ELEC-103 UNIT 3

This unit will be used to involve you with the scientific calculator. It is assumed you know how to use the average four function calculator but, most probably, you do not know how to use many of the additional functions available on a scientific calculator. This unit will help you learn how to use the many new functions available on the scientific calculator. The scientific calculator, hereafter referred to as the calculator, must have certain keys available or it will not be able to be used in conjunction with this unit. The calculator you buy must have, at a minimum, the keys listed below and most likely, it will have many more functions available.

1.Number Entry Keys (10 digits and a decimal point)

0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / .

2.Operation Keys

+ /  / x /  / =

The calculator that you buy should have an equals key. Do not buy a calculator with an ENTER key.

3.Special Function Keys

( / ) /  / / X2 / YX / Ex
log / 10X / sin / cos / tan / lnx / exp
DRG / 1/X / 2nd / INV

4.Memory Keys

XM
X+M
RM
XM
STO
SUM
RCL
EXC

Some calculators

have memory keys

similar to those

on the left and

others have keys

similar to those

on the right.

Either type is

fully acceptable

for this course.

Most calculators have the capability of displaying seven or more digits to the right of the decimal point. However, calculators also have some additional guard digits built into their calculation capability. Thus, the precision of calculation is better than would be expected from the window display. When Pi () is entered on a TI35 PLUS scientific calculator using the Pi key, the display shows 3.141592654. The additional digits can be displayed using the following example:

NOTE:All examples in this unit were performed using the TEXAS INSTRUMENT TI35 PLUS Scientific Calculator. The entries will be shown boxed and all results, or calculator displays, will be shown unboxed.

2nd /  / 3.141592654 /  / 3.14159 / = / 0.000002653 / x / 1000000 / = / 2.65359

In the previous example, the last several digits, when Pi was entered, were 2654. The calculator was then forced into displaying those extra digits we know are there. The last calculator display shows the calculator really knows the last few digits to be 265359. Originally, the calculator did not have space to display the full number of digits so it displayed the digits shown. The last digits in the original display were 654. The four was turned into a four from a three by the calculator. The calculator automatically rounds off, in certain instances, and you will also learn how to round off before you write down your answer to a problem that is being calculated.

2 /  / 3 / = / 0.666666666

In the next example, the calculator does not display the last two digits and does not round off the last digit displayed. Not all calculators work the same way and, therefore, you may not see the same results. Divide 2 by 3. Some calculators will display what is shown on the right, while others will display the last digit as a seven. The program that manufacturers use with their calculators will differ and therefore the displayed answers will differ with the last digit. However, the overall results will be the same, but how the calculator gets there will be different. Since your calculator might not act the same as other calculators being used in class, you should bring your calculator instruction manual to class while working on this unit.

Most calculators have three ways in which you can clear the display resister. The first method is to turn the calculator off, then turn it on again. That procedure clears the internal registers and the display register while the main memory remains untouched. Another way to accomplish the same thing is to press the ON key twice. This also clears the internal registers and the display register while the main memory remains untouched.

The third way to clear the display is to press the ON key once.this clears the display register only. It is called clear entry since it clears only the display register and does not clear anything else. Some calculators have a separate clear entry key labeled, CE. It is a useful key when you are in the middle of a problem and the last entry you made was incorrect. Instead of starting the problem all over again from the beginning, you can clear just the last entry from the calculator, enter the correct information and proceed with the problem.

To clear the main memory register you need only press the On key once and then press the STO key or the XM key, depending upon the type calculator you have. If anything was stored in the main memory register a "M" would have been visible somewhere on the main display. After pressing the STO key or the XM key, as the case may be, the main memory register will contain a zero and the "M" on the main display will no longer be visible.

Since everyone knows how to use the add, subtract, multiply and divide functions, in conjunction with the equals key, those functions will not be covered or reviewed. However, it should be noted that whenever the equals key must be used in conjunction with another key, the original key that was pressed is called a BINARY OPERATION KEY.

Binary operation keys are:

+ /  / x /  / YX

Unary operation keys are:

1/X / sin / cos / tan / log / 10X / lnx / eX / X2 /

other keys are Unary Operation keys too, but are not listed here since they will not be used in conjunction with this unit.

As stated earlier, this unit will be useful only with calculators that have an equals key. Such calculators are called algebraic notation calculators. The other major type of calculator available has no equals key but utilizes the entry key. This is called the RPN, or Reverse Polish Notation type calculator.

