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In this sequel to his award-winning How Mathematics Happened, physicist Peter S. Rudman explores the history of mathematics among the Babylonians and Egyptians, showing how their scribes in the era from 2000 to 1600 BCE used visualizations of how plane geometric figures could be partitioned into squares, rectangles, and right triangles to invent geometric algebra, even solving problems that we now do by quadratic algebra. Using illustrations adapted from both Babylonian cuneiform tablets and Egyptian hieroglyphic texts, Rudman traces the evolution of mathematics from the metric geometric algebra of Babylon and Egypt—which used numeric quantities on diagrams as a means to work out problems—to the nonmetric geometric algebra of Euclid (ca. 300 BCE). Thus, Rudman traces the evolution of calculations of square roots from Egypt and Babylon to India, and then to Pythagoras, Archimedes, and Ptolemy. Surprisingly, the best calculation was by a Babylonian scribe who calculated the square root of two to seven decimal-digit precision. Rudman provocatively asks, and then interestingly conjectures, why such a precise calculation was made in a mud-brick culture. From his analysis of Babylonian geometric algebra, Rudman formulates a "Babylonian Theorem", which he shows was used to derive the Pythagorean Theorem, about a millennium before its purported discovery by Pythagoras.
He also concludes that what enabled the Greek mathematicians to surpass their predecessors was the insertion of alphabetic notation onto geometric figures. Such symbolic notation was natural for users of an alphabetic language, but was impossible for the Babylonians and Egyptians, whose writing systems (cuneiform and hieroglyphics, respectively) were not alphabetic. Rudman intersperses his discussions of early math conundrums and solutions with "Fun Questions" for those who enjoy recreational math and wish to test their understanding. The Babylonian Theorem is a masterful, fascinating, and entertaining book, which will interest both math enthusiasts and students of history.
We can be sure that the place-value, base-10 number system with Hindu-Arabic symbols that we now use globally must be just about the best for everyday use. It is the clear winner of a survival-of-the-fittest number-system game that has been going on for millennia. (The nomenclature base-10 is interchangeable with decimal and place-value with positional.) The victory of base-10 is remarkable because the choice was simply because we happen to have ten fingers. A creationist might see a divine hand literally at work here, but the rest of us require another explanation. What makes base-10 such an ideal number system?
The larger the base, the more compact the number system. For example, the base-10 number 189 (= 1 x [10.sup.2] + 8 x 10 + 9) is written as 10111101 (= [2.sup.7] + [2.sup.5] + [2.sup.4] + [2.sup.3] + [2.sup.2] + 1) as a base-2 number and as 99 (= 9 ??20 + 9) as a base-20 number. Base-10 may not be as compact as base-20 but it is adequately compact for everyday use. Electronic digital computers use base-2 because the two symbols 0 and 1 conveniently represent OFF and ON switches. Base-20 evolved in cultures that first solved the problem of counting to greater than ten by adding toes.
