 5.3.5.1.178: The set of rational numbers is the set of numbers of the form p q ,...
 5.3.5.1.179: For the rational number p q , p is called the .
 5.3.5.1.180: For the rational number p q , q is called the .
 5.3.5.1.181: Rational numbers such as 23 4 and 11 2 are examples of numbers.
 5.3.5.1.182: Rational numbers such as 11 4 and 3 2 are examples of fractions.
 5.3.5.1.183: The number 1 2 can be represented as a(n) decimal number.
 5.3.5.1.184: The number 1 3 can be represented as a(n) decimal number.
 5.3.5.1.185: In the decimal number 0.285714, the 2 is in the position.
 5.3.5.1.186: In the decimal number 0.285714, the 8 is in the position.
 5.3.5.1.187: The product of a number and its must equal 1.
 5.3.5.1.188: When adding or subtracting two fractions with unlike denominators, ...
 5.3.5.1.189: The rational numbers 1 2 and 5 10 are examples of fractions.
 5.3.5.1.190: In Exercises 1322, reduce each fraction to lowest terms. 3 6
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 5.3.5.1.197: In Exercises 1322, reduce each fraction to lowest terms.120 135
 5.3.5.1.198: In Exercises 2126, convert each mixed number to an improper fractio...
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 5.3.5.1.204: In Exercises 2730, write the number of inches indicated by the arro...
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 5.3.5.1.208: In Exercises 3136, convert each improper fraction to a mixed number...
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 5.3.5.1.213: In Exercises 3136, convert each improper fraction to a mixed number...
 5.3.5.1.214: In Exercises 3744, express each rational number as terminating or r...
 5.3.5.1.215: In Exercises 3744, express each rational number as terminating or r...
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 5.3.5.1.220: In Exercises 3744, express each rational number as terminating or r...
 5.3.5.1.221: In Exercises 3744, express each rational number as terminating or r...
 5.3.5.1.222: In Exercises 4552, express each terminating decimal number as a quo...
 5.3.5.1.223: In Exercises 4552, express each terminating decimal number as a quo...
 5.3.5.1.224: In Exercises 4552, express each terminating decimal number as a quo...
 5.3.5.1.225: In Exercises 4552, express each terminating decimal number as a quo...
 5.3.5.1.226: In Exercises 4552, express each terminating decimal number as a quo...
 5.3.5.1.227: In Exercises 4552, express each terminating decimal number as a quo...
 5.3.5.1.228: In Exercises 4552, express each terminating decimal number as a quo...
 5.3.5.1.229: In Exercises 4552, express each terminating decimal number as a quo...
 5.3.5.1.230: In Exercises 5360, express each repeating decimal number as a quoti...
 5.3.5.1.231: In Exercises 5360, express each repeating decimal number as a quoti...
 5.3.5.1.232: In Exercises 5360, express each repeating decimal number as a quoti...
 5.3.5.1.233: In Exercises 5360, express each repeating decimal number as a quoti...
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 5.3.5.1.237: In Exercises 5360, express each repeating decimal number as a quoti...
 5.3.5.1.238: In Exercises 6170, perform the indicated operation and reduce your ...
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 5.3.5.1.248: In Exercises 7180, perform the indicated operation and reduce your ...
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 5.3.5.1.250: In Exercises 7180, perform the indicated operation and reduce your ...
 5.3.5.1.251: In Exercises 7180, perform the indicated operation and reduce your ...
 5.3.5.1.252: In Exercises 7180, perform the indicated operation and reduce your ...
 5.3.5.1.253: In Exercises 7180, perform the indicated operation and reduce your ...
 5.3.5.1.254: In Exercises 7180, perform the indicated operation and reduce your ...
 5.3.5.1.255: In Exercises 7180, perform the indicated operation and reduce your ...
 5.3.5.1.256: In Exercises 7180, perform the indicated operation and reduce your ...
