 1.3.1.1.281: In Exercises 18, write a positive or negative integer that describe...
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 1.3.1.1.289: In Exercises 920, start by drawing a number line that shows integer...
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 1.3.1.1.301: In Exercises 2132, express each rational number as a decimal 3 4
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 1.3.1.1.312: In Exercises 2132, express each rational number as a decimal 7 6
 1.3.1.1.313: In Exercises 3336, list all numbers from the given set that are: a....
 1.3.1.1.314: In Exercises 3336, list all numbers from the given set that are: a....
 1.3.1.1.1107: In Exercises 3536, graph each real number on a number line. 2.5
 1.3.1.1.315: In Exercises 3336, list all numbers from the given set that are: a....
 1.3.1.1.1108: In Exercises 3536, graph each real number on a number line. 4 3 4
 1.3.1.1.316: In Exercises 3336, list all numbers from the given set that are: a....
 1.3.1.1.1109: In Exercises 3738, express each rational number as a decimal. 5 8
 1.3.1.1.317: Give an example of a whole number that is not a natural number.
 1.3.1.1.1110: In Exercises 3738, express each rational number as a decimal. 3 11
 1.3.1.1.318: Give an example of an integer that is not a whole number.
 1.3.1.1.1111: Consider the set 17, 9 13 , 0, 0.75, 2, p, 81 . List all numbers fr...
 1.3.1.1.319: Give an example of a rational number that is not an integer.
 1.3.1.1.1112: Give an example of an integer that is not a natural number.
 1.3.1.1.320: Give an example of a rational number that is not a natural number.
 1.3.1.1.1113: Give an example of a rational number that is not an integer.
 1.3.1.1.321: Give an example of a number that is an integer, a whole number, and...
 1.3.1.1.1114: Give an example of a real number that is not a rational number.
 1.3.1.1.322: Give an example of a number that is a rational number, an integer, ...
 1.3.1.1.1115: In Exercises 4346, insert either or in the shaded area between each...
 1.3.1.1.323: Give an example of a number that is an irrational number and a real...
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 1.3.1.1.324: Give an example of a number that is a real number, but not an irrat...
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 1.3.1.1.1121: In Exercises 4950, find each absolute value. 58
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 1.3.1.1.351: In Exercises 7178, find each absolute value. 6
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 1.3.1.1.359: In Exercises 7986, insert either , , or in the shaded area to make ...
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 1.3.1.1.375: The table shows the record low temperatures for five U.S. states. S...
 1.3.1.1.376: The table shows the record low temperatures for five U.S. states. S...
 1.3.1.1.377: What is a set?
 1.3.1.1.378: What are the natural numbers?
 1.3.1.1.379: What are the whole numbers?
 1.3.1.1.380: What are the integers?
 1.3.1.1.381: How does the set of integers differ from the set of whole numbers?
 1.3.1.1.382: Describe how to graph a number on the number line.
 1.3.1.1.383: What is a rational number?
 1.3.1.1.384: Explain how to express 38 as a decimal.
 1.3.1.1.385: Describe the difference between a rational number and an irrational...
 1.3.1.1.386: If you are given two different real numbers, explain how to determi...
 1.3.1.1.387: Describe what is meant by the absolute value of a number. Give an e...
 1.3.1.1.388: In Exercises 108111, determine whether each statement makes sense o...
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 1.3.1.1.400: In Exercises 120123, use a calculator to find a decimal approximati...
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 1.3.1.1.404: Exercises 124126 will help you prepare for the material covered in ...
 1.3.1.1.405: Exercises 124126 will help you prepare for the material covered in ...
 1.3.1.1.406: Exercises 124126 will help you prepare for the material covered in ...
Solutions for Chapter 1.3: The Real Numbers
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 1.3: The Real Numbers
Get Full SolutionsChapter 1.3: The Real Numbers includes 142 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 142 problems in chapter 1.3: The Real Numbers have been answered, more than 90689 students have viewed full stepbystep solutions from this chapter. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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

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

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

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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.

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

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

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.