 R.1.1: The numbers in the set are called numbers.
 R.1.2: The value of the expression is .
 R.1.3: The fact that is a consequence of the Property.
 R.1.4: The product of 5 and equals 6 may be written as .
 R.1.5: True or False Rational numbers have decimals that either terminate ...
 R.1.6: True or False The ZeroProduct Property states that the product of ...
 R.1.7: True or False The least common multiple of 12 and 18 is 6.
 R.1.8: True or False No real number is both rational and irrational.
 R.1.9: In 920, use U = universal set = and to find each set. A B
 R.1.10: In 920, use U = universal set = and to find each set. A C
 R.1.11: In 920, use U = universal set = and to find each set. A B
 R.1.12: In 920, use U = universal set = and to find each set. A C
 R.1.13: In 920, use U = universal set = and to find each set. 1A B2 C
 R.1.14: In 920, use U = universal set = and to find each set. 1A B2 C
 R.1.15: In 920, use U = universal set = and to find each set. A
 R.1.16: In 920, use U = universal set = and to find each set. C
 R.1.17: In 920, use U = universal set = and to find each set. A B
 R.1.18: In 920, use U = universal set = and to find each set. B C
 R.1.19: In 920, use U = universal set = and to find each set. A B
 R.1.20: In 920, use U = universal set = and to find each set. B C
 R.1.21: In 2126, list the numbers in each set that are (a) Natural numbers,...
 R.1.22: In 2126, list the numbers in each set that are (a) Natural numbers,...
 R.1.23: In 2126, list the numbers in each set that are (a) Natural numbers,...
 R.1.24: In 2126, list the numbers in each set that are (a) Natural numbers,...
 R.1.25: In 2126, list the numbers in each set that are (a) Natural numbers,...
 R.1.26: In 2126, list the numbers in each set that are (a) Natural numbers,...
 R.1.27: In 2738, approximate each number (a) rounded and (b) truncated to t...
 R.1.28: In 2738, approximate each number (a) rounded and (b) truncated to t...
 R.1.29: In 2738, approximate each number (a) rounded and (b) truncated to t...
 R.1.30: In 2738, approximate each number (a) rounded and (b) truncated to t...
 R.1.31: In 2738, approximate each number (a) rounded and (b) truncated to t...
 R.1.32: In 2738, approximate each number (a) rounded and (b) truncated to t...
 R.1.33: In 2738, approximate each number (a) rounded and (b) truncated to t...
 R.1.34: In 2738, approximate each number (a) rounded and (b) truncated to t...
 R.1.35: In 2738, approximate each number (a) rounded and (b) truncated to t...
 R.1.36: In 2738, approximate each number (a) rounded and (b) truncated to t...
 R.1.37: In 2738, approximate each number (a) rounded and (b) truncated to t...
 R.1.38: In 2738, approximate each number (a) rounded and (b) truncated to t...
 R.1.39: In 3948, write each statement using symbols. The sum of 3 and 2 equ...
 R.1.40: In 3948, write each statement using symbols. The product of 5 and 2...
 R.1.41: In 3948, write each statement using symbols. The sum of x and 2 is ...
 R.1.42: In 3948, write each statement using symbols. The sum of 3 and y is ...
 R.1.43: In 3948, write each statement using symbols. The product of 3 and y...
 R.1.44: In 3948, write each statement using symbols. The product of 2 and x...
 R.1.45: In 3948, write each statement using symbols. The difference x less ...
 R.1.46: In 3948, write each statement using symbols. The difference 2 less ...
 R.1.47: In 3948, write each statement using symbols. The quotient x divided...
 R.1.48: In 3948, write each statement using symbols. The quotient 2 divided...
