 2.4.2.4.1: Match the type of complex number with its definition. (a) real numb...
 2.4.2.4.2: The imaginary unit is defined as _______ , where _______ .
 2.4.2.4.3: The set of real multiples of the imaginary unit combined with the s...
 2.4.2.4.4: Complex numbers can be plotted in the complex plane, where the hori...
 2.4.2.4.5: Complex numbers can be plotted in the complex plane, where the hori...
 2.4.2.4.6: In Exercises 514, write the complex number in standard form.2 9
 2.4.2.4.7: In Exercises 514, write the complex number in standard form.6
 2.4.2.4.8: In Exercises 514, write the complex number in standard form.8
 2.4.2.4.9: In Exercises 514, write the complex number in standard form.5i i 2
 2.4.2.4.10: In Exercises 514, write the complex number in standard form.3i 5i i...
 2.4.2.4.11: In Exercises 514, write the complex number in standard form.75
 2.4.2.4.12: In Exercises 514, write the complex number in standard form.4 2 75 7 2
 2.4.2.4.13: In Exercises 514, write the complex number in standard form.0.09
 2.4.2.4.14: In Exercises 514, write the complex number in standard form.0.0004
 2.4.2.4.15: In Exercises 1524, perform the addition or subtraction and write th...
 2.4.2.4.16: In Exercises 1524, perform the addition or subtraction and write th...
 2.4.2.4.17: In Exercises 1524, perform the addition or subtraction and write th...
 2.4.2.4.18: In Exercises 1524, perform the addition or subtraction and write th...
 2.4.2.4.19: In Exercises 1524, perform the addition or subtraction and write th...
 2.4.2.4.20: In Exercises 1524, perform the addition or subtraction and write th...
 2.4.2.4.21: In Exercises 1524, perform the addition or subtraction and write th...
 2.4.2.4.22: In Exercises 1524, perform the addition or subtraction and write th...
 2.4.2.4.23: In Exercises 1524, perform the addition or subtraction and write th...
 2.4.2.4.24: In Exercises 1524, perform the addition or subtraction and write th...
 2.4.2.4.25: In Exercises 25 36, perform the operation and write the result in s...
 2.4.2.4.26: In Exercises 25 36, perform the operation and write the result in s...
 2.4.2.4.27: In Exercises 25 36, perform the operation and write the result in s...
 2.4.2.4.28: In Exercises 25 36, perform the operation and write the result in s...
 2.4.2.4.29: In Exercises 25 36, perform the operation and write the result in s...
 2.4.2.4.30: In Exercises 25 36, perform the operation and write the result in s...
 2.4.2.4.31: In Exercises 25 36, perform the operation and write the result in s...
 2.4.2.4.32: In Exercises 25 36, perform the operation and write the result in s...
 2.4.2.4.33: In Exercises 25 36, perform the operation and write the result in s...
 2.4.2.4.34: In Exercises 25 36, perform the operation and write the result in s...
 2.4.2.4.35: In Exercises 25 36, perform the operation and write the result in s...
 2.4.2.4.36: In Exercises 25 36, perform the operation and write the result in s...
 2.4.2.4.37: In Exercises 37 44, write the complex conjugate of the complex numb...
 2.4.2.4.38: In Exercises 37 44, write the complex conjugate of the complex numb...
 2.4.2.4.39: In Exercises 37 44, write the complex conjugate of the complex numb...
 2.4.2.4.40: In Exercises 37 44, write the complex conjugate of the complex numb...
 2.4.2.4.41: In Exercises 37 44, write the complex conjugate of the complex numb...
 2.4.2.4.42: In Exercises 37 44, write the complex conjugate of the complex numb...
 2.4.2.4.43: In Exercises 37 44, write the complex conjugate of the complex numb...
 2.4.2.4.44: In Exercises 37 44, write the complex conjugate of the complex numb...
 2.4.2.4.45: In Exercises 45 52, write the quotient in standard form.
 2.4.2.4.46: In Exercises 45 52, write the quotient in standard form.
 2.4.2.4.47: In Exercises 45 52, write the quotient in standard form.