Any N-position, base-b integer can be expressed using Hindu-Arabic symbols in algebraic notation as a sequence of N integers: [a.sub.N-1 [a.sub.N-2 ... [a.sub.n] ... [a.sub.2] [a.sub.1] [a.sub.0], where the [a.sub.n]'s are integers from 0 to b - 1. Equation (1.1) gives the base-10 value of an integer in any base.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
The fractional part of any N-position, base-b number can be expressed using Hindu-Arabic symbols in algebraic notation as a sequence of N integers: 0. [a.sub.-1] [a.sub.-2] [a.sub.N]. Equation (1.2) gives the base-10 value of the fractional part of a number in any base:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)
FUN QUESTION 1.1: Write the base-5 number 234 as a base-10 number. __________________________________________________________________ __________________________________________________________________
FUN QUESTION 1.2: Write the base-10 number 189 as a base-5 number. Hint: 189 = 125[a.sub.3] + 25[a.sub.2] + 5[a.sub.1] + [a.sub.0]. Start by finding the largest possible value for [a.sub.3]. __________________________________________________________________ __________________________________________________________________
FUN QUESTION 1.3: Write the base-5 fractional number 0.234 as a base-10 number. __________________________________________________________________ __________________________________________________________________
FUN QUESTION 1.4: Write the base-10 fractional number 0.189 as a base-5 number. __________________________________________________________________ __________________________________________________________________
For the Babylonian base-60 system, various methods enable expression with just the ten Hindu-Arabic symbols. The system I shall use is the same notation generally used to express time (hours:minutes:seconds with 1 hour = 60 minutes = [60.sup.2] seconds), a vestige of the Babylonian base-60 system. Colons separate place values and a period (called a sexagesimal point) separates the integer part from the fractional part (just as a decimal point does in the base-10 system). Thus, the decimally transcribed sexagesimal (base-60) number [1:3:20.15:40.sub.60] has a base-10 value: 1 x [60.sup.2] + 3 x 60 + 20 + 15/60 + 45/[60.sup.2] = 3800.2625. Lacking any generally agreed-upon convention, there is some unavoidable confusion in reading decimally transcribed base-60 numbers. In this book, all calculations presented are done in modern base-10 to avoid the distraction of having to interpret other notations. Whenever there is the possibility of confusion as to whether a number is written in decimally transcribed sexagesimal, I add a subscripted 60 as just previously done.
The larger the base, the larger the memorization burden it imposes. The memorization of the values of only ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) in the base-10 system is no problem and neither would be the memorization of the b symbols in any practical base-b number. However, when arithmetic, and particularly multiplication, is considered, the memorization burden of a number system with a large base looms large. Figure 1.1 presents the base-10 multiplication table that we all boringly learned in elementary school but now barely ever use since electronic calculators became ubiquitous. We do not have to memorize all 102 possible products because most of the entries are eliminated by the rules 0 x n = 0, 1 x n = n, and m x n = n x m, where m and n are any numbers. Without much effort, we can simply add up the entries in figure 1.1 and find that there are 36, not an overwhelming memorization burden.
In order to calculate the number of multiplication table entries for any base-b, we note from figure 1.1 that the number of entries is the arithmetic series 1 + 2 + ... + 7 + 8 = 36, and by generalizing this series we can obtain that for base-b, the number of multiplication table entries is
[N.sub.b] = (b = 2) (b - 1)/2 (1.3)
Using this equation, we can readily calculate that for base-20, [N.sub.20] = 171 is quite a memorization burden, while for the Babylonian base-60, [N.sub.60] = 1,711 imposes an impossible memorization burden, which explains why these number systems did not survive.
Now we can see why base-10 is essentially an optimized number system: the base is large enough to produce adequately compact numbers, but small enough not to impose too heavy a memorization burden. For the Babylonians, multiplication with a base-60 number system was indeed a problem and we shall consider later how they solved it, but now we want to understand why they invented a number system with such a large base. The answer is simply that they invented their number system in a prehistoric (prewriting), prearithmetic era. For counting, base-60 was fine, but when their culture advanced to a stage where arithmetic was required, base-60 was so entrenched in Babylonian habits and records that they preferred to find ways to cope rather than change their number system. The problem is similar to current reluctance in the United States to abandon use of archaic English units and adopt the metric system.
Let us go back to the time of the so-called Neolithic revolution, around 10,000 BCE in Egypt and Babylon, when hunter-gatherer cultures were beginning to transform into materially richer and more complex herder-farmer cultures. Finger counting now no longer satisfied counting requirements, and the ancient solution to its limitations was pebble counting. In its simplest realization, pebble counting requires little memorization and uses only the principle of one-for-one correspondence. Consider a shepherd putting a pebble in a bowl as each sheep goes out to pasture and removing a pebble as each sheep returns. The number of pebbles left in the bowl is then the number of sheep lost. This is counting and arithmetic without a need for names for numbers, or an ability to articulate counting, with no limit to the number counted, and with a permanent record, a significant advance beyond finger counting.
