An exponent is another neat notation. Suppose I want to multiply an expression by itself (called “squaring” the expression):
(j + 7)0 + 7) = (j + 7)2.
The little “2” placed high up in the upper right means “square the expression” or more directly, “write the expression down twice, making it clear that you mean multiplication.” This is also sometimes called “raising the expression to the power of 2. ”
Similarly, I can “cube” the expression
(j + 7)0 + 7)0 + 7) = (j + 7)3
and so on.
In general, (anything)" is called raising the expression “anything” to the nth power.
The following discussion of exponents is not needed in order to understand the book; I just thought that some readers might be curious as to why raising an expression to the power of 2 is called “squa... Read More
Very often, a bank or loan company will charge some sort of loan initiation fee at the outset of the loan (I call these up-front costs). Suppose, still using Table 2.2 as our example, that the bank wants $100 up front for setting up the loan. One way to handle this would be for the bank to just give you $9,900 while recording the loan as $10,000; then Table 2.2 would still be correct.
Usually, however, this is not what happens. When you take a $10,000 loan, you probably want to walk out of the bank with $10,000 for whatever your purpose is. In this case, the bank adds the extra $100 into the loan. That is, it pretends that it really loaned you $10,100. Columns 2 and 3 of Table 2.4 shows the new balance worksheet for this loan... Read More
This topic is not necessary for understanding the rest of the book. It’s an introduction, however, to a powerful notation that allows us to work with numbers from arbitrarily large lists in a very concise manner.
Looking at Table 1.3 again, suppose that I want to add up the values of e from e5 to e8. That is, add up e5 + e6 + e7 + e8. Just for the record, I don’t know why I’d ever want to do this with a Life Table. There are cases, however, such as adding up the interest payments on a loan, where I often want to do this.
The shorthand notation involves the use of the uppercase Greek letter sigma:
How to read this: Beneath the sigma you see “i = 5.” Above the sigma you see “8... Read More
When looking at relationships between variables, the formula tells it all. Very often, however, a picture is indeed worth a thousand words in “giving us a feeling” for what the formula is telling us.
We will often be presented with a graph that we’ll study to gain some insight into the information the graph is presenting. Conversely, we will often need to be able to create a graph to show a formula that we are interested in. I’ll take this latter approach first.
Let’s start with a simple formula:
у = 27,000 – 2,000x.
This formula gives us a value for the variable у when we give it a value for the variable x. These variables might stand for the depreciation of a car’s value, the interest on a loan, the number of years that you will hold a loan, and so on.
Before I draw a graph... Read More
We frequently don ’ t need to know an answer to many decimal places. When we give someone directions to drive to our house, we usually say something like, “Get off the highway at exit 14, go right and follow the road for about 12mi until you see an old church on the right.”
We could have said “Follow the road for 11.87 mi” but “about 12” gives enough information to tell someone when to start looking for the church. I don’t need to delve into the theory of approximations. Instead, I’ll use some commonsense rules, such as “about 14 mi” means that the number is closer to 14 than it is to 13 or 15.
The mathematical expression
means that “x is approximately equal to 14.” Other ways of writing this are x ~ 14 and x = 14.
The number 2,123,774 has seven significant figures... Read More
This section is useful for understanding the mathematics of the average and incremental IRS income tax rates discussed in Chapter 9 ’ It is not necessary for understanding Chapter 9 or anything else in this book.
Suppose I were to take a walk for 25 minutes. I’m walking along a marked track, so I know exactly how far I’ve walked at all times. Every few seconds, I write down how long I’ve been walking and how far I’ve walked. After I’ve finished my walk, I produce the graph shown in Figure 1.7 ’ interpolating my data as described above.
At the end of 25 minutes, I’ve walked about 2,200 feet. As the graph shows, I started off walking at a good pace and then I slowed down. From about 8 minutes to about 13 minutes, I hardly moved at all... Read More