I-8 INDEX simplifying rational expressions with, 78 – 79 Negative of a matrix, 953 Negative of a real number, 32 Negative reciprocals, 256 Negative root of a radical, 83 Negative slope, 243 Newton, 803 Newton’s law of cooling, 505 n-factorial (n!), 1074, 1089 Nonlinear inequalities definition of, 402 in two variables, 940 Nonlinear systems of equations application of, 932–933 with complex solutions, 932 definition of, 928 elimination method for solving, 929 graphing calculator method for solving, 928 with nonreal solutions, 932 substitution method for solving, 928 Nonreal complex number, 124 Nonstrict inequalities, 173, 940 Notation exponential, 29, 46 factorial, 1074, 1089 function, 227 interval, 169 inverse function, 437, 725 Leibniz, 481 limit, 25, 227 radical, 82 – 86 set-builder, 16, 169 for sets of numbers, 16 sigma, 1044 summation, 1044 nth root of a complex number, 833 nth root theorem for complex numbers, 834 nth term of an arithmetic sequence, 1052 of a geometric sequence, 1061 of a sequence, 1040 Null set definition of, 16 symbol for, 16 Number(s) complex, 123 – 129 conjugate of, 90 counting, 15 decimals, 20 exact, 566 factors of, 28 fractions, 2 imaginary, 123 integers, 19 irrational, 20 mixed, 3 natural, 15, 19 negative reciprocals, 256 rational, 20 real. See Real numbers reciprocal of, 4, 26, 32 sets of, 15, 19–20 whole, 19 Number line(s) definition of, 19 Minimum point of a parabola, 327 Minor axis of an ellipse, 1002 Minor of a matrix element, 908 Minute, 528 Mixed numbers converting between improper fractions and, 3 explanation of, 3 Mixture problems, 114–115 Models absolute value, 184 cost-benefit, 82 exponential, 457 linear, 116, 245 linear systems, 881 logarithmic, 478, 480, 495–496 mathematical, 116, 245 matrix, 958 polynomial, 369 – 370 quadratic, 145, 330 rational, 392 – 393 Modulus of a complex number, 822 Monomial(s) definition of, 51 dividing polynomials by, 56 Motion problems, 113–114 Multiple-angle identity, deriving, 714 Multiple angles, trigonometric equations with, 745 – 746 Multiplication of binomials, 53–55 of complex numbers, 126–127 of decimals, 8–9 of fractions, 3–4 of functions, 297 identity property for, 32 of matrices. See Matrix multiplication order of operations for, 29 of polynomials, 53–55 by powers of ten, 8–9 of radicals, 88–89 of rational expressions, 74–75 of real numbers, 26, 28 by zero, 32 Multiplication property of equality, 106 Multiplication property of zero, 32 Multiplicative inverse definition of, 966 of a matrix, 966 of a real number, 26, 32, 966 Multiplicity of zeros, 351 Mutually exclusive events, 1103 N Napier, John, 471 Natural logarithms applications of, 480 definition of, 480 modeling with, 480 Natural numbers, 15, 19 Nautical mile, 606 Negation, order of operations for, 29 Negative angle, 526 Negative exponents definition of, 42–43, 47 factoring expressions with, 67–68 Logistic function, 512 LORAN, 1023 Lowest terms of a fraction, 2 of a rational expression, 72–73 M Mach number, 723 Magnitude of a vector, 801 Major axis of an ellipse, 1002 Mandelbrot set, 837 Mapping, 222 Marginal propensity to consume, 265 Mathematical induction generalized principle of, 1083–1084 principle of, 1080–1081 proof by, 1082 Mathematical model, 116, 245 Matrix (Matrices) addition of, 951 additive inverse of, 953 augmented, 892 cofactor of an element of, 908–909 column of, 892, 951 definition of, 892 determinant of, 907–908 diagonal form of, 893 dimensions of, 892 element of, 892 equal, 951 expansion by a given row or column, 909 identity, 965 minor of an element of, 908 modeling with, 958 multiplication of. See Matrix multiplication multiplicative inverse of, 966 negative of, 953 order of, 892 properties of, 951–959 reduced-row echelon form of, 893 row of, 892, 951 row transformations of, 892 scalar multiplication of, 954 sign array for, 910 singular, 970 size of, 892 square, 951 steps to find inverse of, 968 subtraction of, 953 zero, 952 Matrix inverses method for solving linear systems, 970 Matrix method for solving linear systems, 893 Matrix multiplication applications of, 958–959 definition of, 955 properties of, 954, 957 by scalars, 954 Maturity value, 109 Maximum point of a parabola, 328 Measure of an angle, 526, 556, 557 – 558 Members of a set, 15 Midpoint formula, 205–207
RkJQdWJsaXNoZXIy NjM5ODQ=