With growing understanding of the concept of number, our generic shepherd (now perhaps many generations later) wants to know how many sheep he has. A bowl of hundreds of pebbles is not very defining, so he replaces each group of ten pebbles with one larger pebble. Why ten? Because he had first learned to count using his fingers, just as children still do, and therefore ten is a natural choice. If he still has too many pebbles to define his number of sheep conveniently, he can replace every ten big pebbles with a bigger pebble, and so on. There are now never more than nine pebbles of the same size, so even a quick glance at his collection of pebbles allows him to visualize the number. Unaware though he surely was, he had now invented an additive, base-10 number system. The smallest pebble has a value of one; the next larger pebble has a value of ten; the next larger pebble after that has a value of a hundred; and so on. A sequence of replacements using the same replacement number defines a number system with a base equal to the replacement number. Figure 1.2 illustrates this unrealized and unintended pebble-counting invention of an additive, base-10 number system.
If rather than using pebbles of different sizes in 1-for-10 replacements, ten pebbles in one bowl were replaced by one pebble in a different bowl, a place-value, base-10 number system would have been invented. Even if one starts with an additive system with different-sized pebbles, for easier visualization of quantity one naturally and intuitively tends to gather each size in a separate pile. Now the values of pebbles are redundantly defined both by pebble size and by pebble position. Eventually this unnecessary redundancy tends to be realized, and hence additive systems tend to evolve into place-value systems.
If his hands were man's first calculating machine, then pebbles were his second. Number systems with a base originated in a pebble-counting era. Once a replacement is chosen for some logical reason, such as 1-for-10 because we have ten fingers, then why change it in a sequence of replacement numbers. Since a sequence of the same replacements defines a number system with a base, number systems with a base tend to evolve naturally once use of replacement has begun. The advantage of a number system with a base becomes more apparent when arithmetic other than simple addition of small numbers is required.
We have now developed the three important characteristics of an efficient number system, and they are valid whether the counting is by fingers, pebbles, or written symbols:
1. Replacement-extends the counting limit.
2. Base-enables efficient arithmetic.
3. Position-minimizes the number of different symbols.
Quantitative measurement, in addition to just counting of things, became a requirement after about 10,000 BCE. This imposed new demands on number systems. Consider the following scenario: a woman is making a garment and invents the natural, intuitive measuring system of using the widths of her fingers. She measures something as 19 finger widths. In practice, measurement in units of finger widths is by a series of handbreadth-to-hand-breadth placements, so she defines a handbreadth as a natural, intuitive, new unit, with 1-for-4 replacements of fingers by hands. Now, letting a larger pebble represent a handbreadth, she records her measurement as 4 large pebbles and 3 small pebbles. Our prehistoric seamstress has now invented a measurement system with units of handbreadths and fingers. The 1-for-10 replacement scheme that was so natural and intuitive for counting is no longer obviously the better system for units of measurement. What to do?
Solution 1: Retain the natural 1-for-10 replacements for counting, and retain the various natural replacements for measuring. This method largely accounts for the English system still used in the United States although not officially in England since 1965.
Solution 2: Define new, not-so-natural units of measurement so that 1-for-10 replacements also define measurement units. This is an important component of the present, almost globally adopted metric system and was also the ancient Egyptian solution. It might be an overstatement to say that the Egyptians realized more than 5,000 years ago what the rest of the world is only now realizing. It is possible they simply did what came naturally and used the same intuitive 1-for-10 replacement system for measuring units as they did for counting. It was fortuitously a good choice.
Solution 3: Modify the natural 1-for-10 replacements for counting by adopting the natural replacements for measurements. This was the Babylonian solution. It was fortuitously a bad choice that would only become evident millennia later when required to perform arithmetic that was more complex.
