The square, or second power, is usually the only power to get its own button on a calculator. Too many other powers are used in computing, so the rest are taken care of with a general power button, which looks like one of the following:
, xy, yx /
The two buttons with xraised to the yor yraised to the xare sort of scripted.
You have to enter the numbers exactly in the correct order for the calculator to compute what you mean. To use this button, you put in the xvalue first, you hit the button, and then finally you put in the yvalue.
I’m sure you’re dying to try out these new tricks on your calculator. As prac- tice, determine how to enter the following expressions in a calculator:
72+ 1 35– 41/3 2(58)
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Chapter 5: Working with Formulas
Taking advantage of hand-held
Here’s how you input each of the exponents into your calculator:
x
x
7 1 7 1
7 2 1
7 2 1
or
or y
2
"
2/
+ +
+ +
( )
35-41 3/
"
3 / 5 - 4 / 1 ' 3Notice here that the fractional exponent has to have parentheses around the numbers. Negative exponents also require parentheses around them.
( )
2 5 2 5 8
2 5 8
or
8
"
/# /
_ i
The order of operations dictates that the 5 in this expression is raised to the power first, and then the multiplication takes place. You don’t need to input into your calculator the parentheses to be assured of having the operations done in the correct order. (But they don’t hurt either.)
Distinguishing between subtraction and negativity
The subtraction symbol (–) is understood as being an operation. Subtraction is one of the four basic binary operations. And, in algebra, students are told that subtract, minus, negative, opposite,and lessare all indicated with the same symbol: the subtraction sign. The algebra ruling works fine when deal- ing with algebraic expressions. But calculators are a bit fussy and make a dis- tinction between the operation of subtraction (minusand less) and the condition of being negative (opposite). Most scientific calculators have a sub- traction button. You also find a separate negative button, which is distin- guished from the subtraction button by parentheses: (–).
To get familiar with subtraction and negativity, determine how to enter the following expressions in a calculator:
16 –18 –16 – (–18) –34
Here’s how to enter the previous expressions in your calculator:
16-18
"
16 - 1816 18
"
16 18- - -^ h ^-h - ^-h
You can type in the parentheses, but they aren’t really necessary in this case.
But when in doubt, always use the parentheses to avoid errors. Look at the dif- ference between not using parentheses around a negative number being raised to a power and then using the parentheses. You get two different answers:
( )
3 3 4
3 3 4
vs.
4
4
"
"
/
/
- -
- -
^
^ ^
h
h h
The answer to this first expression is –81, because the calculator raises the 3 to the fourth power and then changes the number to the opposite. If you want the number –3 raised to the fourth power, you have to put parentheses around both the negative and the 3. Notice that the answer is positive. Why?
Raising a negative number to the fourth power gives you a positive result. It’s all tied to the order of operations.
Grouping operations successfully
Usually you won’t run into any difficulties if you use more parentheses than necessary. I’m usually pretty heavy-handed with parentheses in math and commas in writing (but my editor takes care of that). The parentheses help you say what you mean — mathematically. Parentheses are needed if you have more than one term in the denominator (or bottom) of a fraction or more than one term in a radical. They’re supposed to make your intent clearer.
Using the grouping info I provide, determine how to enter the following expressions in a calculator:
3 5
28 and 18 2
2
2
+ -
You input the first expression like this:
( )
3 5
28 28 3 2 5
2+
"
' / +The parentheses ensure that the power and sum are performed in the denom- inator, and then the result divides the 28.
And here’s how to enter the second expression:
( )
18-22
"
18 - 2 / 2You want the root of the result under the radical.
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Chapter 5: Working with Formulas
Going scientific with scientific notation
Scientific notationis used to write very large or very small numbers in a useful, readable, compact form.
A number written in scientific notation consists of [a number between 1 and 10] ×[a power of 10]
For example, the number 234,000,000,000,000,000,000,000,000 written in scien- tific notation looks like this: 2.34 ×1026
The power of 26 on the 10 tells you that the decimal point was moved 26 places, from the end of the last zero to directly behind the 2. The number 2.34 is between 1 and 10.
Now here’s an example at the opposite end of the spectrum — it includes a number that’s quite small. Writing the number
0.000000000000000000000000000000000000234567 in scientific notation gives you: 2.34567 ×10–37.
A negative power is used in this instance, because very small numbers require that the decimal point moves to the right to get the nonzero digit part (the first number) to form a number between 1 and 10.
Calculators go into scientific notation mode when the result of a computation is too large for the screen or has more digits than the calculator can handle.
Instead of displaying scientific notation like I do here, calculators show an E and then, usually, the exponent. So, if you multiply numbers together and get the result 4.3E16, the calculator is reporting that the answer is written in sci- entific notation and is 4.3 ×1016= 43,000,000,000,000,000.
Repeating operations: Simplifying your work with a computer spreadsheet
Computer spreadsheets are truly wonderful tools. You not only get orderly reports of numbers all typed out neatly and in regular rows and columns, but you also have computing capability that’s equal to a calculator’s capability.
For instance, you can direct the computer spreadsheet to do all sorts of com- putations, such as adding all the numbers in a column and performing multi- plications and additions on selected numbers. And here’s the best thing: You can then tell the spreadsheet to repeat these same operations over and over on lots and lots of numbers. Now you can spend all that extra time lying on a beach somewhere . . . .
In this section, I don’t go into too much detail on how to enter formulas or values, because different computer packages have different rules. But after you see how tables of data are produced, hopefully you’ll be inspired to check into the particular spreadsheet program that you have and find the correct commands to type in.