Since the calculator has such a great number of digits after the decimal point, it is quite obvious that all the digits are not necessary. However, there needs to be some procedure for you to follow which will allow some uniformity in the way answers to problems are written down. For this unit, and unless otherwise specified, answers will be written including two places to the right of the decimal point. This is where ROUNDING OFF becomes important since you cannot just throw all the extra digits away.

You must use a ROUNDING OFF procedure and that procedure will be discussed in the following paragraphs along with other related topics such as PLACE VALUES (weighted position values) for the Decimal Numbering System and some discussion about Floating Decimal, one of the systems most commonly used with scientific calculators.

Figure 31 lists the weighted position values for the Decimal Numbering System, the system used by our country. The first place to the left of the decimal point is called the UNITS place.

FIGURE 31

The second place to the left of the decimal point is called the TENS place. The third place to the left of the decimal point is called the HUNDREDS place, and so forth on out to the MILLIONS place. The units place can also be represented in scientific notation by 1x100. Which, as you can see, can be further reduced to 100. The tens place is represented by 101, the hundreds place by 102, the thousands place by 103 and so forth on out to 106 in the millions place.

The first place to the right of the decimal point is called the TENTHS place. The second place to the right of the decimal point is called the HUNDREDTHS place. The third place to the right of the decimal point is called the THOUSANDTHS place. These places can also be represented in scientific notation by 101, 102, etc., as shown in Figure 31.

EXAMPLE:

Write 3729.1608 and underline the digit in the:

a. Tens Placeb. Tenths Placec. Thousandths Place

d. Thousands Placee. Units Placef. Ten Thousandths Place

a. 3729.1608b. 3729.1608c. 3729.1608

d. 3729.1608e. 3729.1608f. 3729.1608

FIGURE 32

Using Floating decimal, the calculator will show as many decimal places as may result from the calculation that can be displayed. Therefore, it is possible to have many digits displayed to the right of the decimal. All those digits are not necessary for a written answer so it is necessary to learn how to ROUND OFF. To round off to the tenths place, using the number 3729.1608, one would write 3729.2 as the answer. To round off to the hundreds place, using the number 3729.1608, one would write 3700 as the answer. There is a procedure you can follow and it can be stated in words or displayed in a visual manner. Both methods will be used so you can refer to the flow chart in Figure 32 as you read the procedure for ROUNDING OFF.

To learn the procedure for ROUNDING OFF, one must first be aware of a few definitions that are important to this procedure.

DEFINITIONS

a. Roundoff Digit The last digit to be kept.

b. Test Digit  The digit immediately to the right of the Roundoff Digit.

Example:Round off 649.03574 to the hundredths place / The Round off digit is three.(the one that is underlined) The Test digit is five.(the digit to the right of the Roundoff digit)

Following the flow chart in Figure 32, you must underline the roundoff digit. The hundredths place is being used for this example since all answers in this unit will be rounded off to the hundredths place, unless otherwise specified. Since the rounding off is to the hundredths place, which is the second digit to the right of the decimal place, a line is placed under the number three. Proceed with the flow of the flow chart, which is from left to right. Enter the flow chart decision box and answer the question in the decision box. The test digit is the digit immediately to the right of the rounding off digit. In this case, the answer to the question in the decision box is yes. If the answer is yes, the action to be taken is stated in the yes action box. Add one to the round off digit. This means the underlined number, three, becomes a four. Proceed with the flow from left to right in the flow chart and another decision box is entered. The question in this decision box means that if the round off number is to the right of the decimal point the answer should be yes. If, however, the round off number is to the left of the decimal point the answer to the question in the decision box is no. Obviously, this round off digit is to the right of the decimal. Therefore, all digits to the right of the round off digit are dropped. The answer is 649.04 when 649.03574 is rounded off to the hundredths place.

EXAMPLES:

a. Round off to the tens;  649.03574  649.03574  650

b. Round off to the tenth;  649.03574  649.03574  649.0

c. Round off to the thousandth; 649.03574  649.03574  649.036

d. Round off to the hundred;  649.03574 649.03574  600

e. Round off to the unit;  649.03574  649.03574  649

Using the five examples listed above, in conjunction with the flow chart in Figure 32, should enable you to grasp the rounding off concept.

FACTORING

When a whole number is divided by a whole number and the result is a whole number, then the divisor and the quotient are factors of the dividend. Let the dividend be 75, the divisor be 3 and the quotient will be 25.

Example: 75  3 = 25 / Factors of 75 are 3 x 25. However, 3 and 25 are not the complete factors of 75, since 25 has factors of 5 x 5. Therefore the complete factors of 75 really consist of 5 x 5 x 3.

