Section 9: Subscripts, Summation, and Factorials

9.1 Subscript and Summation Notation

Subscripts are small "index" numbers on the lower right side of a variable to distingiuish a particular item from a group of similar items. For example, we discussed (x, y) orderd pairs under graphing linear equations. If I wanted to distoinguish the first or-fdered part from the second, I could use subscripts: (x1,y1) and (x2, y2). When physicists want to track velocity of an item at differnt points in time, they often use subscripted variables like v0, v1, v2, v3, v4 for time periods 0 through 4. Then v is the general velocity variable and the subscript is used to distiguich the occurrence for each specific time period.

Subscript notation is also useful for functions with multiple inputs. We can use x as a general label for the input. Then if there are, for example, three inputs we may use x1, x2, and x3 to distiguish the first, second, and third input variable.

Summation notation is cloerly realted to subscript notation. A summation means to add up a set of numbers. If we wanted to know how many times a cuckoo clock "cucked" each day, we would add 1 + 2 + 3 + ... + 12. This is a summation. The summation notation for this would be If we want to sum up a particular set of numbers we would use similar notation. Suppose the set of x's is (x1, x2, x3, x4, x5, x6) = (1, 8, 2, 5, 7, 4). Then Summation notation also occurs when summing formulas for functions. Using the same set of numbers..

Example 39 To sum all the x's squared Example 40 To sum all the x's plus 3 Of course this is a relatively small set of numbers. When there are many numbers in a set, summation notation provides a useful shorthand. In that situation a computer would probably do the arithmetic for us, but the notation tells us what is going on. To sum a set of unknown size (we say size n) we write In the situation where the entire set is summed, some books simplify the notation to 9.2 Factorials

A factorial is a special mathematical function for non-negative integers (also know as the whole numbers). The notation is an exclamation point, !, and it means that we want to take the integer and multiply it by every integer between it and 1.

Definition 16

0! = 1
1! = 1
n! = n(n-1)(n-2)... (3)(2)(1) for n > 2

You can imagine that these get very big for relatively small numbers. For example 5! = 5•4•3•2•1 = 120, 6! = 6•5•4•3•2•1 = 720, and 10! = 3,628,800. Some calculators have ! keys. Most of them can handle numbers somewhere in the range of 69! = 1.71 x 1098 before overflowing their memories.

However, you don't always need a ! key on your calculator to do factorial problems. Factorial problems in probability and statistics usually have several factorials in them and large parts of them cancel out. Notice above that 6! is simply 6(5!) and 5! is 5(4!). If my problem were 6!/4! I could calculate both the numerator and denominator before doing the division and get 720/24 = 30. On the other hand, it might be easier to re-write it (6•5•4!)/4!, cancel out the 4! terms, and end up with 6•5 = 30. Look at the following examples:

Example 41 