle? -- Traveling in the quantum field -- Expedition II: Atom land -- Atoms -- Going subatomic: the electron -- Nuclear options -- The source of chemistry -- Expedition III: The Isle of Leptons, and roads onward -- Electromagnetism -- Invariance and relativity -- The good ship Dirac -- Spin and antimatter -- The electron's overweight siblings -- Rest stop. Gravity: a distant diversion. The weakest force ; Planes and merry-go-rounds ; Different, yet somehow, the same ; Ripples in the space-time continuum -- Expedition IV: Great train journeys -- Protons, neutrons, and the nucleus -- Hadrons -- Quarks and the strong force -- Life beyond the bridge -- Flavors and generations -- Expedition V: The Isles by air -- The weak force -- Parity, helicity, and chirality -- Mixed messages -- North from South -- Expedition VI: The remote neutrino sector -- Massless matter? -- The standard model is dead -- long live the standard model! -- Neutrino badlands -- Expedition VII: Into Bosonia -- Symmetry and conservation -- Symmetry and bosons -- Virtual particles and the defense against infinity -- Mass and hidden symmetry -- Electroweak symmetry breaking -- Hunting the Higgs -- Expedition VIII: Far East -- Why go? -- Clues and constraints -- Sea monsters and dark matters -- Supersymmetry -- Into another dimension? -- Over the edge -- A fifth force -- Into the cosmos.

"This book brings the impossibly small world of particle physics to life, taking readers on a guided journey through the subatomic world. With maps to help "guide" the reader through "Atom Land" along the way, as they learn about "electron ports," "boson continents," "hadron islands," and more"-- Provided by publisher.

Havil, Julian, author.

Princeton : Princeton University Press, ©2012.

ix, 298 pages : illustrations ; 24 cm

Greek beginnings -- The route to Germany -- Two new irrationals -- Irrationals, old and new -- A very special irrational -- From the rational to the transcendental -- Transcendentals -- Continued fractions revisited -- The question and problem of randomness -- One question, three answers -- Does irrationality matter? -- Appendix A: The spiral of Theodorus -- Appendix B: Rational parameterizations of the circle -- Appendix C: Two properties of continued fractions -- Appendix D: Finding the tomb of Roger Apéry -- Appendix E: Equivalence relations -- Appendix F: The mean value theorem.

Annotation The ancient Greeks discovered them, but it wasn't until the nineteenth century that irrational numbers were properly understood and rigorously defined, and even today not all their mysteries have been revealed. InThe Irrationals, the first popular and comprehensive book on the subject, Julian Havil tells the story of irrational numbers and the mathematicians who have tackled their challenges, from antiquity to the twenty-first century. Along the way, he explains why irrational numbers are surprisingly difficult to define--and why so many questions still surround them. That definition seems so simple: they are numbers that cannot be expressed as a ratio of two integers, or that have decimal expansions that are neither infinite nor recurring. But, asThe Irrationalsshows, these are the real "complex" numbers, and they have an equally complex and intriguing history, from Euclid's famous proof that the square root of 2 is irrational to Roger Apéry's proof of the irrationality of a number called Zeta(3), one of the greatest results of the twentieth century. In between, Havil explains other important results, such as the irrationality of e and pi. He also discusses the distinction between "ordinary" irrationals and transcendentals, as well as the appealing question of whether the decimal expansion of irrationals is "random". Fascinating and illuminating, this is a book for everyone who loves math and the history behind it.

Diaconis, Persi, author.

©2018

x, 255 pages : illustrations ; 25 cm

Measurement -- Judgment -- Psychology -- Frequency -- Mathematics -- Inverse inference -- Unification -- Algorithmic randomness -- Physical chance -- Induction.

"In the sixteenth and seventeenth centuries, gamblers and mathematicians transformed the idea of chance from a mystery into the discipline of probability, setting the stage for a series of breakthroughs that enabled or transformed innumerable fields, from gambling, mathematics, statistics, economics, and finance to physics and computer science. This book tells the story of ten great ideas about chance and the thinkers who developed them, tracing the philosophical implications of these ideas as well as their mathematical impact"--Dust jacket front flap.

