3.4 KiB
Page 284
Definition
If m and n are integers and m \leq n, the symbol \sum_{k=m}^{n}{a_k},
read the summation from k equals m to n of $a$-sub-$k$, is the sum of
all the terms a_m, a_{m + 1}, a_{m + 2}, \dots, a_n. We say that
a_m + a_{m + 1} + a_{m + 2} + \dots + a_n is the expanded form of the sum,
and we write
\sum_{k=m}^{n}{a_k} = a_m + a_{m + 1} + a_{m + 2} + \dots + a_n We call k the index of the summation, m the lower limit of the
summation, and n the upper limit of the summation.
Page 287
Definition
If m and n are integers and m \leq n, the symbol \prod_{k = m}^{n}{a_k}
read the product from k equals m to n of $a$-sub-$k$, is the product
of all the terms a_m, a_{m + 1}, a_{m + 2}, \dots, a_n.
We write
\prod_{k = m}^{n}{a_k} = a_m \cdot a_{m + 1} \cdot a_{m + 1} \dots a_n Page 288
Theorem 5.1.1
If a_m, a_{m + 1}, a_{m + 1}, \dots and b_m, b_{m + 1}, b_{m + 1}, \dots are
sequences of real numbers and c is any real number, then the following
equations hold for any integer n \geq m:
-
\sum_{k = m}^{n}{a_k} + \sum_{k = m}^{n}{b_k} = \sum_{k = m}^{n}{(a_k + b_k)} -
c \cdot \sum_{k = m}^{n}{a_k} = \sum_{k = m}^{n}{c \cdot a_k} \quad \text{generalized distributive law} -
\left(\prod_{k = m}^{n}{a_k}\right) \cdot \left(\prod_{k = m}^{n}{b_k}\right) = \prod_{k = m}^{n}{(a_k \cdot b_k)}
Page 291
Definition
For each positive integer n, the quantity n factorial denoted n!, is
defined to be the product of all the integers from 1 to n:
n! = n \cdot (n - 1) \dots 3 \cdot 2 \cdot 1 Zero factorial, denoted 0!, is defined to be 1:
0! = 1 Page 292
Definition
Let n and r be integers with 0 \leq r \leq n. The symbol
\binom{n}{r} is read "n choose $r$" and represents the number of subsets of size r
that can be chosen from a set with n elements.
Page 292
Formula for Computing $\dbinom{n}{r}$
For all integers n and r with 0 \leq r \leq n,
\binom{n}{r} = \frac{n!}{r!(n - r)!} Page 295
Algorithm 5.1.1 Decimal to Binary Conversion Using Repeated Division by $2$
[In Algorithm 5.1.1 the input is a nonnegative integer a. The aim of the
algorithm is to produce a sequence of binary digits $r[0], r[1], r[2], \dots
r[k] so that the binary representation of n is
\left(r[k]r[k - 1] \dots r[2]r[1]r[0]\right)_2 That is,
a = 2^k \cdot r[k] + 2^{k - 1} \cdot r[k - 1] + \dots + 2^3 \cdot r[2] + 2^1 \cdot r[1] + 2^0 \cdot r[0] .]
Input: a [a nonegative integer]
Algorithm Body:
q := a, i := 0
[Repeatedly perform the integer division of q by 2 until q becomes 0.
Store successive remainders in a one-dimensional array
r[0], r[1], r[2], \dots r[k]. Even if the initial-value of q equals 0, the
loop should execute one time (so that r[0] is computed). Thus the guard
condition for the while loop is i = 0 or q \neq 0.]
\text{\textbf{while }}(i = 0 \text{ or } q \neq 0)\\ \ \ r[i] := q \mod 2\\ \ \ q := q \text{ div } 2\\ \ \ \text{[r[i] and q can be obtained by calling the division algorithm.]}\\ \ \ i := i + 1\\ \text{\textbf{end while}}
[After execution of this step, the values of r[0], r[1], \dots, r[i - 1] are
all $0$'s and $1$'s, and
a = \left(r[i - 1]r[i - 2] \dots r[2]r[1]r[0]\right)_2.]
Output: r[0], r[1], r[2], \dots, r[i - 1] [a sequence of integers]