 5.3.5.1.257: In Exercises 7180, perform the indicated operation and reduce your ...
 5.3.5.1.258: In Exercises 8186, use the appropriate formula to evaluate the expr...
 5.3.5.1.259: In Exercises 8186, use the appropriate formula to evaluate the expr...
 5.3.5.1.260: In Exercises 8186, use the appropriate formula to evaluate the expr...
 5.3.5.1.261: In Exercises 8186, use the appropriate formula to evaluate the expr...
 5.3.5.1.262: In Exercises 8186, use the appropriate formula to evaluate the expr...
 5.3.5.1.263: In Exercises 8186, use the appropriate formula to evaluate the expr...
 5.3.5.1.264: In Exercises 8792, evaluate each expression.a 1 3 # 6 7 b + 1 4
 5.3.5.1.265: In Exercises 8792, evaluate each expression.a 1 6 , 2 3 b  1 7
 5.3.5.1.266: In Exercises 8792, evaluate each expression.a 3 4 + 1 6 b , a2  7 6
 5.3.5.1.267: In Exercises 8792, evaluate each expression.a 1 3 # 3 7 b + a 3 5 #...
 5.3.5.1.268: In Exercises 8792, evaluate each expression.a3  4 9 b , a4 + 2 3 b
 5.3.5.1.269: In Exercises 8792, evaluate each expression.a 2 5 , 4 9 b a 3 5 # 6b
 5.3.5.1.270: In Exercises 93104, write an expression that will solve the problem...
 5.3.5.1.271: In Exercises 93104, write an expression that will solve the problem...
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 5.3.5.1.281: In Exercises 93104, write an expression that will solve the problem...
 5.3.5.1.282: Cutting Lumber If a piece of wood 8 3 4 ft long is to be cut into f...
 5.3.5.1.283: Dimensions of a Room A rectangular room measures 8 ft 3 in. by 10 f...
 5.3.5.1.284: Hanging a Picture The back of a framed picture that is to be hung i...
 5.3.5.1.285: Increasing a Book Size The dimensions of the cover of a book have b...
 5.3.5.1.286: Dense Set of Numbers A set of numbers is said to be a dense set if ...
 5.3.5.1.287: Dense Set of Numbers A set of numbers is said to be a dense set if ...
 5.3.5.1.288: Dense Set of Numbers A set of numbers is said to be a dense set if ...
 5.3.5.1.289: Dense Set of Numbers A set of numbers is said to be a dense set if ...
 5.3.5.1.290: Dense Set of Numbers A set of numbers is said to be a dense set if ...
 5.3.5.1.291: Dense Set of Numbers A set of numbers is said to be a dense set if ...
 5.3.5.1.292: Halfway Between Two Numbers To find a rational number halfway betwe...
 5.3.5.1.293: Halfway Between Two Numbers To find a rational number halfway betwe...
 5.3.5.1.294: Halfway Between Two Numbers To find a rational number halfway betwe...
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 5.3.5.1.296: Halfway Between Two Numbers To find a rational number halfway betwe...
 5.3.5.1.297: Halfway Between Two Numbers To find a rational number halfway betwe...
 5.3.5.1.298: Cooking Oatmeal Following are the instructions given on a box of oa...
 5.3.5.1.299: Consider the rational number 0.9. a) Use the method from Example 8 ...
 5.3.5.1.300: Paper Folding Read the Recreational Mathematics box on page 230. Ne...
 5.3.5.1.301: The ancient Greeks are often considered the first true mathematicia...
Solutions for Chapter 5.3: Number Theory and the Real Number System
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 5.3: Number Theory and the Real Number System
Get Full SolutionsThis textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9. Since 124 problems in chapter 5.3: Number Theory and the Real Number System have been answered, more than 150120 students have viewed full stepbystep solutions from this chapter. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. Chapter 5.3: Number Theory and the Real Number System includes 124 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

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).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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.

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·