 11.1: Write the words used to read the number.22,356,027
 11.2: Write the words used to read the number.106,357,291,582
 11.3: Write the words used to read the number.730,531,968
 11.4: Write the words used to read the number.21,000,017
 11.5: Write the words used to read the number.523,800,007,190
 11.6: Write the words used to read the number.713,205,538
 11.7: Fourteen thousand, nine hundred eightyfive.
 11.8: Thirtytwo million, nine hundred fortythree thousand, six hundred ...
 11.9: Seventeen billion, eight hundred three thousand, seventyfive.
 11.10: Fifty million, six hundred twelve thousand, seventyeight.
 11.11: Three hundred six thousand, five hundred fortyone.
 11.12: Three hundred million, seven hundred sixty thousand, five hundred t...
 11.13: Round 483 to tens.
 11.14: Round 3,762 to hundreds.
 11.15: Round 298,596 to tenthousands.
 11.16: Round 57,802 to the first digit.
 11.17: Cisco, the worlds largest Internet equipment maker, recorded earnin...
 11.18: Net income at Levi Strauss, the worlds biggest maker of branded clo...
 11.19: McDonalds produced 86,347,582 Big Macs. How many Big Macs were prod...
 11.20: Oslo, Hong Kong, Tokyo, and New York City are the four most expensi...
 11.21: According to Experian, a creditreporting agency, the average debt ...
 11.22: Experian reported that the average debt for people living in Missis...
 11.23: Experian reported that people in the age range of 18 to 29 had the ...
 11.24: Experian recently reported that people in the 50 to 69 age range ca...
Solutions for Chapter 11: PLACE VALUE AND OUR NUMBER SYSTEM
Full solutions for Business Math,  9th Edition
ISBN: 9780135108178
Solutions for Chapter 11: PLACE VALUE AND OUR NUMBER SYSTEM
Get Full SolutionsChapter 11: PLACE VALUE AND OUR NUMBER SYSTEM includes 24 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Business Math, , edition: 9. Since 24 problems in chapter 11: PLACE VALUE AND OUR NUMBER SYSTEM have been answered, more than 19360 students have viewed full stepbystep solutions from this chapter. Business Math, was written by and is associated to the ISBN: 9780135108178.

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

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.

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.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

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 .

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

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.