Background Mathematics

1.1 ARITHMETIC, NOTATION, AND FORMULAS

Almost all of the mathematics used in this book involve only the four basic opera­tions of addition, subtraction, multiplication, and division. If you can comfortably read about and then actually perform calculations using these four operations, you have all the math background you need. If you have a pocket calculator or a com­puter with a spreadsheet program, then you have the “machine power” to do what­ever you need to do without resorting to pencil and paper.

Mathematical notation, the way we express what we want to calculate, can sometimes be confusing. Mathematical notation is the vocabulary of the language of mathematical concepts. Often a student will think he or she doesn’t understand a concept when he or she simply is not familiar with the notation. It’s like being given driving directions in a foreign language when you don’t know the words for “turn left” or “turn right.” To further complicate things, there is almost always more than one way to write a particular mathematical expression. Often the choice is a matter of style and/or convenience. In this section, I’ll go through the various ways of writing different expressions involving only the four basic operations and explain why I will choose what I choose when I choose it.

I’tl begin with the definition of a variable. A variable, simply speaking, is a letter or a name that represents a number. If I want to say, for example, that an item in a catalog costs the price listed in the catalog plus a $10 shipping and handling fee, I can write, “If the cost of an item is X dollars, then you must pay X + 10 dollars if you want to order any item from this catalog.” It’s a way of generalizing a rela­tionship rather than having to recreate the relationship for each example.

I can get fancier and say that if Y represents the amount you must pay, then

Y = X +10.

Some people like to use letters from the end of the alphabet; some like to use letters from the beginning of the alphabet; some like to use Greek letters. It doesn’t matter which letters are used, as long as you’re clearly told what number each of

Understanding the Mathematics of Personal Finance: An Introduction to Financial Literacy, by

Lawrence N. Dworsky

Copyright © 2009 John Wiley & Sons, Inc.

these letters represent. Some authors of books and computer programs use variables that are case sensitive. That is, X and x represent different numbers. I don’t do this either in this book or in my spreadsheets.

In some situations, it’s convenient to use a whole word as the variable. In the above example, instead of letting Y be the cost, I could have written

Cost = X +10.

The expression

Y = X +10

is called a formula. A formula is a mathematician’s version of a recipe. You put in X (in this case the catalog price for the item) and you get out Y (in this case the amount you must pay to have the item appear on your doorstep). It’s conventional, but not necessary, for the variable that you’re calculating to appear on the left-hand side of the equal sign and the variable(s) that you’re supplying to appear on the right-hand side of the equal sign.

Typically, numbers that don’t change are shown as numbers, such as the 10 in the above formula, and numbers that depend on your particular situation, such as X and Y in the above formula, are represented by letters. This isn’t a law; it’s just a common practice.

If, for example, the shipping cost depends on the item’s weight, I could say that

Y = X + Z,

where Z is the shipping cost. If I do this, then I must refer you to a table or to another formula that explains how to calculate or look up the shipping cost before you can use this formula.

I’ll start my discussion of the four basic operations with addition, spelling out some things that are probably obvious. I’m doing this in order to be able to draw a contrast with the other basic functions and also to start explaining the use of parentheses.

When adding numbers, it doesn’t matter what order you do things in:

3 + 7 + 5 = 3 + 5 + 7 = 7 + 5 + 3 =…

Note also the use of the expression “…”. This means “and so on” and hopefully will be obvious in its intent when I use it.

I could also write the above addition example using parentheses to group some of the operations:

(3 + 7) + 5 =.

Writing it this way means “add 3 to 7 first, then add the result to 5.” In this example, the parentheses don’t contribute any value since the order of the additions doesn’t matter. On the other hand, they don’t introduce any error. In short, in this example, while the parentheses are harmless, they’re also pointless.

Now, let’s look at subtraction:

This is pretty clear so far.

However, while 2 + 6 is the same as 6 + 2,

6 – 2 = 4 and 2 – 6 = -4

are clearly not the same.

Getting a little more complicated,

(7 – 3) + 2 = 6.

The parentheses here mean first evaluate 7 – 3 (= 4) and then add the result to 2, yielding 6.

This is not the same as

7-(3 + 2) = 2.

In this case, the instructions are to first add 3 + 2 (= 5) and then subtract the result from 7, yielding 2.

Subtraction differs from addition in the importance of notation because the order in which things are calculated matters.

