# Strange Curves, Counting Rabbits, & Other Mathematical Explorations

Keith Ball
Pages: 272
https://www.jstor.org/stable/j.ctt7s6cg

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xii)
4. Acknowledgements
(pp. xiii-xvi)
5. CHAPTER ONE Shannon’s Free Lunch
(pp. 1-24)

Pick up a paperback book, any book which was published fairly recently, and on the back you will find a number—the ISBN or International Standard Book Number. (Hardback books are sometimes numbered inside the front cover.) The ISBN identifies the title among all titles published internationally. The ISBN sequence of this book is

0-691-11321-1.

(The hyphens are not important for our purposes but I left them in to make the number easier to read.) This number has a surprising property. In Table 1.1 the ISBN digits are written vertically in the first column. The numbers from 1 to 10...

6. CHAPTER TWO Counting Dots
(pp. 25-40)

When I was about 11 or 12 years old, my classmates and I were set a lot of problems in mathematics classes in which we had to calculate the areas of various regions in the plane, something like the ones in Figure 2.1. These shapes were always polygons and their corners were points in the plane whose coordinates were whole numbers: ‘lattice points’. The idea was that we had to cut the regions up into rectangles and triangles, or else surround them with rectangles and triangles, so as to find the area. Cutting up works fine for the example on...

7. CHAPTER THREE Fermat’s Little Theorem and Infinite Decimals
(pp. 41-62)

At the age of about 8 or 9, most of us are introduced to the idea of representing fractions in decimal notation, usually with examples like the following:

$1/2\, = \,0.5, \quad 1/4\, = \,0.25, \quad 1/5\, = \,0.2.$

Soon afterwards things get more complicated. When you write the fraction 1/3 as a decimal, it does not terminate like the ones above, but instead you get a recurring decimal

$1/3 = 0.333\,333\,3 \ldots .$

At this point the teacher usually mentions thateveryfraction has a decimal expansion that either terminates or recurs, and nothing more is said on the matter. That is rather a pity because there is a great deal going on...

8. CHAPTER FOUR Strange Curves
(pp. 63-82)

During the last 20 years or so, there has been a lot of publicity for fractals: strange curves and shapes which look a bit like weird sea creatures. Figure 4.1 is typical of the things you see in glossy picture books on the subject, and even on wall calendars.

This chapter is about some of the oldest fractals of them all. These are the ‘space-filling’ curves, originally invented by the Italian mathematician Giuseppe Peano in 1890. Although they date back to long before the current interest in fractals, they exhibit many of the characteristic properties of their modern cousins.

To...

9. CHAPTER FIVE Shared Birthdays, Normal Bells
(pp. 83-108)

Suppose there are 23 people in a room. What is the chance that among them are two people with the same birthday: two people who celebrate their birthdays on the same day of the year? There are 23 names distributed at random among 365 (or 366) boxes, one for each day of the year. We want to know the likelihood that there will be a box containing more than one name. Clearly, 23 names distributed among 365 boxes will produce a pretty ‘thin’ distribution: the overwhelming majority of boxes will not getanynames in them. So we might expect...

10. CHAPTER SIX Stirling Works
(pp. 109-126)

If I havendifferent objects, the number of different ways in which I can place them in order isnfactorial, the product of the firstnintegers:$n! = 1 \times 2 \times 3 \times \cdots \times n.$The reason is that I havenpossible choices for the first object and, for each of these,n− 1 choices for the second object, and so on. Because of this, factorials turn up a good deal in mathematics, especially in probability theory and combinatorial (or counting) problems. For many reasons it is useful to be able to estimate the value ofn!.

At first sight you might find that...

11. CHAPTER SEVEN Spare Change, Pools of Blood
(pp. 127-152)

A few months ago, my brother-in-law Carl Wittwer, who is a biologist, asked me a nice mathematical question. He had developed a blood test for detecting a certain minor abnormality in infants. The presence of a particular substance in the blood would indicate the presence of the condition and vice versa. His test was sufficiently sensitive that it would be possible to detect the presence of one or more affected children within a group of 100 or so by testing a mixture containing a small amount of blood from each child, i.e. by testing a pooled sample of blood.¹

Obviously,...

12. CHAPTER EIGHT Fibonacci’s Rabbits Revisited
(pp. 153-188)

Probably the most famous sequence of numbers in mathematics is theFibonacci sequence, which begins$1, 1, 2, 3, 5, 8, 13, 21, \ldots,$and in which each term is the sum of the two previous terms. The sequence has an almost unlimited number of entertaining properties, so many that there is an entire (slightly eccentric) journal devoted to publishing new discoveries about this one sequence.¹

The name Fibonacci was a kind of nickname used by Leonardo of Pisa who lived at the end of the 12th century and the first half of the 13th. He introduced the sequence by way of a recreational question concerning the breeding patterns...

13. CHAPTER NINE Chasing the Curve
(pp. 189-218)

Soon after meeting calculus at school, one is introduced to the idea of Taylor series. Starting with a functionf, you try to use the derivatives offat a particular point to approximate the function by a polynomial. The first derivative at 0, for example, tells you how to approximate the curvey=f(x) by a straight line—its tangent line at 0. Figure 9.1 shows the example of the tangent line toy= ln(1+x), which we already met in Chapter 5.

For a generalf, the equation of the tangent line is$y = f(0) + f^{\prime}(0)x,$so the derivative provides...

14. CHAPTER TEN Rational and Irrational
(pp. 219-246)

Mathematics originally grew out of the study of practical problems. Nomadic bands of hunters can survive without mathematics, but once agriculture starts it is important to be able to predict the seasons by counting days. As society develops, and adopts a monetary system, arithmetic is needed to handle it. Geometry is needed to measure land and to build any reasonably elaborate building. Perhaps because of this, the mathematicians of classical antiquity felt that mathematics was embedded in the world in a way that today might seem fanciful. It is true that mathematics can be used to build startlingly accurate models...

15. Index
(pp. 247-251)