By Isaac Asimov
Seventeen essays on numbers and quantity thought and the connection of numbers to size, the calendar, biology, astronomy, and the earth.
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Extra resources for Asimov on Numbers
If a ring has only a finite number of units, and exactly one unit u, other than 1, whose square is 1, then the product of all the units in the ring is u. Chapter 4. Units 54 Proof. The proof is the same. The unit 1 doesn't affect the product. Each unit different from 1 and u can be paired with its inverse in the product, a mutual annihilation. The only unit that remains in the product is u. Exercises 1. Let p denote a prime integer. Prove that the only solutions to the 2 quadratic equation X == 1 in Zp are 1 and -1.
A basic fact about relatively prime integers is the following theorem. Theorem 9. If a and b are relatively prime, and a divides be, then a divides c. Proof. As a and b are relatively prime, there exist integers s and t such that sa+ tb = 1 (Bezout's equation). Multiplying both sides of this equation by c we get sac+ tbc =c. As a divides be, there exists an integer r so that be = ra. Thus we can rewrite the above equation in the form c = sea + tra = ( sc + tr )a showing that c is a multiple of a.
Orders of elements: Fermat and Euler 55 unit -1 has order 2, and the unit 1 has order 1. In the ring Z 7 we can compute the order of a unit by simply calculating its powers until we hit 1, as shown in the following table. u 1 2 3 4 5 6 Powers of u Order of u 1 1 2, 4, 8 = 1 3 3, 9 = 2, 6, 18 = 4, 12 = 5, 15 = 1 6 4 16 = 2 8 = 1 3 ' ' 5, 25 = 4, 20 = 6, 30 = 2, 10 = 3, 15 = 1 6 2 6, 36 = 1 Notice that all the orders of units in Z 7 are factors of 6. It follows that 6 every unit u in has the property that u = 1.
Asimov on Numbers by Isaac Asimov