Adding a Dimension: Mathematics
January 9, 1987
A review of "Adding a Dimension: Mathematics" by Isaac Asimov.
Copyright © 1998 Property of Deborah K. Fletcher. All rights reserved.
Adding a Dimension: Mathematics is a collection of essays dealing with mathematical principles. It was published by Lancer Books, Inc., of New York, in 1969. It consists on ninety-six pages, which are divided into seven chapters.
The first chapter is entitled "T-Formation." It deals with extremely large numbers, and systems by which to classify and use them. The first number to be introduced is the googol, which is defined as the number 1 followed by a hundred zeroes.
The googol opens up a subject of extremely large numbers, specifically multiples of a trillion, which are abbreviated into a system called T-formation, in which a trillion is denoted by T-1; a trillion trillion T-2; a trillion trillion trillion T-3; and so forth. In order to visualize T-1 (bearing in mind the publication date), Asimov has estimated that T-1 is the number of dimes it takes to run the United States for one year ($100,000,000,000). To further visualize T-formation (or "Asimovian") numbers, he gives that T-2 nucleons make up about 1/16 of an ounce of mass, while T-3 nucleons have a mass of about 1.67 trillion grams, T-4 nucleons equals the mass of the earth's oceans, T-5 nucleons equal the mall of a thousand solar systems, T-6 nucleons equal the mass of ten thousand galaxies the size of our own, and T-7 nucleons are far more massive than the entire universe. Considering the size of a nucleon (proton or neutron), such dimensions are more than a little staggering.
From T-formations and googols, Asimov moves on to more familiar numbers, such as the Fibonacci series. The numbers of this series begin small (1, 1, 2, 3, 5, 8, ...), but by the fifty-fifth number is greater than T-1.
The next move is to prime numbers. Again, the early numbers of this set are easy: 2, 3, 5, 7, 13, 17, 19, .... Higher primes, however, are difficult to ascertain quickly. Marin Mersenne developed a formula about 1600 which would calculate any given prime:
2p - 1, where p is a prime.
This formula seems to work, and the numbers obtained by it become known as Mersenne numbers. However, M67 and M257 are not primes, while M61, M89, and M107 (which he did not list as primes) are prime.
Finally, in this chapter, is the introduction of the googolplex, which is defined as 10 raised to the googol. For this, the T-formation system is expanded, and a googolplex is just over T-(T-8). [T-(T-1)] is equal to 1012000000000000, which is more than 101013. A googolplex is equal to 101034.
The second chapter is entitled "One, Ten, Buckle My Shoe." This chapter introduces several number systems, including: two-based, three-based, four-based, five-based, and so on. The two-based system is the familiar binary system. The first seven numerals in the binary system are:
1 equals 1
10 equals 2
11 equals 3
100 equals 4
101 equals 5
110 equals 6
111 equals 7.
A system for converting ordinary numbers to binary numbers is given by the following example, representing 131 in binary form:
131 / 2 = 65 remainder 1
65 ÷ 2 = 32 remainder 1
32 ÷ 2 = 16 remainder 0
16 ÷ 2 = 8 remainder 0
8 ÷ 2 = 4 remainder 0
4 ÷ 2 = 2 remainder 0
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1.
Therefore, 131 is written 11000001. In order to compare this with the other systems, the following table is provided for 131:
| 2-based system |
| 11000001 |
| 3-based system |
| 11212 |
| 4-based system |
| 2003 |
| 5-based system |
| 1011 |
| 6-based system |
| 335 |
| 7-based system |
| 245 |
| 8-based system |
| 203 |
| 9-based system |
| 155 |
The remainder of the chapter discusses examples of converting between differing systems.
The third chapter is entitled "Varieties of the Infinite." This chapter begins by defining infinity as a non-number. In this chapter, rules of infinite arithmetic are given:
ì - n = ì, where n is an integer;
ì - ì = ì;
ì + n = ì;
ì + ì = ì;
ì / ì = ì; and
ì * ì = ì2 = ì.
It then looks at infinite series, and discusses why the infinite series of odd integers is the same size as the infinite series of all integers.
Next, infinity is done one better by continuum, represented by C. It is given that ìì = C. Even this is a fairly small number, relatively speaking, since George Cantor discovered the transfinite numbers in 1895. These are represented by the Hebrew letter aleph: À. The lowest transfinite number is aleph-null, or À0. The single arithmetic rule for transfinite numbers is that
>À0À0 = À1; À1À1 = À2; ....
Also, infinity ie equal to alpha-null, and the continuum is aleph-one.
