Calendrical Calculations by N. Dershowitz, E. Reingold

By N. Dershowitz, E. Reingold

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B 68. 090909 . . into a fraction. ) 61. The sum of any two rational numbers is a rational number. 62. The sum of any two irrational numbers is an irrational number. 13 Exponents Reason 1. 2. 3. 4. ” This notation as well as other improvements in algebra may be found in his Geometry, published in 1637. If n is a natural number, x n denotes the product of n factors, each equal to x. In this section the meaning of x n will be expanded to allow the exponent n to be any rational number. 140 61ր2 14Ϫ5ր3 bar67388_chR_014-035 14 11/02/06 CHAPTER R 00:52 Page 14 BASIC ALGEBRAIC OPERATIONS Z Integer Exponents Definition 1 generalizes exponent notation to include 0 and negative integer exponents.

EXAMPLE 1 Polynomials and Nonpolynomials (A) Polynomials in one variable: x2 Ϫ 3x ϩ 2 6x3 Ϫ 12x Ϫ 1 3 (B) Polynomials in several variables: 3x2 Ϫ 2xy ϩ y2 4x3y2 Ϫ 13xy2z5 (C) Nonpolynomials: 12x Ϫ 3 ϩ5 x x2 Ϫ 3x ϩ 2 xϪ3 2x2 Ϫ 3x ϩ 1 (D) The degree of the first term in 6x3 Ϫ 12x Ϫ 13 is 3, the degree of the second term is 1, the degree of the third term is 0, and the degree of the whole polynomial is 3. (E) The degree of the first term in 4x3y2 Ϫ 13xy2 is 5, the degree of the second term is 3, and the degree of the whole polynomial is 5.

Of 81. b negative No real nth root One real nth root ؊9 has no real square roots. ؊2 is the only real cube root of ؊8. *In this section we limit our discussion to real roots of real numbers. After the real numbers are extended to the complex numbers (see Section 1-4), additional roots may be considered. For example, it turns out that 1 has three cube roots: in addition to the real number 1, there are two other cube roots of 1 in the complex number system. bar67388_chR_014-035 11/02/06 00:52 Page 19 S E C T I O N R–2 Exponents 19 Thus, 4 and 5 have two real square roots each, and Ϫ9 has none.

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