Units of linear measurement started just as simple counting did, by using body parts. Using the example of English units, with which we are most familiar, figure 1.3 shows the replacement numbers as they were defined up to the fifteenth century. To convert from a larger unit to a smaller unit it is necessary to multiply by the product of all the replacement numbers between the respective units. Thus, from figure 1.3 we can calculate that the number of inches in a fathom is 4 x 3 x 3 x 2 = 72. Compared to converting units within the modern metric system where conversion is only a matter of moving a decimal point (for example,1.75 m = 175 cm), English units are a bother. Nowadays, with the ready availability of rulers and tape measures there is no logical reason to continue to use body-parts units, but conversion is a question of politics, not mathematics, and hence is beyond the scope of this book.
For longer distances, the fundamental English unit was based on farming practice, the furlong, literally a furrow long. Another example of an English unit based on farming practice is the acre, which legend has it was the area plowed by a pair of oxen in one day. Eventually, as measurements that were more precise were required, a government had to set standards to reconcile farming-practice units and body-parts units. Thus, up to the fifteenth century in England the furlong was defined as 1 furlong = 40 rods = 600 feet. However, England had also inherited from the Romans the unit of the mile, which the Romans had defined as 1,000 double paces. A double pace is 5 feet, so the mile, clearly derived from the Latin word for 1,000, was 5,000 feet. This had the unfortunate affect of inconveniently making the mile a nonintegral (5,000/600 = 8.333...) multiple of the rod, so in the sixteenth century Queen Elizabeth redefined many units to essentially their present values, which among other things redefined the mile as exactly 8 furlongs. The details of this redefinition need not concern us here. The essential point is that every culture goes through a process similar to what occurred centuries ago in England. Units of measurement from different sources must be reconciled by standards set by the government.
For volume measurements, for which body parts do not provide convenient units, ancient practice was that some frequently used container size was chosen as a reference and other units were defined as multiples of this. What is interesting about this method is the universal tendency to choose multiples of two. Figure 1.4 illustrates such practice with English volume units.
In ancient Egypt the basic body-parts unit was the cubit, the distance from the elbow to the tip of the middle finger, about 46 cm. Figure 1.5 illustrates its subdivision into fingers, handbreadths, and feet.
Figure 1.5 also illustrates the peculiar definition of a royal cubit that is equal to 7 handbreadths, whereas the cubit is equal to 6 handbreadths. No surviving document explains the need for the royal cubit unit. We can only guess what the reason was. Perhaps the unit was to differentiate between properties of the pharaoh and properties of common people. However, a clue is provided by the definition of another anomalous unit, the remen: 1 remen = 5 handbreadths. My guess is that the role of these apparently anomalous units was to enable easy halving or doubling of land areas. If the linear dimensions of an area were first measured as a certain number of royal cubits, a half-size area would be one with its linear dimensions given as the same number of remens. If the linear dimensions of an area were first measured as a certain number of remens, a doubled area would be one with its linear dimensions given as the same number of royal cubits. Figure 1.6 illustrates the process for various shapes of areas: proceeding from left to right the area is reduced by [(5/7).sup.2] = 25/49 [congruent to] 1/2; proceeding from right to left the area is increased by [(7/5).sup.2] = 49/25 [congruent to] 2.
For longer distances and larger areas, the Egyptians mimicked their base-10 counting system as illustrated in figure 1.7. The tA is both a distance and an area unit since it is a line with a finite width. Such use of a unit for both distance and area was general ancient practice.
Volume units for liquids were also base-10: hin [congruent to] 0.5 liter, hqAt [congruent to] 5 liter, Xar [congruent to] 50 liter. However, volume units for grain were based on successive halving. Starting with a hqAt, fractional volumes were called Horus-eye fractions and were written with special symbols as illustrated in figure 1.8. It was documented practice in Babylon and Canaan to use the same unit for a land area and the volume of grain required to seed the area. Perhaps the system of halving land area as illustrated in figure 1.6 and the system of halving grain volumes exhibited by the Horus-eye fractions of figure 1.8 are related.
Excerpted from THE BABYLONIAN THEOREM by PETER S. RUDMAN Copyright © 2010 by Peter S. Rudman. Excerpted by permission.
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