1. 12 P
Prologue: Calculator Arithmetic

2. Typing Mathematical Expressions

3. Typing Mathematical Expressions
When we write expressions such as using pen and paper, the paper serves as a two-dimensional display, and we can express fractions by putting one number on top of another and exponents by using a superscript.
When we enter such expressions on a computer, calculator, or typewriter, however, we must write them on a single line, using special symbols and (often) additional parentheses.
The caret symbol ∧ is commonly used to denote an exponent, so in typewriter notation comes out as
71  ∧ 2  5 .

4. Typing Mathematical Expressions
In Figure P.1, we have entered (0.1 TG) this expression, and the resulting answer is shown in Figure P.2.
You should use your calculator to verify that we did it correctly.
Entering
The value of
Figure P.1
Figure P.2

5. Rounding

6. Rounding
When a calculation yields a long answer such as the , we will commonly shorten it to a more manageable size by rounding.
Rounding means that we keep a few of the digits after the decimal point, possibly changing the last one, and discard the rest.
There is no set rule for how many digits after the decimal point you should keep; in practice, it depends on how much accuracy you need in your answer, as well as on the accuracy of the data you input.

7. Rounding
As a general rule, in this text we will round to two decimal places. Thus for
we would report the answer as
In order to make the abbreviated answer more accurate, it is standard practice to increase the last decimal entry by 1 if the following entry is 5 or greater. Verify with your calculator that

8. Rounding
In the answer the next digit after 1 is 7, which indicates that we should round up, so we would report the answer rounded to two decimal places as 9.32.
Note that in reporting as the rounded answer above, we followed this same rule. The next digit after 4 in is 2, which does not indicate that we should round up.

9. Rounding
To provide additional emphasis for this idea, Figure P.3 shows a calculation where rounding does not change the last reported digit, and Figure P.4 shows a calculation where rounding requires that the last reported digit be changed.
An answer that will be reported as 1.71
An answer that will be reported as 1.89
Figure P.3
Figure P.4

10. Parentheses and Grouping

11. Parentheses and Grouping
When parentheses appear in a calculation, the operations inside are to be done first.
Thus 4(2 + 1) means that we should first add and then multiply the result by 4, getting an answer of 12.
This is correctly entered (0.2 TG) and calculated in Figure P Where parentheses appear, their use is essential.
A correct calculation of 4(2 + 1) when
parentheses are properly entered
Figure P.5

12. Parentheses and Grouping
If we had entered the expression as 4  2 + 1, leaving out the parentheses, the calculator would have interpreted it to mean first to multiply 4 times 2 and then to add 1 to the result, giving an incorrect answer of 9.
This incorrect entry is shown in Figure P.6.
An incorrect calculation of 4(2 + 1)
caused by omitting parentheses
Figure P.6

13. Parentheses and Grouping
Sometimes parentheses do not appear, but we must supply them.
For example, means 17  (5 + 3). The parentheses are there to show that the whole expression goes in the denominator.
To do this on the calculator, we must supply (0.3 TG) these parentheses.
Figure P.7 shows the result.
Proper use of parentheses in the calculation of
Figure P.7

14. Parentheses and Grouping
If the parentheses are not used, and is entered as 17  5 + 3, the calculator will interpret it to mean that only the 5 goes in the denominator of the fraction.
This error is shown in Figure P.8.
An incorrect calculation of
caused by omitting parentheses
Figure P.8

15. Parentheses and Grouping
In general, we advise that if you have trouble entering an expression into your calculator, or if you get an answer that you know is incorrect, go back and re-enter the expression after first writing it out in typewriter notation, and be careful to supply all needed parentheses.

16. Minus Signs

17. Minus Signs
The minus sign used in arithmetic calculations actually has two different meanings.
If you have $9 in your wallet and spend $3, then you will have 9 − 3 = 6 dollars left.
Here the minus sign means that we are to perform the operation of subtracting 3 from 9.

18. Minus Signs
Suppose in another setting that you receive news from the bank that your checking account is overdrawn by $30, so your balance is −30 dollars.
Here the minus sign is used to indicate that the number we are dealing with is negative; it does not signify an operation between two numbers.

19. Minus Signs
Once the problem is recognized, it is usually easy to spot when the minus sign denotes subtraction (when two numbers are involved) and when it indicates a change in sign (when only one number is involved). The following examples should help clarify the situation:
−8 − means negative 8 subtract 4 (0.7 TG)
means (0.8 TG)
2−3 means negative 3.

