 2.5.1: Represent the decimal integers in 16 in binary notation.
 2.5.2: Represent the decimal integers in 16 in binary notation.
 2.5.3: Represent the decimal integers in 16 in binary notation.
 2.5.4: Represent the decimal integers in 16 in binary notation.
 2.5.5: Represent the decimal integers in 16 in binary notation.
 2.5.6: Represent the decimal integers in 16 in binary notation.
 2.5.7: Represent the integers in 712 in decimal notation.
 2.5.8: Represent the integers in 712 in decimal notation.
 2.5.9: Represent the integers in 712 in decimal notation.
 2.5.10: Represent the integers in 712 in decimal notation.
 2.5.11: Represent the integers in 712 in decimal notation.
 2.5.12: Represent the integers in 712 in decimal notation.
 2.5.13: Perform the arithmetic in 1320 using binary notation.
 2.5.14: Perform the arithmetic in 1320 using binary notation.
 2.5.15: Perform the arithmetic in 1320 using binary notation.
 2.5.16: Perform the arithmetic in 1320 using binary notation.
 2.5.17: Perform the arithmetic in 1320 using binary notation.
 2.5.18: Perform the arithmetic in 1320 using binary notation.
 2.5.19: Perform the arithmetic in 1320 using binary notation.
 2.5.20: Perform the arithmetic in 1320 using binary notation.
 2.5.21: Give the output signals S and T for the circuit in the right column...
 2.5.22: Add 111111112 + 12 and convert the result to decimal notation, to v...
 2.5.23: Find the 8bit twos complements for the integers in 2326.
 2.5.24: Find the 8bit twos complements for the integers in 2326.
 2.5.25: Find the 8bit twos complements for the integers in 2326.
 2.5.26: Find the 8bit twos complements for the integers in 2326.
 2.5.27: Find the decimal representations for the integers with the 8bit re...
 2.5.28: Find the decimal representations for the integers with the 8bit re...
 2.5.29: Find the decimal representations for the integers with the 8bit re...
 2.5.30: Find the decimal representations for the integers with the 8bit re...
 2.5.31: Use 8bit representations to compute the sums in 3136.
 2.5.32: Use 8bit representations to compute the sums in 3136.
 2.5.33: Use 8bit representations to compute the sums in 3136.
 2.5.34: Use 8bit representations to compute the sums in 3136.
 2.5.35: Use 8bit representations to compute the sums in 3136.
 2.5.36: Use 8bit representations to compute the sums in 3136.
 2.5.37: Show that if a, b, and a + b are integers in the range 1 through 12...
 2.5.38: Convert the integers in 3840 from hexadecimal to decimal notation.
 2.5.39: Convert the integers in 3840 from hexadecimal to decimal notation.
 2.5.40: Convert the integers in 3840 from hexadecimal to decimal notation.
 2.5.41: Convert the integers in 4143 from hexadecimal to binary notation
 2.5.42: Convert the integers in 4143 from hexadecimal to binary notation
 2.5.43: Convert the integers in 4143 from hexadecimal to binary notation
 2.5.44: Convert the integers in 4446 from binary to hexadecimal notation
 2.5.45: Convert the integers in 4446 from binary to hexadecimal notation
 2.5.46: Convert the integers in 4446 from binary to hexadecimal notation
 2.5.47: In addition to binary and hexadecimal, computer scientists also use...
Solutions for Chapter 2.5: Application: Number Systems and Circuits for Addition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 2.5: Application: Number Systems and Circuits for Addition
Get Full SolutionsChapter 2.5: Application: Number Systems and Circuits for Addition includes 47 full stepbystep solutions. Since 47 problems in chapter 2.5: Application: Number Systems and Circuits for Addition have been answered, more than 56301 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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.

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

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.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.