 5.5.1: In 14, list all numbers from the given set that are a. natural numb...
 5.5.2: In 14, list all numbers from the given set that are a. natural numb...
 5.5.3: In 14, list all numbers from the given set that are a. natural numb...
 5.5.4: In 14, list all numbers from the given set that are a. natural numb...
 5.5.5: Give an example of a whole number that is not a natural number
 5.5.6: Give an example of an integer that is not a whole number.
 5.5.7: Give an example of a rational number that is not an integer
 5.5.8: Give an example of a rational number that is not a natural number.
 5.5.9: Give an example of a number that is an integer, a whole number, and...
 5.5.10: Give an example of a number that is a rational number, an integer, ...
 5.5.11: Give an example of a number that is an irrational number and a real...
 5.5.12: Give an example of a number that is a real number, but not an irrat...
 5.5.13: Complete each statement in 1315 to illustrate the commutative prope...
 5.5.14: Complete each statement in 1315 to illustrate the commutative prope...
 5.5.15: Complete each statement in 1315 to illustrate the commutative prope...
 5.5.16: Complete each statement in 1617 to illustrate the associative prope...
 5.5.17: Complete each statement in 1617 to illustrate the associative prope...
 5.5.18: Complete each statement in 1820 to illustrate the distributive prop...
 5.5.19: Complete each statement in 1820 to illustrate the distributive prop...
 5.5.20: Complete each statement in 1820 to illustrate the distributive prop...
 5.5.21: Use the distributive property to simplify the radical expressions i...
 5.5.22: Use the distributive property to simplify the radical expressions i...
 5.5.23: Use the distributive property to simplify the radical expressions i...
 5.5.24: Use the distributive property to simplify the radical expressions i...
 5.5.25: Use the distributive property to simplify the radical expressions i...
 5.5.26: Use the distributive property to simplify the radical expressions i...
 5.5.27: Use the distributive property to simplify the radical expressions i...
 5.5.28: Use the distributive property to simplify the radical expressions i...
 5.5.29: In 2944, state the name of the property illustrated.6 + (4) = (4)...
 5.5.30: In 2944, state the name of the property illustrated.11 # (7 + 4) = ...
 5.5.31: In 2944, state the name of the property illustrated.6 + (2 + 7) = (...
 5.5.32: In 2944, state the name of the property illustrated.6 # (2 # 3) = 6...
 5.5.33: In 2944, state the name of the property illustrated.(2 + 3) + (4 + ...
 5.5.34: In 2944, state the name of the property illustrated.7 # (11 # 8) = ...
 5.5.35: In 2944, state the name of the property illustrated.2(8 + 6) = 16...
 5.5.36: In 2944, state the name of the property illustrated.8(3 + 11) = 2...
 5.5.37: In 2944, state the name of the property illustrated.12232 # 25 = 21...
 5.5.38: In 2944, state the name of the property illustrated.22p = p22
 5.5.39: In 2944, state the name of the property illustrated.217 # 1 = 217
 5.5.40: In 2944, state the name of the property illustrated.. 217 + 0 = 217
 5.5.41: In 2944, state the name of the property illustrated.217 + 1  2172 = 0
 5.5.42: In 2944, state the name of the property illustrated.. 217 # 1217 = 1
 5.5.43: In 2944, state the name of the property illustrated.22 + 271 22 + 2...
 5.5.44: In 2944, state the name of the property illustrated. 22 + 272 +  1...
 5.5.45: In 4549, use two numbers to show thatthe natural numbers are not cl...
 5.5.46: In 4549, use two numbers to show thatthe natural numbers are not cl...
 5.5.47: In 4549, use two numbers to show thatthe integers are not closed wi...
 5.5.48: In 4549, use two numbers to show thatthe irrational numbers are not...
 5.5.49: In 4549, use two numbers to show thatthe irrational numbers are not...
 5.5.50: Shown in the figure is a 7hour clock and the table for clock addit...
 5.5.51: Shown in the figure is an 8hour clock and the table for clock addi...
 5.5.52: In 5255, determine whether each statement is true or false. Do not ...
 5.5.53: In 5255, determine whether each statement is true or false. Do not ...