The fourth chapter is entitled "A Piece of Pi." This chapter describes attempts to calculate pi, discovering that 355/113 is as close to p as it is necessary to get under usual circumstances.
Next, Asimove differentiates between rational and irrational numbers. The standard irrational, ¸, is given, then it is shown that p is irrational.
The fifth chapter, "Tools of the Trade," is almost a continuation of chapter four. It begins by defining ideals of geometry: lines, points, and circles. It also introduces the basic tools of plane geometry: the straightedge and the compass. It is shown that even lines of irrational lengths, such as ¸, can be constructed with just these tools.
"Squaring the circle" is an ancient construction which cannot be performed with just the basic, or "gentlemanly," tools. It is impossible, because p is a number which cannot be measured precisely with a compass and straightedge, and which is basic to the circle.
Eqations are considered next. The quadratic equation is given:
(-B ± ¶(B2 - 4AC)) / 2A
This is done to show that a polynomial equation can be manipulated.
Chapter six is entitled "The Imaginary That Isn't." Obviously, it deals with complex numbers.
First, it is postulated that every equation of the nth degree has exactly n solutions. This is easily seen with equations such as
x + 3 = 5
2 + 3 = 5
x = 2
or
x2 + 4x - 5 = 0
1 + 4 - 5 = 0
x=1
and
25 - 20 - 5 = 0
x = -5.
It is not so easy, however, with
x2 + 1 = 0,
because that involves a (¶-1). This is handled with imaginary numbers (i-numbers), which are given, with their arithmetic rules, in the table:
| +i |
| times |
| +i |
| = |
| -1 |
| -i |
| times |
| -i |
| = |
| -1 |
| +i |
| times |
| -i |
| = |
| 1 |
This gives the solutions to x2 + 1 = 0 as ±i.
The seventh, and final, chapter is entitled "Pre-fixing It Up." It deals with units of measure, and with the magnitudes thereof.
The chapter begins with the metric system. It explains the advantages of a ten-based system of measure, as opposed to a set of systems using miles, feet, inches, rods, furlongs, pecks, bushels, pints, drams, ounces, pounds, tons, and grains. In that system, it is not even possible to cross over from one system to another, while it is common to do so in the metric system (one cubic centimeter equals one milliliter). A table of the system of lengths is given for the metric system:
| 1 kilometer |
| = |
| 1000 |
| meters |
| 1 hectometer |
| = |
| 100 |
| meters |
| 1 dekameter |
| = |
| 10 |
| meters |
| 1 meter |
| = |
| 1 |
| meter |
| 1 decimeter |
| = |
| 0.1 |
| meter |
| 1 centimeter |
| = |
| 0.01 |
| meter |
| 1 millimeter |
| = |
| 0.001 |
| meter |
The qualifying portion of each term is the prefix, which is built upon the root, in this case the meter.
With advancing technology, larger and smaller prefixes are necessary. A table of these prefixes is also given:
| Size |
| Prefix |
| trillion |
| (1012) |
| tera- |
| billion |
| (109) |
| giga- |
| million |
| (106) |
| mega- |
| thousand |
| (103) |
| kilo- |
| hundred |
| (102) |
| hekto- |
| ten |
| (101) |
| deka- |
| one |
| (100) |
|
|
| tenth |
| (10-1) |
| deci- |
| hundredth |
| (10-2) |
| centi- |
| thousandth |
| (10-3) |
| milli- |
| millionth |
| (10-6) |
| micro- |
| billionth |
| (10-9) |
| nano- |
| trillionth |
| (10-12) |
| pico- |
| quadrillionth |
| (10-15) |
| femto- |
| quintillionth |
| (10-18) |
| atto- |
The chapter continues with examples of the uses of the extremely large and small measurements of length, volume, mass, and time. One time-table is interesting:
| 1 |
| second |
| = |
| 1 |
| second |
| 1 |
| kilosecond |
| = |
| 16 2/3 |
| minutes |
| 1 |
| megasecond |
| = |
| 11 2/3 |
| days |
| 1 |
| gigasecond |
| = |
| 32 |
| years |
| 1 |
| terasecond |
| = |
| 32000 |
| years |
The chapter ends with a summarization of the ranges of the systems of measure to the effect that the measurable lengths cover 41 orders of magnitude, the measurable masses 83 orders of magnitude, and the measurable times 40 orders of magnitude. The metric system covers 30 orders of magnitude.
Over-all Adding a Dimension: Mathematics was a fascinating work. It presented many topics, which are usually tedious to learn, in an interesting manner. I have gained a clearer understanding of infinity, particularly, through this book.
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