 5.8.5.1.691: The sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, p is known as the...
 5.8.5.1.692: The irrational number 25 + 1 2 is known as the number.
 5.8.5.1.693: In Exercises 3 4, use the following diagram and assume that the rat...
 5.8.5.1.694: In Exercises 3 4, use the following diagram and assume that the rat...
 5.8.5.1.695: In medieval times, people referred to the golden proportion as the ...
 5.8.5.1.696: A rectangle whose ratio of its length to its width is equal to the ...
 5.8.5.1.697: a) To what decimal value is (15 + 1)/2 approximately equal? b) To w...
 5.8.5.1.698: The sum of any six consecutive Fibonacci numbers is always divisibl...
 5.8.5.1.699: The sum of any 10 consecutive Fibonacci numbers is always divisible...
 5.8.5.1.700: The greatest common factor of any two consecutive Fibonacci numbers...
 5.8.5.1.701: For any four consecutive Fibonacci numbers, the difference of the s...
 5.8.5.1.702: Twice any Fibonacci number minus the next Fibonacci number equals t...
 5.8.5.1.703: Determine the ratio of the length to width of various photographs a...
 5.8.5.1.704: Determine the ratio of the length to the width of your television s...
 5.8.5.1.705: Determine the ratio of the length to the width of a computer screen...
 5.8.5.1.706: Find three physical objects whose dimensions are very close to a go...
 5.8.5.1.707: In Exercises 1724, determine whether the sequence is a Fibonaccity...
 5.8.5.1.708: In Exercises 1724, determine whether the sequence is a Fibonaccity...
 5.8.5.1.709: In Exercises 1724, determine whether the sequence is a Fibonaccity...
 5.8.5.1.710: In Exercises 1724, determine whether the sequence is a Fibonaccity...
 5.8.5.1.711: In Exercises 1724, determine whether the sequence is a Fibonaccity...
 5.8.5.1.712: In Exercises 1724, determine whether the sequence is a Fibonaccity...
 5.8.5.1.713: In Exercises 1724, determine whether the sequence is a Fibonaccity...
 5.8.5.1.714: In Exercises 1724, determine whether the sequence is a Fibonaccity...
 5.8.5.1.715: a) Select any two onedigit (nonzero) numbers and add them to obtai...
 5.8.5.1.716: a) Select any three consecutive terms of a Fibonacci sequence. Subt...
 5.8.5.1.717: Pascals Triangle One of the most famous number patterns involves Pa...
 5.8.5.1.718: Lucas Sequence a) A sequence related to the Fibonacci sequence is t...
 5.8.5.1.719: Explain how to construct the Fibonacci sequence.
 5.8.5.1.720: Explain the relationship between the golden number, golden ratio, g...
 5.8.5.1.721: Describe three examples of where Fibonacci numbers can be found a) ...
 5.8.5.1.722: The eleventh Fibonacci number is 89. Examine the first six digits i...
 5.8.5.1.723: Find the ratio of the second to the first term of the Fibonacci seq...
 5.8.5.1.724: A musical composition is described as follows. Explain why this pie...
 5.8.5.1.725: FibonacciType Sequence The following sequence represents a Fibonac...
 5.8.5.1.726: Draw a line of length 5 in. Determine and mark the point on the lin...
 5.8.5.1.727: The divine proportion is (a + b)>a = a/b (see Fig. 5.13), which can...
 5.8.5.1.728: Pythagorean Triples A Pythagorean triple is a set of three whole nu...
 5.8.5.1.729: Reflections When two panes of glass are placed face to face, four i...
 5.8.5.1.730: Write a report on the history and mathematical contributions of Fib...
 5.8.5.1.731: The digits 1 through 9 have evolved considerably since they appeare...
 5.8.5.1.732: Write a report indicating where the golden ratio and golden rectang...
Solutions for Chapter 5.8: Number Theory and the Real Number System
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 5.8: Number Theory and the Real Number System
Get Full SolutionsChapter 5.8: Number Theory and the Real Number System includes 42 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. Since 42 problems in chapter 5.8: Number Theory and the Real Number System have been answered, more than 80163 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.

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.)

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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].

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 .

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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 picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).