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Unit 1

The Primes

1.3 Number for Number's Sake: The Greeks

PLAYING WITH NUMBERS

Prime numbers may have been of interest to the people of Ishango, but it was the Greeks who began to ask deep questions about them. The Greeks held high esteem for the pursuit of knowledge and, in particular, mathematical truth. One would not have expected such a high-level fascination from the illiterate and innumerate people who conquered the Aegean peninsula. Their conquests and the knowledge that flowed along their trade routes, however, enabled them to catch up quickly with the rest of the mathematical world.

While the rest of humanity was seemingly occupied with the more practical uses of mathematics, the Greeks were among the first to develop a mathematical world that was not necessarily tied to real-world applications. It was during this time that the concept of axiomatic structure emerged—mathematical proof, in other words. Thales, a mathematician who is said to have astonished his countrymen by correctly predicting a solar eclipse in the year 585 BC, is generally credited as taking the first steps toward focusing on the logical structure and principles behind mathematics. Described as the first philosopher and the first mathematician, he is definitely the first person to whom a specific mathematical "discovery" is ascribed, namely that an angle inscribed in a semicircle is a right angle. As is often the case with the emergence of new ideas, there is some dispute on this, however. Thales is generally given the credit for this discovery, although some claim that he simply re-packaged a previous Babylonian finding.

There is some debate as to whether it was Thales or the Pythagoreans who presided over the shift in mathematics from practical concerns to the development of general principles and ideas. The supposed motto of the Pythagorean group, "All Is Number," encapsulates their preoccupation with both mathematical and numerological concepts. For example, they ascribed a gender to numbers, odd numbers being male and even numbers being female. Much of the mathematical tradition of ancient Greece, and, thus, of the civilizations that followed, stemmed from the obsessions of the Pythagoreans.

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THE PRIMACY OF PROOF

Chief among the Pythagorean concerns was the notion of proof. In philosophy it was possible to argue, as the Sophists did, both sides of a scenario and see that neither was a clear winner. Math, however, is different in that "truth" can be proved through a system of assumptions and allowed actions that show that a given statement must follow from initial postulates. In other words, in mathematics at least, there is indisputably a right answer, although it may not always be obvious. This clarity, and the comfort that it often brings, was of central importance to the Pythagoreans, and it represents a distinguishing feature of the field of mathematics.

Theorems proved by, or at least attributed to, early Greeks, such as the Pythagoreans, remain as true today as they were in ancient times. The same cannot be said for any host of other Greek beliefs from non-mathematical disciplines. One of the alluring features of mathematics is that it enables one to say definite things about reality. This aspect was, and continues to be, a major reason why people choose to study mathematics.

FIGURATE NUMBERS

Playing with numbers—the exploration of numbers for their own sake—is perhaps the first step towards mathematical sophistication. The Greeks were fascinated by the different properties that certain numbers exhibited geometrically. If we represent each whole number by a collection of pebbles equal in count to that number, then many interesting relations and properties of numbers can be found by looking at the shapes that one is able to make with different arrangements of the collections.

Square Numbers

Square numbers are whole numbers that, when represented as a collection of pebbles, can form a square array. The first square number is 1, the second is 4, which can be portrayed as a 2 × 2 array, the third square number, 9, forms a 3 × 3 array, etc. The triangular numbers can also be represented by an interesting sequence of dot patterns, which is, in fact, the basis of their classification as "triangular" numbers.

Square Numbers

The squares and triangular numbers, in this context, are examples of "figurate" numbers—numbers that have "shapes." Figurate numbers hold many interesting properties; for example, consider the 5 × 5 square.

5 x 5 square

Is it evident that the fifth square number, 25, represents the sum of the first five odd numbers?

first five odd numbers

We can decompose the square into nested L-shapes, also called gnomons, as shown above. It is, hopefully, straightforward to state that any square can be broken down in this way. It should also be clear that every gnomon representsan odd number, and that nested gnomons represent consecutive odds. Adding a gnomon to an n × n square increases the dimensions to (n+1) × (n+1), which is still a square. It is reasonable, then, to conclude that the sum of the first n odd numbers is n2.

In the online interactive, you will have the opportunity to play with figurate numbers, such as those we have discussed, and to discover interesting relationships between them. The study of figurate numbers leads quite naturally to primes, which we will investigate further in the next section.

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Next: 1.4 Primes


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