# Projective Varieties and Modular Forms by M. Eichler

By M. Eichler

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Definition of b m/n For all positive integers m and n such that m>n is in simplest form, and for all real numbers b for which b1>n is a real number, b m>n = (b1>n ) m = (b m )1>n Because bm>n is defined as (b1>n)m and as (bm)1>n, we can evaluate expressions such as 8 in more than one way. For example, because 81>3 is a real number, 84>3 can be evaluated in either of the following ways. 4>3 84>3 = (81>3)4 = 24 = 16 84>3 = (84)1>3 = 40961>3 = 16 Of the two methods, the bm>n = (b1>n)m method is usually easier to apply, provided you can evaluate b1>n.

That is, if the exponent is positive, move the decimal point to the right the same number of places as the exponent. 5 * 105 = 350,000 m 5 places If the exponent is negative, move the decimal point to the left the same number of places as the absolute value of the exponent. 1 * 106 orchid seeds weigh 1 ounce. ■ Computer scientists measure an operation in nanoseconds. 7 * 109 years to travel from one end of the universe to the other. ■ m Math Matters 8 places Most calculators display very large and very small numbers in scientific notation.

7 2 17 In Exercises 9 to 18, perform the operations given that A { ؍؊3, ؊2, ؊1, 0, 1, 2, 3}, B { ؍؊2, 0, 2, 4, 6}, C { ؍0, 1, 2, 3, 4, 5, 6}, and D { ؍؊3, ؊1, 1, 3}. 9. A ´ B 10. C ´ D 11. A ¨ C 12. C ¨ D 13. B ¨ D 14. B ´ (A ¨ C) 15. D ¨ (B ´ C) 16. (A ¨ B) ´ (A ¨ C) 17. (B ´ C) ¨ (B ´ D) 18. (A ¨ C) ´ (B ¨ D) 5 In Exercises 3 to 8, list the four smallest elements of each set. 3. 52x ƒ x ʦ positive integers6 4. 5 ƒ x ƒ ƒ x ʦ integers6 5. 5 y ƒ y = 2x + 1, x ʦ natural numbers6 6. 5 y ƒ y = x2 - 1, x ʦ integers6 In Exercises 19 to 24, perform the operation, given A is any set.