 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 36679 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra, edition: 9.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.