 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 8744 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra, edition: 9.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Iterative method.
A sequence of steps intended to approach the desired solution.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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