 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
 5.3.5.1.191: In Exercises 1322, reduce each fraction to lowest terms. 15 20
 5.3.5.1.192: In Exercises 1322, reduce each fraction to lowest terms. 28 63
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 5.3.5.1.194: In Exercises 1322, reduce each fraction to lowest terms.95 125
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 5.3.5.1.196: In Exercises 1322, reduce each fraction to lowest terms.112 176
 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...
 5.3.5.1.199: In Exercises 2126, convert each mixed number to an improper fractio...
 5.3.5.1.200: In Exercises 2126, convert each mixed number to an improper fractio...
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 5.3.5.1.203: In Exercises 2126, convert each mixed number to an improper fractio...
 5.3.5.1.204: In Exercises 2730, write the number of inches indicated by the arro...
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 5.3.5.1.207: In Exercises 2730, write the number of inches indicated by the arro...
 5.3.5.1.208: In Exercises 3136, convert each improper fraction to a mixed number...
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 5.3.5.1.212: In Exercises 3136, convert each improper fraction to a mixed number...
 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...
 5.3.5.1.216: In Exercises 3744, express each rational number as terminating or r...
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 5.3.5.1.218: In Exercises 3744, express each rational number as terminating or r...
 5.3.5.1.219: In Exercises 3744, express each rational number as terminating or r...
 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...
 5.3.5.1.234: In Exercises 5360, express each repeating decimal number as a quoti...
 5.3.5.1.235: In Exercises 5360, express each repeating decimal number as a quoti...
 5.3.5.1.236: In Exercises 5360, express each repeating decimal number as a quoti...
 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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 5.3.5.1.247: In Exercises 6170, perform the indicated operation and reduce your ...
 5.3.5.1.248: In Exercises 7180, perform the indicated operation and reduce your ...
 5.3.5.1.249: In Exercises 7180, perform the indicated operation and reduce your ...
 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...
 5.3.5.1.272: In Exercises 93104, write an expression that will solve the problem...
 5.3.5.1.273: In Exercises 93104, write an expression that will solve the problem...
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 5.3.5.1.279: In Exercises 93104, write an expression that will solve the problem...
 5.3.5.1.280: In Exercises 93104, write an expression that will solve the problem...
 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...
 5.3.5.1.295: Halfway Between Two Numbers To find a rational number halfway betwe...
 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 74082 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.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

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.

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

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.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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.

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).