Mitchell, Alanna, author.

New York, New York : Dutton, an imprint of Penguin Random House, LLC, [2018]

ix, 323 pages ; 24 cm

Preface: Playing with the universe -- Part I. Magnet -- The beginning of things -- The unpaired spinning electron -- Parking in the shadow of magnetism's forgotten man -- Into whose embrace iron leaps -- Revolutions on paper -- The Earth's magnetic soul -- Voyage into the underworld -- The greatest scientific undertaking the world had ever seen -- The rock that turned the world upside down -- Part II. Current -- Experiment in Copenhagen -- A very intimate relationship -- Jars full of lightning -- The apothecary's son -- The bookbinder's apprentice -- Magnets making currents -- The lines that fill the air -- Part III. Core -- The contorting gyre -- Shocks inside the Earth -- Pharaohs, fairies, and a tar-paper shack -- Zebra skins under the sea -- At the outer edge of the dynamo -- Anomaly to the South -- The worst physics movie ever -- The great hazardous spinning sphere of sodium -- Part IV. Switch -- Looking up -- Horrors the lights foretold -- Lethal patches -- The cost of catastrophe -- Trout noses and pigeon beaks -- A suit of stiff black crayon.

"A cataclysmic planetary phenomenon is gathering force deep within the Earth. The magnetic North Pole will eventually trade places with the South Pole. Satellite evidence suggests to some scientists that the move has already begun, but most still think it won't happen for many decades. All agree that it has happened many times before and will happen again. But this time it will be different. It will be a very bad day for modern civilization."--Amazon.com.

Yomtov, Nelson, author.

©2018

112 pages : illustrations (some color) ; 24 cm.

Madame physicist -- A star is born -- Coming to the United States -- Going east -- Beta decay -- Parity breakthrough -- New horizons -- The later years.

Women scientists have made key contributions to the pursuit of science and some of the most important discoveries of all time. In Chien-Shiung Wu, learn how the Chinese nuclear physicist chose to pursue a career in science and made breakthrough discoveries in nuclear fission and the scientific understanding of atoms. Features include a timeline, a glossary, essential facts, references, websites, source notes, and an index. Aligned to Common Core Standards and correlated to state standards."--Publisher's website.

Brundle, Joanna, author.

New York : KidHaven Publishing, 2018.

24 pages : color illustrations ; 23 x 29 cm

Measuring is a challenging math skill, but it is also an important one. Early learners are introduced to basic measurement concepts that they can explore on their own or with an adult. Simple, clear text explains different units of measurement, supporting essential math curriculum topics. Familiar examples presented with full-color photographs help readers understand how to apply what they have learned in the world around them. Young readers will become measuring masters in no time thanks to these engaging and relatable examples. Activities, Detailed Table of Contents, Full-Color Photographs.

McCune, Sandra K., author.

©2018

ix, 289 pages : illustrations ; 24 cm

Includes index.

Numbers of algebra -- Computation with real numbers -- Roots and radicals -- Exponentiation -- Order of operations -- Algebraic expressions -- Rules for exponents -- Adding and subtracting polynomials -- Multiplying polynomials -- Simplifying polynomial expressions -- Dividing polynomials -- Factoring polynomials -- Rational expressions -- Solving linear equations and inequalities -- Solving quadratic equations -- The cartesian coordinate plane -- Graphing linear equations -- The equation of a line -- Basic function concepts -- Systems of equations -- Signal words and phrases -- Word problems.

The fastest way to learn algebra is to build a solid foundation in the basics. Inside this book you won't find a lot of endless drills. Instead, you get an original, step-by-step approach to learning algebra. In your first steps, you are introduced to essential concepts, allowing you to grasp the subject almost immediately. You will gradually progress to more challenging skills. Along the way, the authors show you how to solve practical problems using clear, step-by-step instructions. Exercises for each section, with detailed, worked-out solutions, will let you check your progress. In no time at all, you will have acquired the knowledge and skills you need to solve algebraic problems with confidence. --Publisher

Schwartz, David N., 1956- author.

