Proceed with care, because some infinities are larger than others

“Proceed with care, because some infinities are larger than others” explains in detail why the infinite set of real numbers—all of the numbers on the number line—represents a far larger infinity than the infinite set of natural numbers—the counting numbers. Readers learn to distinguish between countable infinity and uncountable infinity by way of a method known as a “one-to-one correspondence.” Mathematics students and enthusiasts are encouraged to proceed with care in both mathematics and life, lest they confuse countable infinity with uncountable infinity, large with unfathomably large, or order with disorder. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Go outside your realm of experience, on a hypercube

“Go outside your realm of experience, on a hypercube” explains how and why mathematicians conceive of cubes in many dimensions, including a four-dimensional hypercube. Einstein’s special theory of relativity and the mathematics of string theory—a subfield of physics that seeks to understand the structure of the universe—both require more than the three dimensions with which we are familiar. The discussion, which focuses on how to make a four-dimensional hypercube, is enhanced with numerous hand-drawn sketches. Mathematics students and enthusiasts are encouraged to go outside their realm of experience in both mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Get disoriented, on a Klein bottle

“Get disoriented, on a Klein bottle” offers a basic introduction to a four-dimensional mathematical object known as a Klein bottle. The discussion is enhanced with numerous hand-drawn sketches. Although a Klein bottle appears to have an inside and an outside, it is an object with only one side and no edges. Readers will learn why a walk on a Klein bottle may turn them upside down. Klein bottles embody the art, science, seriousness, and even playfulness of mathematics. Mathematics students and enthusiasts are encouraged to embrace a Klein-bottle-like disorientation in mathematical and life pursuits as a way of exposing themselves to new ideas. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Be contradictory, because of the infinitude of primes

“Be contradictory, because of the infinitude of primes” offers encouragement and practice with the “proof-by-contradiction” method of mathematical proof. Any mathematician will tell you that the collection of prime numbers is infinite. However, readers are guided in proving this statement by contradicting it. The activity pushes readers to engage deeply with the reasons supporting the fact that there are an infinite number of primes. Mathematics students and enthusiasts are encouraged to debate with enthusiasm in their mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Explore, on a Mobius strip

“Explore, on a Mobius strip” offers an introduction to the mathematical subfield of topology by way of numerous hand-drawn sketches and an accessible discussion of going for a “walk” on a one-sided, one-edged Mobius strip—also known as a Mobius band. The chapter provides directions for making a Mobius strip out of paper and examining its mathematical properties. Mathematics students and enthusiasts are encouraged to explore more in both mathematics and life in order to expand their worldview. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Keep it simple whenever possible, since 0.999…=1

“Keep it simple whenever possible, since 0.999…=1” presents and discusses a very short mathematical proof demonstrating the long-known result that 0.999…=1. The ellipsis in the number 0.999… indicates that this number repeats in an infinite decimal expansion. As such, this number is unwieldy to lug around, insert into equations, and even describe. However, the number 1 is not simply a good approximation for 0.999…., but rather the number 1 may be used in place of 0.999… without loss of information. Mathematics students and enthusiasts are encouraged to keep their mathematical and life pursuits simple whenever possible. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Enjoy the pursuit, as Andrew Wiles did with Fermat’s Last Theorem

“Enjoy the pursuit, as Andrew Wiles did with Fermat’s Last Theorem” recounts the story of how mathematician Andrew Wiles was undaunted in the face of 350 years’ worth of mathematicians’ failed efforts at attempting to solve Fermat’s Last Theorem. He believed in his abilities and ultimately succeeded in providing a proof. Along the way, he satisfied a human longing to seek knowledge and energized the mathematics community. Mathematics students and enthusiasts are encouraged to remember to value the journey in mathematical and life pursuits, even when they struggle. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Look all around, as Archimedes did in life

“Look all around, as Archimedes did in life” offers a range of tales about Archimedes doing mathematics in unusual places, including in the bathtub, under starry skies, and at the beach. Out in the world, he identified a method for computing the volume of an irregular shape, invented a parabolic death ray, explored properties of levers, and argued that a “very large number” is not the same as infinity. The discussion is illustrated with numerous hand-drawn sketches. Mathematics students and enthusiasts are encouraged draw inspiration from Archimedes by doing math wherever they are while going about their daily lives. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Fail more often, just like Albert Einstein did with E=mc2

“Fail more often, just like Albert Einstein did with not only explains Einstein’s famous equation but tells the little-known story of how he experienced failure along the way. As Einstein worked to correct his mistakes, he not only gained a deeper understanding into the equation, but the entire field of special relativity. Many years later, in a letter to a young girl included in this chapter, he referenced his struggles when he wrote, “Do not worry about your difficulties in mathematics; I can assure you that mine are still greater.” Mathematics students and enthusiasts learn that mistakes are an essential part of the mathematical process. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Draw a picture, as in proofs without words

“Draw a picture, as in proofs without words” provides a short, wordless illustration proving that. The hand-drawn sketches invoke an earlier time in the reader’s life spent playing with blocks and making quiet yet important mathematical observations. Readers also learn of the story of young Carl Friedrich Gauss whose elementary teacher tried to keep the class busy by asking that they sum the first one hundred integers. Most of the students began the time-consuming process of adding but Gauss drew a few quick sketches that offered a shortcut to the solution: 5,050. Mathematics students and enthusiasts are encouraged to consider play and drawing as methods for eliciting mathematical thoughts. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.