8 Chapter 1Numbers, variables, and units
cancellation of the common factor 6 is equivalent to dividing both numerator and
denominator by 6, and such an operation does not change the value of the fraction.
EXAMPLES 1.7Simplification of fractions
(1)
(2)
(3)
0 Exercises 26–
In general, the purpose of factorization is to express a quantity in terms of simpler
quantities (see Section 2.3 for factorization of algebraic expressions).
Factorials
The factorial of nis the number whose factors are the first nnatural numbers:
n! 1 = 111 × 121 × 131 ×1-1× 1 n
(1.12)
= 1 n 1 × 1 (n 1 − 1 1) 1 × 1 (n 1 − 1 2) 1 ×1-1× 121 × 11
(read as ‘n factorial’). Consecutive factorials are related by the recurrence relation
(n 1 + 1 1)! 1 = 1 (n 1 + 1 1) 1 × 1 n!
for example, 3! 1 = 131 × 121 × 111 = 16 and 4! 1 = 141 × 131 × 121 × 111 = 141 × 13! 1 = 124. In addition, the
factorial of zero is defined as 0! 1 = 1 1.
EXAMPLES 1.8Factorials
(1) 1! 1 = 111 × 1 0! 1 = 111 × 111 = 11
(2) 5! 1 = 151 × 1 4! 1 = 151 × 141 × 1 3! 1 = 151 × 141 × 131 × 1 2! 1 = 151 × 141 × 131 × 121 = 1120
(3)
(4)
0 Exercises 30–
7
34
7654
324
7535
!
!!
!
!
=
×××
××
=×=
5
3
543
3
54 20
!
!
!
!
=
××
=×=
105
1470
357
2357
1
27
1
14
2=
××
×××
=
×
=
15
25
35
5
3
5
2=
×
=
4
24
2
23
22
2223
1
6
23=
×
=
×
×××
=