 2.4.2.4.48: In Exercises 45 52, write the quotient in standard form.
 2.4.2.4.49: In Exercises 45 52, write the quotient in standard form.
 2.4.2.4.50: In Exercises 45 52, write the quotient in standard form.
 2.4.2.4.51: In Exercises 45 52, write the quotient in standard form.
 2.4.2.4.52: In Exercises 45 52, write the quotient in standard form.
 2.4.2.4.53: In Exercises 5356, perform the operation and write the result in st...
 2.4.2.4.54: In Exercises 5356, perform the operation and write the result in st...
 2.4.2.4.55: In Exercises 5356, perform the operation and write the result in st...
 2.4.2.4.56: In Exercises 5356, perform the operation and write the result in st...
 2.4.2.4.57: In Exercises 57 62, simplify the complex number and write it in sta...
 2.4.2.4.58: In Exercises 57 62, simplify the complex number and write it in sta...
 2.4.2.4.59: In Exercises 57 62, simplify the complex number and write it in sta...
 2.4.2.4.60: In Exercises 57 62, simplify the complex number and write it in sta...
 2.4.2.4.61: In Exercises 57 62, simplify the complex number and write it in sta...
 2.4.2.4.62: In Exercises 57 62, simplify the complex number and write it in sta...
 2.4.2.4.63: Cube each complex number. What do you notice? (a) 2 (b)1 3i 1 (c) 1...
 2.4.2.4.64: Raise each complex number to the fourth power and simplify. (a) 2 (...
 2.4.2.4.65: In Exercises 6570, determine the complex number shown in the comple...
 2.4.2.4.66: In Exercises 6570, determine the complex number shown in the comple...
 2.4.2.4.67: In Exercises 6570, determine the complex number shown in the comple...
 2.4.2.4.68: In Exercises 6570, determine the complex number shown in the comple...
 2.4.2.4.69: In Exercises 6570, determine the complex number shown in the comple...
 2.4.2.4.70: In Exercises 6570, determine the complex number shown in the comple...
 2.4.2.4.71: In Exercises 7176, plot the complex number in the complex plane.
 2.4.2.4.72: In Exercises 7176, plot the complex number in the complex plane.
 2.4.2.4.73: In Exercises 7176, plot the complex number in the complex plane.
 2.4.2.4.74: In Exercises 7176, plot the complex number in the complex plane.
 2.4.2.4.75: In Exercises 7176, plot the complex number in the complex plane.
 2.4.2.4.76: In Exercises 7176, plot the complex number in the complex plane.
 2.4.2.4.77: Fractals In Exercises 77 and 78, find the first six terms of the se...
 2.4.2.4.78: Fractals In Exercises 77 and 78, find the first six terms of the se...
 2.4.2.4.79: mpedance In Exercises 79 and 80, use the following information. The...
 2.4.2.4.80: mpedance In Exercises 79 and 80, use the following information. The...
 2.4.2.4.81: True or False? In Exercises 81 86, determine whether the statement ...
 2.4.2.4.82: True or False? In Exercises 81 86, determine whether the statement ...
 2.4.2.4.83: True or False? In Exercises 81 86, determine whether the statement ...
 2.4.2.4.84: True or False? In Exercises 81 86, determine whether the statement ...
 2.4.2.4.85: True or False? In Exercises 81 86, determine whether the statement ...
 2.4.2.4.86: True or False? In Exercises 81 86, determine whether the statement ...
 2.4.2.4.87: In Exercises 87 90, perform the operation and write the result in s...
 2.4.2.4.88: In Exercises 87 90, perform the operation and write the result in s...
 2.4.2.4.89: In Exercises 87 90, perform the operation and write the result in s...
 2.4.2.4.90: In Exercises 87 90, perform the operation and write the result in s...
Solutions for Chapter 2.4: Complex Numbers
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 2.4: Complex Numbers
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Since 90 problems in chapter 2.4: Complex Numbers have been answered, more than 32478 students have viewed full stepbystep solutions from this chapter. Chapter 2.4: Complex Numbers includes 90 full stepbystep solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(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 picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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