 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 22287 students have viewed full stepbystep solutions from this chapter.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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.