 1.4.1: In Exercises 14, use an integer to express each number in boldface ...
 1.4.2: In Exercises 14, use an integer to express each number in boldface ...
 1.4.3: In Exercises 14, use an integer to express each number in boldface ...
 1.4.4: In Exercises 14, use an integer to express each number in boldface ...
 1.4.5: The number of bachelors degrees in computer and information science...
 1.4.6: Between 2006 and 2007, print advertising revenue in the United Stat...
 1.4.7: On Tuesday, August 18, 2009, the Dow JonesIndustrial Average (DJIA)...
 1.4.8: On Monday, August 17, 2009, the NASDAQclosed at 1930.84. On the pre...
 1.4.9: Concept Check In Exercises 914, give a number that satisfies the gi...
 1.4.10: Concept Check In Exercises 914, give a number that satisfies the gi...
 1.4.11: Concept Check In Exercises 914, give a number that satisfies the gi...
 1.4.12: Concept Check In Exercises 914, give a number that satisfies the gi...
 1.4.13: Concept Check In Exercises 914, give a number that satisfies the gi...
 1.4.14: Concept Check In Exercises 914, give a number that satisfies the gi...
 1.4.15: Concept Check In Exercises 1520, decide whether each statement is t...
 1.4.16: Concept Check In Exercises 1520, decide whether each statement is t...
 1.4.17: Concept Check In Exercises 1520, decide whether each statement is t...
 1.4.18: Concept Check In Exercises 1520, decide whether each statement is t...
 1.4.19: Concept Check In Exercises 1520, decide whether each statement is t...
 1.4.20: Concept Check In Exercises 1520, decide whether each statement is t...
 1.4.21: Concept Check Give three numbers between and 6 that satisfy each gi...
 1.4.22: Concept Check Give three numbers between and 6 that satisfy each gi...
 1.4.23: Concept Check Give three numbers between and 6 that satisfy each gi...
 1.4.24: Concept Check Give three numbers between and 6 that satisfy each gi...
 1.4.25: Concept Check Give three numbers between and 6 that satisfy each gi...
 1.4.26: Concept Check Give three numbers between and 6 that satisfy each gi...
 1.4.27: For Exercises 27 and 28, see Example 2. List all numbers from each ...
 1.4.28: For Exercises 27 and 28, see Example 2. List all numbers from each ...
 1.4.29: Graph each group of numbers on a number line. See FIGURE 4 and FIGU...
 1.4.30: Graph each group of numbers on a number line. See FIGURE 4 and FIGU...
 1.4.31: Graph each group of numbers on a number line. See FIGURE 4 and FIGU...
 1.4.32: Graph each group of numbers on a number line. See FIGURE 4 and FIGU...
 1.4.33: Graph each group of numbers on a number line. See FIGURE 4 and FIGU...
 1.4.34: Graph each group of numbers on a number line. See FIGURE 4 and FIGU...
 1.4.35: Concept Check Match each expression in Column I with its value in C...
 1.4.36: Concept Check Fill in the blanks with the correct values: The oppos...
 1.4.37: Find (a) the opposite (or additive inverse) of each number and (b) ...
 1.4.38: Find (a) the opposite (or additive inverse) of each number and (b) ...
 1.4.39: Find (a) the opposite (or additive inverse) of each number and (b) ...
 1.4.40: Find (a) the opposite (or additive inverse) of each number and (b) ...
 1.4.41: Find (a) the opposite (or additive inverse) of each number and (b) ...
 1.4.42: Find (a) the opposite (or additive inverse) of each number and (b) ...
 1.4.43: Simplify by finding the absolute value. See Example 4. 6 
 1.4.44: Simplify by finding the absolute value. See Example 4. 14 
 1.4.45: Simplify by finding the absolute value. See Example 4. 12 
 1.4.46: Simplify by finding the absolute value. See Example 4. 19 
 1.4.47: Simplify by finding the absolute value. See Example 4. `  ` 23 `
 1.4.48: Simplify by finding the absolute value. See Example 4.  45
 1.4.49: Simplify by finding the absolute value. See Example 4. 6  3  
 1.4.50: Simplify by finding the absolute value. See Example 4. 6  3 
 1.4.51: Students often say Absolute value is always positive. Is this true?...
 1.4.52: Concept Check True or false: If a is negative, then  a  = a.
 1.4.53: Select the lesser of the two given numbers. See Examples 3 and 4.1...
 1.4.54: Select the lesser of the two given numbers. See Examples 3 and 4.8...
 1.4.55: Select the lesser of the two given numbers. See Examples 3 and 4.7...
 1.4.56: Select the lesser of the two given numbers. See Examples 3 and 4.1...
 1.4.57: Select the lesser of the two given numbers. See Examples 3 and 4.4,...
 1.4.58: Select the lesser of the two given numbers. See Examples 3 and 4.4,...
 1.4.59: Select the lesser of the two given numbers. See Examples 3 and 4.3...
 1.4.60: Select the lesser of the two given numbers. See Examples 3 and 4. ...
 1.4.61: Select the lesser of the two given numbers. See Examples 3 and 4....
 1.4.62: Select the lesser of the two given numbers. See Examples 3 and 4....
 1.4.63: Select the lesser of the two given numbers. See Examples 3 and 4. ...
 1.4.64: Select the lesser of the two given numbers. See Examples 3 and 4. 7...
 1.4.65: Decide whether each statement is true or false. See Examples 3 and ...
 1.4.66: Decide whether each statement is true or false. See Examples 3 and ...
 1.4.67: Decide whether each statement is true or false. See Examples 3 and ...
 1.4.68: Decide whether each statement is true or false. See Examples 3 and ...
 1.4.69: Decide whether each statement is true or false. See Examples 3 and ...
 1.4.70: Decide whether each statement is true or false. See Examples 3 and ...
 1.4.71: Decide whether each statement is true or false. See Examples 3 and ...
 1.4.72: Decide whether each statement is true or false. See Examples 3 and ...
 1.4.73: Decide whether each statement is true or false. See Examples 3 and ...
 1.4.74: Decide whether each statement is true or false. See Examples 3 and ...
 1.4.75: Decide whether each statement is true or false. See Examples 3 and ...
 1.4.76: Decide whether each statement is true or false. See Examples 3 and ...
 1.4.77: The table shows the percent change in the Consumer Price Index (CPI...
 1.4.78: The table shows the percent change in the Consumer Price Index (CPI...
 1.4.79: The table shows the percent change in the Consumer Price Index (CPI...
 1.4.80: The table shows the percent change in the Consumer Price Index (CPI...
Solutions for Chapter 1.4: Real Numbers and the Number Line
Full solutions for Beginning Algebra  11th Edition
ISBN: 9780321673480
Solutions for Chapter 1.4: Real Numbers and the Number Line
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.4: Real Numbers and the Number Line includes 80 full stepbystep solutions. Beginning Algebra was written by and is associated to the ISBN: 9780321673480. Since 80 problems in chapter 1.4: Real Numbers and the Number Line have been answered, more than 36435 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Beginning Algebra, edition: 11.

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.

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!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

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).

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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.

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.

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

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.

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

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.

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