 5.5.54: In 5255, determine whether each statement is true or false. Do not ...
 5.5.55: In 5255, determine whether each statement is true or false. Do not ...
 5.5.56: In 5657, name the property used to go from step to step each time t...
 5.5.57: In 5657, name the property used to go from step to step each time t...
 5.5.58: The tables show the operations n and ^ on the set {a, b, c, d, e}. ...
 5.5.59: The tables show the operations n and ^ on the set {a, b, c, d, e}. ...
 5.5.60: The tables show the operations n and ^ on the set {a, b, c, d, e}. ...
 5.5.61: The tables show the operations n and ^ on the set {a, b, c, d, e}. ...
 5.5.62: The tables show the operations n and ^ on the set {a, b, c, d, e}. ...
 5.5.63: The tables show the operations n and ^ on the set {a, b, c, d, e}. ...
 5.5.64: The tables show the operations n and ^ on the set {a, b, c, d, e}. ...
 5.5.65: The tables show the operations n and ^ on the set {a, b, c, d, e}. ...
 5.5.66: If c a b c d d * c e f g h d = c ae + bg af + bh ce + dg cf + dh d ...
 5.5.67: In 6770, use the definition of vampire numbers from the Blitzer Bon...
 5.5.68: In 6770, use the definition of vampire numbers from the Blitzer Bon...
 5.5.69: In 6770, use the definition of vampire numbers from the Blitzer Bon...
 5.5.70: In 6770, use the definition of vampire numbers from the Blitzer Bon...
 5.5.71: A narcissistic number is an ndigit number equal to the sum of each...
 5.5.72: A narcissistic number is an ndigit number equal to the sum of each...
 5.5.73: A narcissistic number is an ndigit number equal to the sum of each...
 5.5.74: A narcissistic number is an ndigit number equal to the sum of each...
 5.5.75: The algebraic expressions D(A + 1) 24 and DA + D 24 describe the dr...
 5.5.76: Closure illustrates that a characteristic of a set is not necessari...
 5.5.77: Name the kind of rotational symmetry shown in 7778.
 5.5.78: Name the kind of rotational symmetry shown in 7778.
 5.5.79: What does it mean when we say that the rational numbers are a subse...
 5.5.80: What does it mean if we say that a set is closed under a given oper...
 5.5.81: State the commutative property of addition and give an example.
 5.5.82: State the commutative property of multiplication and give an example.
 5.5.83: State the associative property of addition and give an example.
 5.5.84: State the associative property of multiplication and give an example.
 5.5.85: State the distributive property of multiplication over addition and...
 5.5.86: Does 7 # (4 # 3) = 7 # (3 # 4) illustrate the commutative property ...
 5.5.87: Explain how to use the 8hour clock shown in Exercise 51 to find 6 5.
 5.5.88: Make Sense? In 8891, determine whether each statement makes sense o...
 5.5.89: Make Sense? In 8891, determine whether each statement makes sense o...
 5.5.90: Make Sense? In 8891, determine whether each statement makes sense o...
 5.5.91: Make Sense? In 8891, determine whether each statement makes sense o...
 5.5.92: In 9299, determine whether each statement is true or false. If the ...
 5.5.93: In 9299, determine whether each statement is true or false. If the ...
 5.5.94: In 9299, determine whether each statement is true or false. If the ...
 5.5.95: In 9299, determine whether each statement is true or false. If the ...
 5.5.96: In 9299, determine whether each statement is true or false. If the ...
 5.5.97: In 9299, determine whether each statement is true or false. If the ...
 5.5.98: In 9299, determine whether each statement is true or false. If the ...
 5.5.99: In 9299, determine whether each statement is true or false. If the ...
Solutions for Chapter 5.5: Real Numbers and Their Properties; Clock Addition
Full solutions for Thinking Mathematically  6th Edition
ISBN: 9780321867322
Solutions for Chapter 5.5: Real Numbers and Their Properties; Clock Addition
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.5: Real Numbers and Their Properties; Clock Addition includes 99 full stepbystep solutions. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Since 99 problems in chapter 5.5: Real Numbers and Their Properties; Clock Addition have been answered, more than 70939 students have viewed full stepbystep solutions from this chapter. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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