 R.1.49: In 4986, evaluate each expression. 9  4 + 2
 R.1.50: In 4986, evaluate each expression. 6  4 + 3
 R.1.51: In 4986, evaluate each expression. 6 + 4 # 3
 R.1.52: In 4986, evaluate each expression. 8  4 # 2
 R.1.53: In 4986, evaluate each expression. 4 + 5  8
 R.1.54: In 4986, evaluate each expression. 8  3  4
 R.1.55: In 4986, evaluate each expression. 4 +13
 R.1.56: In 4986, evaluate each expression. 2  12
 R.1.57: In 4986, evaluate each expression. 6  33 # 5 + 2 # 13  224
 R.1.58: In 4986, evaluate each expression. 2 # 38  314 + 224  3
 R.1.59: In 4986, evaluate each expression. 2 # 13  52 + 8 # 2  1
 R.1.60: In 4986, evaluate each expression. 1  14 # 3  2 + 22
 R.1.61: In 4986, evaluate each expression. 10  36  2 # 2 + 18  324 # 2
 R.1.62: In 4986, evaluate each expression. 2  5 # 4  36 # 13  424
 R.1.63: In 4986, evaluate each expression. 15  32 1 2
 R.1.64: In 4986, evaluate each expression. 15 + 42 1 3
 R.1.65: In 4986, evaluate each expression. 4 + 8 5  3
 R.1.66: In 4986, evaluate each expression. 2  4 5  3
 R.1.67: In 4986, evaluate each expression. 3 5 # 10 21
 R.1.68: In 4986, evaluate each expression. 5 9 # 3 10
 R.1.69: In 4986, evaluate each expression. 6 25 # 10 27
 R.1.70: In 4986, evaluate each expression. 21 25 # 100 3
 R.1.71: In 4986, evaluate each expression. 3 4 + 2 5
 R.1.72: In 4986, evaluate each expression. 4 3 + 1 2
 R.1.73: In 4986, evaluate each expression. 5 6 + 9 5
 R.1.74: In 4986, evaluate each expression. 8 9 + 15 2
 R.1.75: In 4986, evaluate each expression. 5 18 + 1 12
 R.1.76: In 4986, evaluate each expression. 2 15 + 8 9
 R.1.77: In 4986, evaluate each expression. 1 30  7 18
 R.1.78: In 4986, evaluate each expression. 3 14  2 21
 R.1.79: In 4986, evaluate each expression. 3 20  2 15
 R.1.80: In 4986, evaluate each expression. 6 35  3 14
 R.1.81: In 4986, evaluate each expression. 5 18 11 27
 R.1.82: In 4986, evaluate each expression. 5 21 2 35
 R.1.83: In 4986, evaluate each expression. 1 2 # 3 5 + 7 10
 R.1.84: In 4986, evaluate each expression. 2 3 + 4 5 # 1 6
 R.1.85: In 4986, evaluate each expression. 2 # 3 4 + 3 8
 R.1.86: In 4986, evaluate each expression. 3 # 5 6  1 2
 R.1.87: In 8798, use the Distributive Property to remove the parentheses. 6...
 R.1.88: In 8798, use the Distributive Property to remove the parentheses. 4...
 R.1.89: In 8798, use the Distributive Property to remove the parentheses. x...
 R.1.90: In 8798, use the Distributive Property to remove the parentheses. 4...
 R.1.91: In 8798, use the Distributive Property to remove the parentheses. 2...
 R.1.92: In 8798, use the Distributive Property to remove the parentheses. 3...
 R.1.93: In 8798, use the Distributive Property to remove the parentheses. 1...
 R.1.94: In 8798, use the Distributive Property to remove the parentheses. 1...
 R.1.95: In 8798, use the Distributive Property to remove the parentheses. 1...
 R.1.96: In 8798, use the Distributive Property to remove the parentheses. 1...
 R.1.97: In 8798, use the Distributive Property to remove the parentheses. 1...
 R.1.98: In 8798, use the Distributive Property to remove the parentheses. 1...
 R.1.99: Explain to a friend how the Distributive Property is used to justif...
 R.1.100: Explain to a friend why whereas 12 + 32 # 4 = 20.
 R.1.101: Explain why is not equal to
 R.1.102: Explain why is not equal to 4 2 + 3 5 .
 R.1.103: Is subtraction commutative? Support your conclusion with an example.
 R.1.104: Is subtraction associative? Support your conclusion with an example.
 R.1.105: Is division commutative? Support your conclusion with an example.
 R.1.106: Is division associative? Support your conclusion with an example.
 R.1.107: If why does
 R.1.108: If why does
 R.1.109: Are there any real numbers that are both rational and irrational? A...
 R.1.110: Explain why the sum of a rational number and an irrational number m...
 R.1.111: A rational number is defined as the quotient of two integers. When ...
 R.1.112: The current time is 12 noon CST. What time (CST) will it be 12,997 ...
 R.1.113: Both and are undefined, but for different reasons. Write a paragrap...
Solutions for Chapter R.1: Real Numbers
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter R.1: Real Numbers
Get Full SolutionsCollege Algebra was written by and is associated to the ISBN: 9780321716811. This expansive textbook survival guide covers the following chapters and their solutions. Chapter R.1: Real Numbers includes 113 full stepbystep solutions. Since 113 problems in chapter R.1: Real Numbers have been answered, more than 29651 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra, edition: 9.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Solvable system Ax = b.
The right side b is in the column space of A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.