 5.5.5.1.401: The union of the rational numbers and the irrational numbers is the...
 5.5.5.1.402: The symbol used to represent the set of real numbers is .
 5.5.5.1.403: If an operation is performed on any two elements of a set and the r...
 5.5.5.1.404: The equation 2 + 3 = 3 + 2 demonstrates the property of addition
 5.5.5.1.405: The equation 4 # 5 = 5 # 4 demonstrates the property of multiplicat...
 5.5.5.1.406: The equation (2 + 3) + 4 = 2 + (3 + 4) demonstrates the property of...
 5.5.5.1.407: The equation (5 # 6) # 7 = 5 # (6 # 7) demonstrates the property of...
 5.5.5.1.408: The equation 2 # (3 + 4) = 2 # 3 + 2 # 4 demonstrates the property ...
 5.5.5.1.409: In Exercises 912, determine whether the natural numbers are closed ...
 5.5.5.1.410: In Exercises 912, determine whether the natural numbers are closed ...
 5.5.5.1.411: In Exercises 912, determine whether the natural numbers are closed ...
 5.5.5.1.412: In Exercises 912, determine whether the natural numbers are closed ...
 5.5.5.1.413: In Exercises 1316, determine whether the integers are closed under ...
 5.5.5.1.414: In Exercises 1316, determine whether the integers are closed under ...
 5.5.5.1.415: In Exercises 1316, determine whether the integers are closed under ...
 5.5.5.1.416: In Exercises 1316, determine whether the integers are closed under ...
 5.5.5.1.417: In Exercises 1720, determine whether the rational numbers are close...
 5.5.5.1.418: In Exercises 1720, determine whether the rational numbers are close...
 5.5.5.1.419: In Exercises 1720, determine whether the rational numbers are close...
 5.5.5.1.420: In Exercises 1720, determine whether the rational numbers are close...
 5.5.5.1.421: In Exercises 2124, determine whether the irrational numbers are clo...
 5.5.5.1.422: In Exercises 2124, determine whether the irrational numbers are clo...
 5.5.5.1.423: In Exercises 2124, determine whether the irrational numbers are clo...
 5.5.5.1.424: In Exercises 2124, determine whether the irrational numbers are clo...
 5.5.5.1.425: In Exercises 2528, determine whether the real numbers are closed un...
 5.5.5.1.426: In Exercises 2528, determine whether the real numbers are closed un...
 5.5.5.1.427: In Exercises 2528, determine whether the real numbers are closed un...
 5.5.5.1.428: In Exercises 2528, determine whether the real numbers are closed un...
 5.5.5.1.429: Does (5 + x) + 6 = (x + 5) + 6 illustrate the commutative property ...
 5.5.5.1.430: Does (x + 5) + 6 = x + (5 + 6) illustrate the commutative property ...
 5.5.5.1.431: Give an example to show that the commutative property of addition m...
 5.5.5.1.432: Give an example to show that the commutative property of multiplica...
 5.5.5.1.433: Does the commutative property hold for the integers under the opera...
 5.5.5.1.434: Does the commutative property hold for the rational numbers under t...
 5.5.5.1.435: Give an example to show that the associative property of addition m...
 5.5.5.1.436: Give an example to show that the associative property of multiplica...
 5.5.5.1.437: Does the associative property hold for the integers under the opera...
 5.5.5.1.438: Does the associative property hold for the integers under the opera...
 5.5.5.1.439: Does the associative property hold for the real numbers under the o...
 5.5.5.1.440: Does a + (b # c) = (a + b) # (a + c)? Give an example to support yo...
 5.5.5.1.441: In Exercises 4152, state the name of the property illustrated.3(y +...
 5.5.5.1.442: In Exercises 4152, state the name of the property illustrated.6 + 7...
 5.5.5.1.443: In Exercises 4152, state the name of the property illustrated.(7 # ...
 5.5.5.1.444: In Exercises 4152, state the name of the property illustrated.c + d...
 5.5.5.1.445: In Exercises 4152, state the name of the property illustrated.(24 +...
 5.5.5.1.446: In Exercises 4152, state the name of the property illustrated.4 # (...
 5.5.5.1.447: In Exercises 4152, state the name of the property illustrated.13 # ...
 5.5.5.1.448: In Exercises 4152, state the name of the property illustrated.3 8 +...
 5.5.5.1.449: In Exercises 4152, state the name of the property illustrated.1(x ...