If I were to just write

7 – 3 + 2 = ??,

I wouldn’t know how to evaluate this because without the instructions added by the parentheses, I just don’t know what to do first.[1]

Moving on to multiplication, the simplest notation (and one that’s hardly ever used) is to use an “x” to indicate multiplication. Using this notation, it’s clear to see that, as in the case of addition, order doesn’t matter:

3 x 2 x 6 = 3 x 6 x 2 = 6 x 2 x 3 =… = 36

One good reason why the “x” is hardly used to signify multiplication is that, once you’re expecting formulas, you don’t know whether this x means multiplica­tion or is itself a variable representing another number.

For better or worse, there are many notations for multiplication. The important consideration is that the chosen notation must be clear and unambiguous.

When it is clear what I mean, I will just write the two numbers (and/or variables) that I want to multiply next to each other: 3x, or xy. Obviously this won’t work for multiplying 3 by 2, because 32 (or 23) would be interpreted as a two-digit number, not instructions to multiply the two single-digit numbers together.

When multiplying a number by a variable, it’s common to put the number first: 3x means the same as x3 but is almost always written as 3x.

This example also shows why multiplication is almost never written using an “x” to signify multiplication—the “x” is probably the most common choice of a letter for a variable, and writing 3 x x to mean “multiply 3 by the variable x” is just a confusing mess.

The expression

3x (y + 7)

can be interpreted two ways. The two ways are equivalent and both are valid.

The first interpretation is that you should do what’s inside the parentheses first. That is, if y represents some cost or payment or whatever, let’s say y = $15.50, then add y to 7, giving 15.50 + 7 = 22.50. Then we have

3x (22.50).

This is simply three numbers multiplied together. The parentheses now are used just to keep the 3 from being tangled up with the 22.50. Since numbers multiplied together can be multiplied in any order, we have

3x (22.50) = 3(22.50)x = 67.50x

At this point we need a value for x or we just have to stop.

In order to do what I just did, I needed a value for the variable y. If I don’t have a value for y, I can either leave things as they are for the time being, or I can “expand” the expression. This is the second interpretation: What’ s outside the parentheses multiplies everything that’s inside the parentheses. Therefore,

3x (y + 7) = 3x (y) + 3 x (7) = 3xy + 21x

Whether the latter way of writing things is any clearer, or more useful, than the original expression is in the eye of the beholder.

Taking this one step further, what if I have

(12 + 4)(3 + 6).

The same rules apply; you just have to do a little more work:

(12 + 4)(3 + 6) = (16)(9) = 144.

This type of expression, when there are algebraic variables involved, often trips up students. Another correct way of evaluating this expression is to use the second interpretation above: What is outside the parentheses multiplies everything that is inside the parentheses. This time we have to remember that there are two sets of parentheses, so we have

(12 + 4)(3 + 6) = 12 (3 + 6) + 4 (3 + 6) = 12 (3) +12 (6) + 4 (3) + 4 (6)

= 36 + 72 +12 + 24 = 144.

This is sometimes called “expanding” the expression.

An example of the same expression with algebraic variables is

(a + b)(c + d ) = ac + ad + bc + bd.

In this book, I’ll often be presenting formulas for use in calculating a number, typically a dollar value. This isn ’ t an algebra book. You will have the working knowledge that you need as long as you understand the first interpretation, that is, put in the values for the variables then evaluate what’s inside the parentheses (or sets of parentheses).

The last of the four basic operations is division. First, notation: The elementary school notation 6 +- 3 is pretty much never used. Instead, recognizing that a division expression is the same as a fraction expression, 6 divided by 3 will be written as one of the following:

6 3 = 6 = 2.

3

If I want to use the first of these forms for multiple operations, then I have to get involved with parentheses, because order counts. That is,

18 6 3 =

is ambiguous because I don’t know what to do first. My choices are

(18 6) 3 = 3 3 = 1 or

18 (6 3) = 18 2 = 9,

and I have no way of knowing which interpretation was intended.

Using the fraction notation, I can finesse the parentheses issue by working with different size fraction lines. That is,

18

6 = 3 = 1 3 3

while

18 = 18 = 9 6 = 2 = .

3

I could keep going with compound expressions—say, fractions involving sums or differences of numbers and variables inside the parentheses, and so on. But again, this is not an algebra book and my aim is not to trip you up but to give you clear rules for evaluating (finding the numeric value of) a formula when you’re presented with it.

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