20. Minus Signs The calculation (0.9 TG) of 2−3 is shown in Figure P.9.
If we try to use the calculator’s subtraction key (0.10 TG), the calculator will not understand the input and will produce an error message such as the one in Figure P.10.
Calculation of 2−3 using the negative key
Syntax error when subtraction operation is used in 2−3
Figure P.9
Figure P.10

21. Special Numbers  and e

22. Special Numbers  and e
Two numbers,  and e, occur so often in mathematics and its applications that they deserve special mention.
The number  is familiar from the formulas for the circumference and area of a circle:
Area of a circle of radius r =  r2
Circumference of a circle of radius r = 2 r .
The approximate value of  is , but its exact value cannot be expressed by a simple decimal, and that is why it is normally written using a special symbol.

23. A decimal approximation of 
Special Numbers  and e
Most calculators allow you to enter (0.13 TG) the symbol  directly, as shown in Figure P.13.
When we ask the calculator for a numerical answer (0.14 TG), we get the decimal approximation of  shown in Figure P.13.
A decimal approximation of 
Figure P.13

24. A decimal approximation of e
Special Numbers  and e
The number e may not be as familiar as , but it is just as important.
Like , it cannot be expressed exactly as a decimal, but its approximate value is
In Figure P.14 we have entered (0.15 TG) e, and the calculator has responded with the decimal approximation shown.
A decimal approximation of e
Figure P.14

25. Special Numbers  and e
Often expressions that involve the number e include exponents, and most calculators have features to make entering such expressions easy.
For example, when we enter (0.16 TG) e1.02 we obtain 2.77 after rounding.

26. Chain Calculations

27. Chain Calculations
Some calculations are most naturally done in stages.
Many calculators have a special key (0.17 TG) that accesses the result of the last calculation, allowing you to enter your work in pieces.
To show how this works, let’s look at

28. The first step in a chain calculation
Chain Calculations
We will make the calculation in pieces.
First we calculate
Enter (0.18 TG) this to get the answer
in Figure P.15.
To finish the calculation, we need to add this answer to 17/(2 + ):
The first step in a chain calculation
Figure P.15

29. Completing a chain calculation
Chain Calculations
In Figure P.16 we have used the answer (0.19 TG) from Figure P.15 to complete the calculation.
Accessing the results of one calculation for use in another can be particularly helpful when the same thing appears several times in an expression.
For example, let’s calculate
Completing a chain calculation
Figure P.16

30. Accessing previous results to get an
Chain Calculations
Since 7/9 occurs several times, we have calculated it first in Figure P.17.
Then we have used the results to complete (0.20 TG) the calculation.
We would report the final answer rounded to two decimal places as 7.30.
Accessing previous results to get an
accurate answer
Figure P.17

31. Inaccurate answer caused by early rounding
Chain Calculations
There is an additional advantage to accessing directly the answers of previous calculations.
It might seem reasonable to calculate 7/9 first, round it to two decimal places, and then use that to complete the calculation.
Thus we would be calculating
This is done in Figure P.18,
which shows the danger
in this practice.
Inaccurate answer caused by early rounding
Figure P.18

32. Chain Calculations
We got an answer, rounded to two decimal places, of 7.27—somewhat different from the more accurate answer, 7.30, that we got earlier.
In general, if you are making a calculation in several steps, you should not round until you get the final answer.
An important exception to this general rule occurs in applications where the result of an intermediate step must be rounded because of the context.
For example, in a financial computation dollar amounts would be rounded to two decimal places.

33. Example P.2 – Compound Interest and APR
There are a number of ways in which lending institutions report and charge interest.
Part 1: Paying simple interest on a loan means that you wait until the end of the loan before calculating or paying any interest. If you borrow $5000 from a bank that charges 7% simple interest, then after t years you will owe
5000 × ( t) dollars.
Under these conditions, how much money will you owe after 10 years?
Part 2: Banks more commonly compound the interest. That is, at certain time periods the interest you have incurred is calculated and added to your debt.