 5.5.5.1.450: In Exercises 4152, state the name of the property illustrated.(r + ...
 5.5.5.1.451: In Exercises 4152, state the name of the property illustrated.(r + ...
 5.5.5.1.452: In Exercises 4152, state the name of the property illustrated.g # (...
 5.5.5.1.453: In Exercises 5364, use the distributive property to multiply. Then,...
 5.5.5.1.454: In Exercises 5364, use the distributive property to multiply. Then,...
 5.5.5.1.455: In Exercises 5364, use the distributive property to multiply. Then,...
 5.5.5.1.456: In Exercises 5364, use the distributive property to multiply. Then,...
 5.5.5.1.457: In Exercises 5364, use the distributive property to multiply. Then,...
 5.5.5.1.458: In Exercises 5364, use the distributive property to multiply. Then,...
 5.5.5.1.459: In Exercises 5364, use the distributive property to multiply. Then,...
 5.5.5.1.460: In Exercises 5364, use the distributive property to multiply. Then,...
 5.5.5.1.461: In Exercises 5364, use the distributive property to multiply. Then,...
 5.5.5.1.462: In Exercises 5364, use the distributive property to multiply. Then,...
 5.5.5.1.463: In Exercises 5364, use the distributive property to multiply. Then,...
 5.5.5.1.464: In Exercises 5364, use the distributive property to multiply. Then,...
 5.5.5.1.465: In Exercises 6570, name the property used to go from step to step. ...
 5.5.5.1.466: In Exercises 6570, name the property used to go from step to step. ...
 5.5.5.1.467: In Exercises 6570, name the property used to go from step to step. ...
 5.5.5.1.468: In Exercises 6570, name the property used to go from step to step. ...
 5.5.5.1.469: In Exercises 6570, name the property used to go from step to step. ...
 5.5.5.1.470: In Exercises 6570, name the property used to go from step to step. ...
 5.5.5.1.471: In Exercises 7174, determine whether the activity can be used to il...
 5.5.5.1.472: In Exercises 7174, determine whether the activity can be used to il...
 5.5.5.1.473: In Exercises 7174, determine whether the activity can be used to il...
 5.5.5.1.474: In Exercises 7174, determine whether the activity can be used to il...
 5.5.5.1.475: In Exercises 7580, determine whether the activity can be used to il...
 5.5.5.1.476: In Exercises 7580, determine whether the activity can be used to il...
 5.5.5.1.477: In Exercises 7580, determine whether the activity can be used to il...
 5.5.5.1.478: In Exercises 7580, determine whether the activity can be used to il...
 5.5.5.1.479: In Exercises 7580, determine whether the activity can be used to il...
 5.5.5.1.480: In Exercises 7580, determine whether the activity can be used to il...
 5.5.5.1.481: Describe two other activities that can be used to illustrate the co...
 5.5.5.1.482: Describe three other activities that can be used to illustrate the ...
 5.5.5.1.483: Does 0 , a = a , 0 (assume a 0)? Explain.
 5.5.5.1.484: KenKen Refer to the Recreational Mathematics box on page 254. Compl...
 5.5.5.1.485: a) Consider the three words man eating tiger. Does (man eating) tig...
 5.5.5.1.486: A set of numbers that was not discussed in this chapter is the set ...
Solutions for Chapter 5.5: Number Theory and the Real Number System
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 5.5: Number Theory and the Real Number System
Get Full SolutionsA Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. Since 86 problems in chapter 5.5: Number Theory and the Real Number System have been answered, more than 70959 students have viewed full stepbystep solutions from this chapter. Chapter 5.5: Number Theory and the Real Number System includes 86 full stepbystep solutions. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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).

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.

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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

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

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

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.

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.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).