Prime numbers are whole numbers that are not factorable. For a prime number there is no whole number greater than 1 that can be divided into the prime number that will result in another whole number. The first 16 Prime Numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, and 53

Composite numbers are whole numbers greater than 1, which are not prime numbers. Therefore, composite numbers consist of factors and the complete factors of a composite number are all prime numbers.

THE SQUARE ROOT () KEY

One of the methods that can be used to determine whether a number is a prime number or not involves using the square root () key. If the number is not a prime number, it must be a composite number. This method will provide you with the complete factors of the composite number, if the number is not a prime number. The procedure requires you to take the square root of the number in question. Compare the square root of the number to the list of prime numbers. Choose a prime number that is the next prime number smaller than the square root of the original number.

EXAMPLE: What are factors of 143?

The square root of 143 () = 11.958 and if you look at the list of prime numbers you will see that 11 is the first prime number smaller than the square root of 143.

NOTE
On some calculators the square root key symbol () does not appear on a key but appears above the key. If that is the case for your calculator you should press the second (2nd) or inverse (INV) key then press the key that the symbol is over. More than likely it will be over the X2 key.

Divide 143 by the first prime number smaller than the square root of 143. Therefore, you will divide 143 by 11 and 143  11 = 13. Since the quotient (13) is a whole number, the divisor (11) is one of the factors of 143, the original number. Since the quotient is also a prime number, the factors of 143, (the prime numbers, that when multiplied together equal 143) are 11 x 13.

PROCEDURE FOR FACTORING

EXAMPLE: What are factors of the number 99?

Step 1 / 99 / / 9.95

The first step is to take the square root of the original number. Next, look at the list of prime numbers and pick the prime number just smaller than 9.95, the square root of 99.

Step 2 / 99 /  / 7 / 14.14

In this instance that specific prime number is 7. Divide 99 by 7 to determine if the result (quotient) is a whole number or a decimal number.

Step 3 / 99 /  / 5 / 19.80

When the quotient is not a whole number you will continue to divide the original number by the next lower prime number. This continues until you divide the original number by a prime number and the result is a whole number.

Step 4 / 99 /  / 3 / 33

Step 4 shows the original number being divided by the prime number 3 with a quotient of 33. Since the quotient is a whole number, the prime number just used is a factor of the original number. Therefore, 3 is one of the factors of 99, the original number. The quotient will now be used in place of the original number to continue the problem. As a matter of fact, the number 33, in this case, is going to be treated the same as if it was the original number. Before proceeding with the problem, write down the number 3 as being one of the factors of the original number, 99.

Step 5 / 33 / / 5.74

Continue the problem in step 5 by taking the square root of 33, the whole number quotient of the previous step. Look at the list of prime numbers for the prime number just lower than 5.74, the square root of 33. That number will be 5 and in step 6, 33 is divided by 5 to check for a whole number as a quotient.

Step 6 / 33 /  / 5 / 6.60

Step 6 results in a quotient that is not a whole number and therefore, the next smaller prime number is used in step 7 as the divisor.

Step 7 / 33 /  / 3 / 11

Step 7 results in a quotient that is a whole number, so the divisor, the prime number 3, is one of the factors of the original number, 99. The quotient, the number 11, should now be used as the new number in the procedure, as was done with the number 33 just a few steps back in the procedure.

However, the number 11 is itself a prime number. Therefore, 3 x 3 x 11 are the complete factors of 99 and, when multiplied together, will equal 99. The factors of 99 are all prime numbers and factors of composite numbers will always be prime numbers as they are in this instance.

EXAMPLE: What are factors of the number 97?

Step 1 / 97 / / 9.85

The first step is to take the square root of the original number. Next, look at the list of prime numbers and pick the prime number just smaller than 9.85, the square root of 97.

Step 2 / 97 /  / 7 / 13.86

In this instance that prime number is 7. Divide 97 by 7 to determine if the result (quotient) is a whole number or a decimal number.

Step 3 / 97 /  / 5 / 19.40

When the quotient is not a whole number you will continue to divide the original number by the next lower prime number.

Step 4 / 97 /  / 3 / 32.33

This continues until you divide the original number by a prime number and the result is a whole number.

Step 5 / 97 /  / 2 / 48.50

However, in this instance, the original number, 97, has been divided by all the prime numbers smaller than the square root of the original number and no quotient has been a whole number. This means the original number, 97, is a prime number and cannot be divided by another whole number and produce a quotient that is a whole number.