CALCULATOR ARITMETIC

Objectives:

• To evaluate rational expressions using a calculator.
• To get used to using a calculator (please bring one in class).

An implicit objective is: To use the calculator as a tool for developing familiarity with the order of arithmetic operations in complicated expressions.

Addition, subtraction, multiplication and division are arithmetic operations that you can perform if the numbers involved are not too messy. If you are asked to evaluate the expression

6(12-4)/3

you probably have no trouble in calculating it to be 16. However, if you are asked to evaluate the same type of rational expression with the numbers more complicated you may experience some difficulty. Try evaluating

3.27(4.67-0.98)/75.34

How long will it take you to find that it is 0.1601579 ? By hand, it may take you more than 5 minutes; by calculator it will take you less than 30 seconds. The hand computation is more tedious, lengthy and prone to error. Why not delegate the computation to a machine?

Recall 1 The number 0.1601579 is called a decimal representation of a number. It is a short-hand notation for

Similarly, the number 354.73 is short-hand notation that stands for the number

You may recall that 10^0=1 and that 10^1=10.

Recall 2: There are five important operation keys on your calculator.

To calculate

3.27(4.67-0.98)/75.34

start with the numbers inside the parentheses, multiply by 3.27 and divide by 75.34. The sequence of entries is as follows:

4.67 0.983.27 75.34

By pressing the  key you are telling the calculator that you have finished entering the number that precedes that key.

You have learned how to arithmetically combine fractional and rational expressions. The expressions you encountered did not involve numbers that were too difficult to handle. However, what would you do if you were given an expression such as

Clearly you would hope to rely on a calculator to perform the tedious arithmetic. Before working on this expression, let's work on a simpler one.

Take

and find the rational number that represents it. You would probably first combine the 2 + 1. This would give you 1/(2+1/3). Next, you would add the 2 and the 1/3 to get 1/(7/3), which you would recognize as 3/7.
It would be silly to use a calculator to evaluate the expression ; however, it is instructive to see how we would use a calculator to evaluate such an expression.

Recall 3: Your calculator probably has the additional key . This is a very useful key. Here is how it works in evaluating the expression

The following sequence of keys will give you the correct real number approximation to 3/7, or 0.4285714.

21 2

Think of the sequence above as an arithmetic sentence, composed of phrases which tell the calculator to perform operations as follows:

21 says: add 2 and 1 and display the result (the result is 3),

2 says: put the result in the denominator, add 2 and calculate the new result (the new result is 7/3),

says: compute 1 over the previous result (the new result is 3/7).

Of course the calculator does not display 3/7. Rather, it displays the digital approximation to 3/7,. which is 0.4285714.

Recall 4: Many calculators have parentheses keys. and

If your calculator has such keys, then you may calculate the expression in Recall 2 as

3.27 4.67 0.98 75.34 .

Note

What happens when we see an expression like

4*3 + 7*5 ?

Does it mean multiply 4 and 3, add 7 and multiply the result by 5? Or does it mean multiply 4 and 3, multiply 7 and 5, and then add the result? In the first case you get 95, in the second 47.

The general rule of arithmetic is that when there are no parentheses, multiplication and division are performed before addition and subtraction. In the above example the implied meaning is that

4*3 + 7*5 = (4*3) + (7*5).

Warning

Be sure to clear the calculator's memory of previous calculations before performing new ones; otherwise, you may be carrying a previous calculation which will combine with your new calculation. Press  for the clear button, or turn the calculator off and on again.

Note

Not all calculators are alike. We have given methods that are widely accepted by the most popular calculators. Consult your users manual to find out about specific differences.

Examples

Example 1 Describe the sequence of keys you would use to evaluate the expression
4/2.36+5/4.39

Solution: If your calculator has parentheses keys, then enter the sequence

4 2.364.39 .

Another way is

2.364.395 .

Notice that the same number of keys are used in both sequences. This shows that there are many ways of getting the same result. The result is 2.833867

Example 2. Calculate the expression without using parentheses.

55.87[(52.3-34.5)/43.87]

Solution: Start from within the parentheses. Follow the sequence

52.3 3.45 43.87 55.87 .

The result is 62.212205

Example 3 Using parentheses, calculate
(4.21+5.27*6.53)/(3.33-1.56)

Solution:

4.21 5.27 6.53 3.33 1.56 .

The result is 21.82096.

Notice that it is not necessary to place parentheses before the 5.27 and after the 6.53 because -- in the absence of parentheses -- multiplication is always performed before addition.

Example 4. Evaluate the expression using the least number of keys.

Solution: Enter the following sequence of keys:

2.35 1.67 1.67 2.35 1.672.35 1.67 2.35

Example 5. Evaluate the expression using the least number of keys.
1.63/(2.86-3.44)

Solution: Enter the following sequence of keys:

3.44 2.86 1.63

Notice that we have introduced a new key here--the plus/minus key . This key has the effect of changing the sign of the number that is displayed. For example, if 6 is displayed, then pressing  changes the 6 to -6. It is easy to avoid using this key to evaluate 1/(2.86-3.44); however, it is not so easy to avoid it in the next example.

Example 6. Evaluate the expression using the least number of keys.

Solution: Enter the following sequence of keys:

5.4 4.5 4.5 5.4 4.5

Warning

It is easy to make a mistake in entering numbers, especially when the numbers are long. To guard against such errors, always do the following: Before pressing any operation key, check that the displayed number is indeed the number that you intended to enter. Get into the habit of doing this and you will make less errors.

EXERCISES

1. Use a calculator to evaluate 34.8 - 53.96 + 67.33

2. Describe the sequence of keys that you would use to compute

3.24/(4.63-2.36)

3 Describe the sequence of keys that you would use to compute

4.75/(5.36-4.89)
without using the parentheses keys.

4 Describe the sequence of keys that you would use to compute

3.66/5.32+4.67/3.22

5 Describe the sequence of keys that you would use to compute using the parentheses keys.

(3.14+4.97)(3.67-1.95)/(5.36-4.89)

6 Describe the sequence of keys that you would use to compute without using the parentheses keys.

[(3.14+4.97)*3.67]/(5.36*4.89)

7 Evaluate

[(47.34-57.92)/(99.74-29.87)]*4.75*3.14

8 Describe the sequence of keys that you would use to compute using the parentheses keys
[(47.34-57.92)/(99.74-29.87)]*4.75*3.14

9 Describe the sequence of keys that you would use to compute

without using the parentheses keys.