34. Example P.2 – Compound Interest and APR
cont’d
From that time on, you incur interest not only on your principal (the original debt) but on the added interest as well. Suppose the interest is compounded yearly, but you make no payments and there are no finance charges. Then, again with a principal of $5000 and 7% interest, after t years you will owe
5000 × 1.07t dollars.
Under these conditions, how much will you owe after 10 years?
Part 3: For many transactions such as automobile loans and home mortgages, interest is compounded monthly rather than yearly. In this case, the amount owed is calculated each month using the monthly interest rate.

35. Example P.2 – Compound Interest and APR
cont’d
If r (as a decimal) is the monthly interest rate, then after m months, the amount owed is
5000 × (1 + r)m dollars,
assuming the principal is $5000.
The value of r is usually not apparent from the loan agreement. But lending institutions are required by the Truth in Lending Act to report the annual percentage rate, or APR, in a prominent place on all loan agreements. The same statute requires that the value of r be calculated using the formula
If the annual percentage rate is 7%, what is the amount owed after 10 years?

36. Example P.2 – Solution Part 1:
To find the amount owed after 10 years, we use t = 10 to get
5000 × ( × 10) .
Entering (0.21 TG) this on the calculator as we have done in Figure P.19 reveals that the amount owed in 10 years will be $8500.
Balance after 10 years using simple interest
Figure P.19

37. Example P.2 – Solution Part 2: This time we use (0.22 TG)
cont’d
Part 2:
This time we use (0.22 TG)
5000 ×
From Figure P.20 we see that, rounded to the nearest cent, the amount owed will be $
Comparison with part 1 shows the effect of compounding interest. We should note that at higher interest rates the effect is more dramatic.
Balance after 10 years using yearly compounding
Figure P.20

38. Example P.2 – Solution Part 3: The first step is to use the formula
cont’d
Part 3:
The first step is to use the formula
to get the value of r as we have done in Figure P.21.
Ten years is 120 months, and this is the value we use for m.
Getting the monthly interest rate from the APR
Figure P.21

39. Example P.2 – Solution
cont’d
Using this value for m and incorporating the value of r that we just calculated, we entered (0.23 TG) 5000 × (1 + r)120 in Figure P.22, and we conclude that the amount owed will be $10,
Balance after 10 years using monthly compounding
Figure P.22

40. Example P.2 – Solution
cont’d
Comparing this with the answer from part 2, we see that the difference between yearly and monthly compounding is significant.
It is important that you know how interest on your loan is calculated, and this may not be easy to find out from the paperwork you get from a lending institution.
The APR will be reported, but the compounding periods may not be shown at all.

41. Scientific Notation

42. Scientific Notation
It is cumbersome to write down all the digits of some very large or very small numbers.
A prime example of such a large number is Avogadro’s number, which is the number of atoms in 12 grams of carbon 12.
Its value is about
602,000,000,000,000,000,000,000 .
An example of a small number that is awkward to write is the mass in grams of an electron:
gram .

43. Scientific Notation
Scientists and mathematicians usually express such numbers in a more compact form using scientific notation.
In this notation, numbers are written in a form with one nonzero digit to the left of the decimal point times a power of 10.
Examples of numbers written in scientific notation are × 104 and 2.7 × 10−4.
The power of 10 tells how the decimal point should be moved in order to write the number out in longhand.

44. Scientific Notation
The 4 in 2.7 × 104 means that we should move the decimal point four places to the right. Thus
2.7 × 104 = 27,000
since we move the decimal point four places to the right.
When the exponent on 10 is negative, the decimal point should be moved to the left. Thus
2.7 × 10–4 =
since we move the decimal point four places to the left.
With this notation, Avogadro’s number comes out as × 1023, and the mass of an electron as × 10−31 gram.

45. Scientific notation for a large number
Many times calculators display numbers like this but use a different notation for the power of 10.
For example, Avogadro’s number 6.02 × 1023 is displayed as 6.02E23, and the mass in grams of an electron × 10–31 is shown as 9.11E–31.
In Figure P.23 we have calculated
The answer reported by the calculator written in longhand is ,125,899,907,000,000.
Scientific notation for a large number
Figure P.23

46. Scientific notation for a small number
In presenting the answer in scientific notation, it would in many settings be appropriate to round to two decimal places as 1.13 × 1015.
In Figure P.24 we have calculated 7/320. The answer reported there equals
If we write it in scientific notation and round to two decimal places, we get × 10–9.
Scientific notation for a small number
Figure P.24