In the following sections, I show you two examples of spreadsheets that you can create. One shows you how to sum up the revenue from different sources, and another determines the payment of an amortized loan.
Summing across rows or down columns
A useful type of table is one that shows the revenue from sales, where some of the money from the month’s sales come in that month, another percentage of those sales comes in the second month, and the rest (that actually does come in) is received in the third month. Every month, the total amount of rev- enue has to be tallied from the different sources — from previous months and from the current billing.
Guess what? You can use a spreadsheet to determine the amounts of money coming in each month as a percentage of particular sales. You also can use the spreadsheet to find the sum of all the revenue sources for the month.
Show the entries for the columns of a spreadsheet where you expect to col- lect 50% of the revenue from sales during the first month, 45% of the revenue during the second month, 4% during the third month, and write off the last 1% as a bad debt. Table 5-1 shows the setup. You copy the three percentage entries into the corresponding cells for each month. If you type in formulas and refer to sales amounts in the first column, the spreadsheet does the com- putations for you.
Table 5-1 Creating a Spreadsheet
Projected January February March April
Sales
January: 0.5($400,000) = 0.45($400,000) = 0.04($400,000) =
$400,000 $200,000 $180,000 $16,000
February: = 0.5($500,000) = 0.45($500,000) = .04($500,000)
$500,000 = $250,000 = $225,000 = $20,000
March: = 0.5($700,000) = .45($700,000)
$700,000 = $350,000 = $315,000
April: = 0.5($900,000)
$900,000 = $450,000
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Chapter 5: Working with Formulas
The table, which extends down for each month’s revenue, could be extended across for a full year or for several years. The spreadsheet does the computa- tion for you, rounded to the number of decimal places you indicate in your setup.
With one of the table commands available, you can find the sum of the num- bers going across the rows of the table or those going down the columns. Just highlighting and dragging across the values you want usually does the trick.
So the sum of the entries in the April column will come out to be 4% of February’s revenue, 45% of March’s revenue, and 50% of April’s revenue. If you change the sales amount (you find an extra $1,000 in April perhaps), the formula structure of the table will not only adjust the percentage amount across the row, but it also takes care of all the columns that are affected, too.
Creating an amortized loan schedule
You can determine the monthly payment of a particular loan using one of the appropriate formulas (see Chapter 12 for more on loan formulas). But it gets a bit tedious if you want to find the monthly payments involved in more than one or two scenarios — where you change the interest rate a bit or the length of the loan a bit. You don’t want to have to type the numbers into the calcula- tor over and over again.
Thankfully, a computer spreadsheet makes a table of loan payments quickly and accurately. You do have to type in the equation and set up the input values, but after doing the preliminaries, you can immediately copy, drag, and observe all the possibilities. The following example will walk you through the process.
Create a chart of the monthly loan payments on a $100,000 loan where you compare interest rates of 8%, 8.25%, 8.5% and 8.75%. Compare these rates at 15 years, 20 years, 25 years, and 30 years.
You start with a spreadsheet that has the interest rates, as decimals, at the top of consecutive columns and the years at the beginning of rows. You can see what I mean in Figure 5-1.
1
2 15
0.08 0.0825 0.085 0.0875 20
25 30 3
4 5 6 7 8
A B C D E F
Figure 5-1:
Setting up a spreadsheet for loan payments.
I typed in the interest rates and years, but you can have the computer spread- sheet fill in a bunch of consecutive amounts for you. By simply entering in a command to add a certain amount to each successive value, you save yourself a lot of typing.
For instance, in Figure 5-1 you enter the 15 for 15 years. Then in the cell directly below that one, you enter a command such as “= A2+5.” The number 20 should appear in that cell. Now you can copy the command and drag down through as many cells as you want, and each will show a number that’s 5 more than the previous cell. The same thing works when moving across the interest rates. You can increase by 0.25%, as shown in the figure, or you can pick some other increment. Also, you can format the cells to produce just about any number of decimal points. For example, you can set the format to show you four decimal places; the numbers are then rounded to that many places.
Now you’re ready for the fun part: entering the formula referencing the cell positions. The variables in your formula will be entries such as A2 or B3, telling the formula to look at that particular row and column. To begin, remem- ber that the formula for the monthly payment amount of an amortized loan is
R r
P r
1 1
12 12
t
= 12
- +
-
c c
m m
where Ris the amount of the regular payment, Pis the amount borrowed, r⁄12
is the interest rate each month, and tis the number of years. (You can find all the details on this formula in Chapter 12.)
Now you type the formula into cell B2 of the table. The following is one possi- bility for the format of the formula (which is the one I use in my spreadsheet).
However, you need to check the instructions and help menu with your particu- lar computer spreadsheet. Here’s the formula I used:
100,000* B1/12 / 1 1 B1/12 / 12*15
=a _ ik a -_ + i ^- hk
Yes, this all fits into the one tiny cell. Actually, what you’ll see is just the numerical answer in the cell. The formula should be available in the editing box. You can then copy that cell and hold and drag to the right until the whole row is highlighted. The spreadsheet picks up the interest values from the respective column heads.
Figure 5-2 shows what your row in the spreadsheet should look like.
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Chapter 5: Working with Formulas
When it comes to spreadsheets, you really need to just play around with them.
Say to yourself, “I wonder if I can get the spreadsheet to . . .” And, amazingly, you usually canget the spreadsheet to do what you want. When in doubt, con- sult the program manual for help.
1
2 15
0.08
955.6521 970.1404 984.7396 999.4487 0.0825 0.085 0.0875 20
25 30 3
4 5 6 7 8
A B C D E F
Figure 5-2:
Part of the loan payment schedule.