©2017

xxiii, 453 pages : illustrations ; 25 cm

Part one: Becoming Fermi. Prodigy ; Pisa ; Germany and Holland ; Quantum breakthroughs ; Of geckos and men -- Part two: The Rome years. Family life ; The Rome School ; Beta rays ; Goldfish ; Physics as soma ; The Nobel Prize -- Part three: The Manhattan Project. The New World ; Splitting the atom ; Fermi meets the Navy ; Piles of graphite ; The move to Chicago ; "We're cookin'!" ; Xenon-135 ; On a mesa ; An unholy Trinity -- Part four: The Chicago years. Return to Chicago ; In the public eye ; A patent fight ; Brilliant teacher, beloved mentor ; Travels abroad ; Home to die ; Fermi's legacy.

"In December 1942, a team at the University of Chicago achieved a milestone in human history: a nuclear chain reaction. At the forefront of this breakthrough stood Enrico Fermi, the father of the nuclear age. But as David N. Schwartz shows in this groundbreaking biography, Fermi's impact goes well beyond this epochal event. With his theory of beta decay and his development of quantum statistics, Fermi revolutionized modern physics. Straddling the classical and quantum ages, equally at ease with elegant mathematics and grubby experiments, Fermi truly was the last man who knew everything--at least about physics. In [this book], Schwartz draws from newly discovered archival material and exclusive interviews with those who knew Fermi to reveal the complex figure behind these historic contributions. A reluctant member of the Italian Fascist party, Fermi escaped to New York when Mussolini promulgated a series of anti-Semitic laws that put his wife, Laura, at risk. A citizen of an Axis power at the heart of the US government's most secret war effort, the Manhattan Project, he became one of its leading lights. A less-than-ideal father and husband, he was nevertheless one of history's greatest scientific mentors and teachers. He was also a deep thinker, as perspicacious about extraterrestrial life as he was about quantum field theory. The Last Man Who Knew Everything brings Fermi's brilliant, complex genius to life in a profound and consuming read."--Dust jacket flap.

Stewart, Ian, 1945- author.

©2017

vi, 303 pages : illustrations, portraits ; 24 cm

Do not disturb my circles : Archimedes -- Master of the Way : Liu Hui -- Dixit algorismi : Muhammad al-Khwarizmi -- Innovator of the infinite : Madhava of Sangamagrama -- The gambling astrologer : Girolamo Cardano -- The last theorem : Pierre de Fermat -- System of the world : Isaac Newton -- Master of us all : Leonhard Euler -- The heat operator : Joseph Fourier -- Invisible scaffolding : Carl Friedrich Gauss -- Bending the rules : Nikolai Ivanovich Lobachevsky -- Radicals and revolutionaries : Évariste Galois -- Enchantress of a number : Augusta Ada King -- The laws of thought : George Boole -- Musician of the primes : Bernhard Riemann -- Cardinal of the continuum : Georg Cantor -- The first great lady : Sofia Kovalevskaia -- Ideas rose in crowds : Henri Poincaré -- We must know, we shall know : David Hilbert -- Overthrowing academic order : Emmy Noether -- The formula man : Srinivasa Ramanujan -- Incomplete and undecidable : Kurt Gödel -- The machine stops : Alan Turing -- Father of fractals : Benoit Mandelbrot -- Outside in : William Thurston -- Mathematical people.

A celebrated mathematician traces the history of math through the lives and work of twenty-five pioneering mathematicians. In Significant Figures, acclaimed mathematician Ian Stewart introduces the visionaries of mathematics throughout history. Delving into the lives of twenty-five great mathematicians, Stewart examines the roles they played in creating, inventing, and discovering the mathematics we use today. Through these short biographies, we get acquainted with the history of mathematics from Archimedes to Benoit Mandelbrot, and learn about those too often left out of the cannon, such as Muhammad ibn Musa al-Khwarizmi (c. 780-850), the creator of algebra, and Augusta Ada King (1815-1852), Countess of Lovelace, the world's first computer programmer. Tracing the evolution of mathematics over the course of two millennia, Significant Figures will educate and delight aspiring mathematicians and experts alike.