 P.1.1: A real number is ________ if it can be written as the ratio of two ...
 P.1.2: ________ numbers have infinite nonrepeating decimal representations.
 P.1.3: The point 0 on the real number line is called the ________.
 P.1.4: The distance between the origin and a point representing a real num...
 P.1.5: A number that can be written as the product of two or more prime nu...
 P.1.6: An integer that has exactly two positive factors, the integer itsel...
 P.1.7: An algebraic expression is a collection of letters called ________ ...
 P.1.8: The ________ of an algebraic expression are those parts separated b...
 P.1.9: The numerical factor of a variable term is the ________ of the vari...
 P.1.10: The ________ ________ states that if
 P.1.11:
 P.1.12:
 P.1.13: 2.01, 0.666 . . . , 13, 0.010110111 . . . , 1, 6 5
 P.1.14: 2.3030030003 . . . , 0.7575, 4.63,10, 75, 4 2.
 P.1.15: , 1 3, 6 3, 1 22, 7.5, 1, 8, 22 2.30
 P.1.16: 25, 17, 7, 11.1, 1312
 P.1.17: (a) 3 (b) (c
 P.1.18: (a) 8.5 (b) (c)
 P.1.19: 5
 P.1.20: 1
 P.1.21: 333
 P.1.22: 11
 P.1.23: 3 2 10123
 P.1.24: 7 6 5 4 3 2 1 0
 P.1.25: 4, 8 3.
 P.1.26: 3.5, 1
 P.1.27: 2, 7
 P.1.28: 1, 16
 P.1.29: 5 6, 2 3
 P.1.30: 8 7, 3 7
 P.1.31: x 5
 P.1.32: x 2
 P.1.33: x < 0
 P.1.34: x > 3
 P.1.35: 4,
 P.1.36: , 2
 P.1.37: 2 < x < 2
 P.1.38: 0 x 5
 P.1.39: 1 x < 0
 P.1.40: 0 < x 6
 P.1.41: 2, 5
 P.1.42: 1, 2
 P.1.43: is nonnegative.
 P.1.44: is no more than 25.
 P.1.45: is greater than and at most 4.
 P.1.46: is at least and less than 0.
 P.1.47: is at least 10 and at most 22.
 P.1.48: s less than 5 but no less than
 P.1.49: The dogs weight is more than 65 pounds
 P.1.50: The annual rate of inflation is expected to be at least 2.5% but no...
 P.1.51:
 P.1.52:
 P.1.53:
 P.1.54:
 P.1.55:
 P.1.56:
 P.1.57:
 P.1.58:
 P.1.59: 2 x
 P.1.60: 1
 P.1.61: 3 3 <
 P.1.62: 4 4
 P.1.63: 5 5
 P.1.64: 6 6 2005
 P.1.65: 2 2 2006
 P.1.66: (2)22
 P.1.67: a 126, b 75
 P.1.68: a 126, b 75
 P.1.69: a 5 2, b 0
 P.1.70: a 1 4, b 11 4
 P.1.71: a 16 5 , b 112
 P.1.72: a 9.34, b 5.65
 P.1.73: The distance between and 5 is no more than 3.
 P.1.74: The distance between and is at least 6.
 P.1.75: is at least six units from 0.
 P.1.76: is at most two units from
 P.1.77: While traveling on the Pennsylvania Turnpike, you pass milepost 57 ...
 P.1.78: The temperature in Bismarck, North Dakota was at noon, then at midn...
 P.1.79: Wages $112,700 $113,356
 P.1.80: Utilities $9,400 $9,772
 P.1.81: Taxes $37,640 $37,335
 P.1.82: Insurance $2,575 $2,613
 P.1.83: 1996 $1560.6 billion
 P.1.84: 1998 $1652.7 billion
 P.1.85: 2000 $1789.2 billion
 P.1.86: 2002 $2011.2 billion
 P.1.87: 2004 $2293.0 billion
 P.1.88: 2006 $2655.4 billion
 P.1.89: 7x 4
 P.1.90: 6x
 P.1.91: 3x
 P.1.92: 33x
 P.1.93: 4x3 x 2 5
 P.1.94: 3x4 x2 4
 P.1.95: 4x 6 x 1 x 0
 P.1.96: 9 7x
 P.1.97: x x 2 x 2 2 3x 4
 P.1.98: x x 1 x 1 2 5x 4 x
 P.1.99: x 1 x 1
 P.1.100: x x 2
 P.1.101: x 9 9 x
 P.1.102: 2 1 21
 P.1.103: 6 1
 P.1.104: x 3 x 30
 P.1.105: 2 x 32 x 2 3
 P.1.106: z 2 0 z 2
 P.1.107: 1 1 x1 x
 P.1.108: z 5x z x 5 x
 P.1.109: x y 10 x y 10
 P.1.110: x 3y x 3y 3xy
 P.1.111: 3 t 43 t 3 4
 P.1.112: 7 7 12 1 7 712 1 12 12
 P.1.113: 3 16 5 16
 P.1.114: 7 4
 P.1.115: 5 8 5 12 1 6
 P.1.116: 10 11 6 33 13 66
 P.1.117: 12 1 4
 P.1.118: 6 4 2008
 P.1.119: 2x 3 x 4
 P.1.120: 5x 6 2 9
 P.1.121: (
 P.1.122: (a)
 P.1.123: CONJECTURE (a) Use a calculator to complete the table. (b) Use the ...
 P.1.124: CONJECTURE (a) Use a calculator to complete the table. (b) Use the ...
 P.1.125: If and
 P.1.126: If and
 P.1.127: If then
 P.1.128: Because
 P.1.129: THINK ABOUT IT Consider and where and (a) Are the values of the exp...
 P.1.130: THINK ABOUT IT Is there a difference between saying that a real num...
 P.1.131: THINK ABOUT IT Because every even number is divisible by 2, is it p...
 P.1.132: THINK ABOUT IT Is it possible for a real number to be both rational...
 P.1.133: WRITING Can it ever be true that for a real number Explain
 P.1.134: CAPSTONE
Solutions for Chapter P.1: Review of Real Numbers and Their Properties
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter P.1: Review of Real Numbers and Their Properties
Get Full SolutionsAlgebra and Trigonometry was written by and is associated to the ISBN: 9781439048474. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. Chapter P.1: Review of Real Numbers and Their Properties includes 134 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 134 problems in chapter P.1: Review of Real Numbers and Their Properties have been answered, more than 51797 students have viewed full stepbystep solutions from this chapter.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.