PSAT Test (Day 44)

Today is PSAT day across America. In the district whose calendar we're following, there is only the test and breaks today, with no periods or classes at all. Therefore there is no worksheet to post today.

Because of Yom Kippur, the PSAT was delayed a week -- from Day 40 to Day 44. This means that instead of falling between Chapters 3 and 4, the test now occurs on the day of Lesson 4-4. In other words, Lesson 4-4 will not be taught on the blog this year. Instead, we go directly from Lesson 4-3 yesterday to Lesson 4-5 tomorrow. (After all, real Geometry students in our classes will miss a day of math due to the test.)

Instead, the main part of today's post is about our side-along reading book. Today I made the decision to stick to tradition and read Ian Stewart's The Story of Mathematics: From Babylonian Numerals to Chaos Theory, rather than watch the DVD or read the computer book that I purchased. As usual, I'll cover a chapter in every post and add the new "Stewart" label. (By the way, the British mathematician Stewart is still with us -- he just turned 74 last month.)

Here is the table of contents:

1. Tokens, Tallies and Tablets
2. The Logic of Shape
3. Notations and Numbers
4. Lure of the Unknown
5. Eternal Triangles
6. Curves and Coordinates
7. Patterns in Numbers
8. The System of the World
9. Patterns in Nature
10. Impossible Quantities
11. Firm Foundations
12. Impossible Triangles
13. The Rise of Symmetry
14. Algebra Comes of Age
15. Rubber Sheet Geometry
16. The Fourth Dimension
17. The Shape of Logic
18. How Likely is That?
19. Number Crunching
20. Chaos and Complexity

Without further ado, let's begin.

Chapter 1 of Ian Stewart's The Story of Mathematics is called "Tokens, Tallies and Tablets: The Birth of Numbers." Here's how it begins:

"Mathematics began with numbers, and numbers are still fundamental, even though the subject is no longer limited to numerical calculations."

As the title of our new book tells us, this book is all about the history of math. And as we'll soon see, the three objects in the chapter title will be used to represent numbers. Stewart tells us that math, as we might expect, all started with numbers. But he warns us:

"Numbers seem very simple and straightforward, but appearances can be deceptive. Calculations with numbers can be hard; getting the right number can be difficult."

The author describes the history of writing numbers. He tells us that our current numbering system, including the decimal point, has only been around for 450 years. He compares this to the age of the technology we use to make calculations:

"Computers, which have embedded mathematical calculations so deeply into our culture that we no longer notice their presence, have been with us for a mere 50 years; computers powerful enough and fast enough to be useful in our homes and offices first became widespread about 20 years ago."

Notice that Stewart wrote this book in 2007, so by now it's more like 30 years of home computers. He now asks:

"How did this truly enormous numerical industry arise? It all began with little clay tokens, 10,000 years ago in the Near East."

These tokens were used to count things -- bushels of grain, animals, and jars of oil. Then ancient storekeepers kept these tokens in clay envelopes:

"If there were seven spheres inside, the accountants would draw seven pictures of spheres in the wet clay of the envelope. At some point the Mesopotamian bureaucrats realized that once they had drawn the symbols on the outside of the envelope, they didn't actually need the contents, so they didn't need to break open the envelope to find out which tokens were in it."

Stewart now moves on to tally marks. Some prehistoric tallies have been discovered on a baboon's leg bone:

"The bone was found in a cave in the Lebombo mountains, on the border between Swaziland and South Africa, so the cave is known as the border cave, and the bone is the Lebombo bone. In the absence of a time machine, there is no way to be certain what the marks represented, but we can make informed guesses."

And one possible guess is that they represent the days of a lunar month. Cue the "Earth, Moon, and Sun" song now. Meanwhile the author moves on to the first numerals:

"By 3000 BC, the Sumerians had developed an elaborate form of writing, now called cuneiform or wedge-shaped. The history of this period is complicated, with different cities becoming dominant at different times."

Of course, it was Babylon that would eventually become the capital. I won't attempt to represent Babylonian numbers in ASCII, but the important part is that they were base 60:

"That is, the value of a symbol may be some number, or 60 times such a number, or 60 times 60 times such a number, depending on the symbol's position."

And Stewart tells us that we have the Babylonians to thank for having 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a turn:

"And something like 23,11,14 would indicate the Babylonian symbols for 23, 11 and 14 written in order, with the numerical value (23 * 60 * 60) + (11 * 60) + 14, which comes to 83,474 in our notation. Not only do we [in modern times] use ten symbols to represent arbitrarily large numbers; we also use the same symbols to represent arbitrarily small ones."

The author explains that just as we use a decimal point to represent small numbers, the Babylonians used a similar symbol to represent a "sexagesimal point." Stewart now proceeds to describe another numeration system used by the ancient Egyptians:

"Repeating these symbols up to nine times, and then combining the results, can represent any whole number."

The ancient Egyptians also had symbols to represent fractions. We've discussed these on the blog before -- they are unit fractions:

"For instance, 5/6 = 1/2 + 1/3. Interestingly, the Egyptians did not write 2/5 as 1/5 + 1/5. Their rule seems to have been: use different unit fractions."

Stewart now moves on to discuss the significance of mathematics on our culture -- something that our young students need to appreciate:

"The evolution of culture, and that of mathematics, has gone hand in hand for the last four millennia. It would be difficult to disentangle cause and effect -- I would hesitate to argue that mathematical innovation drives cultural change, or that cultural needs determine the direction of mathematical progress."

And indeed, the author explains the difference between mathematical developments and other cultural changes:

"New kinds of housing, new kinds of transport, even new ways to organize government bureaucracies, are relatively obvious to every citizen. Mathematics, however, mostly takes place behind the scenes."

And this is why many students in our classes wonder why they have to study math or succeed on the math section of today's PSAT -- the math that drives modern technology is invisible. Stewart concludes the chapter as follows:

"The inventions of number notation and arithmetic rank alongside those of language and writing as some of the innovations that differentiate us from trainable apes."

And thus concludes the first chapter. I hope you will enjoy our reading of Stewart's book.