Session Nine

Welcome to the ninth section of MTH140, Number bases.  In this session you will learn about using the TI-85 graphing calculator to convert numbers from one base to another.

Begin this session by watching the video tape.  If you have purchased a textbook, follow along in your book. If you are using the Internet version, follow along on the webpage. If you have any difficulties, contact your instructor.  After the explanation of using your TI-85, turn off the video player and try to work the practice problems.  Once you have worked the problems, check your answers in the answer key listed in the back of your book or click on the Answer Key links at the end of every section.

Section Nine

Using the TI-85 to Add in Different Number Bases

The TI-85 is pre-set to work with decimal numbers. However, the calculator is able to work with other number systems as well. Since the digits used in both the binary number system and the octal number are subsets of the digits used in the decimal system, the calculator just needs to be told what number system you are working with. This is done on the MODE screen.

To access the MODE screen, use the following keystrokes:

The screen appears as follows:

 

9.1~ Adding in the Octal System

To work with octal numbers, we must access the Oct function on the calculator. After accessing the MODE screen, do this by using the following keystrokes:

¯ ¯ ¯ ¯ ¯ ® ®

The screen appears as follows:

Press the EXIT key to return to the home screen and you can now perform any operation in the octal system.

Example: Add 2477₈ + 3541₈ .

Solution: Make sure the calculator is in octal mode. Use the following keystrokes:

2 4 7 7 + 3 5 4 1

The screen appears as follows:

Notice the shaded letter "o" that appears after the number. This is a reminder to you that the sum that you have just found is an octal number. The calculator has shown that 2477₈ + 3541₈ = 6240 ₈

9.1~ Practice Problems

Directions: Turn off your VCR.  Add the following numbers in the octal system using your calculator.  When you have completed the problems, check your answers in the back of the book or by clicking the Answer key. button found on the left hand side of your screen.  Turn on your VCR to see the solutions to the practice problems worked on the board by a teacher.

1. 1345₈ + 5674₈
2. 324₈ + 432₈
3. 7777₈ + 1111₈
4. 76543₈ + 7777₈
5. 767676₈ + 767676₈

9.1~ Homework Problems

Directions: Turn off your VCR.  Add the following numbers in the octal system using your calculator.  Check your answers by using the Answer Key.   If you have any questions, contact your instructor.

1. 754₈ + 111₈
2. 754754₈ + 754754₈
3. 123234₈ + 123435₈
4. 547421₈ + 3456₈
5. 75747372₈ + 47474₈
6. 32743₈ + 13242₈
7. 176453₈ + 472174₈
8. 33334₈ + 3213222277₈
9. 7654321₈ + 1234567₈
10. 777777₈ + 7777777₈

 Turn on your VCR and continue with this section.  If you have purchased a textbook, follow along in your book. If you are using the Internet version, follow along on the webpage. If you have any difficulties, contact your instructor.  After the explanation of adding binary numbers, turn off the video player and try to work the practice problems.  Once you have worked the problems, check your answers in the answer key listed in the back of your book or click on the Answer Key links at the end of every section.

 9.2 ~ Adding Binary Numbers

Working with binary numbers is similar to working with octal numbers on the calculator. To work with binary numbers, we must access the Bin function on the calculator. After accessing the MODE screen (2^nd, MORE) do this by using the following keystrokes:

¯ ¯ ¯ ¯ ¯ ®

The screen appears as follows:

Press the EXIT key to return to the home screen and you can now perform any operation in the octal system.

Example 1: Add 1₂ + 1₂ .

Solution: Make sure the calculator is in binary mode. Use the following keystrokes:

1 + 1

The screen appears as follows:

Notice the shaded letter "b" that appears after the number. This is a reminder to you that the sum that you have just found is an binary number. The calculator is showing that 1₂ + 1₂ = 10₂

Example 2: Add 111₂ + 1011₂ .

Solution: Make sure the calculator is in binary mode. Use the following keystrokes:

1 1 1 + 1 0 1 1 1

The screen appears as follows:

Notice the shaded letter "b" that appears after the number. This is a reminder to you that the sum that you have just found is a binary number. The calculator is showing that 111₂ + 10111₂ = 11110₂

Any binary number can be added very easily in this manner. Remember after working with binary numbers to change your calculator back to decimal notation by using the MODE screen.

9.2 ~ Practice Problems

Directions: Turn off your VCR.  Add the following numbers in the binary system using your calculator.  Check your answers in the Answer Key. Turn your VCR back on to view the solutions to the practice problems.

1. 1111₂ + 11₂
2. 100011₂ + 111₂
3. 100₂ + 10111₂
4. 101010₂ + 11111₂
5. 101010₂ + 1010100₂

9.2 ~ Homework Problems

Directions: Turn off your VCR.  Add the following numbers in the binary system using your calculator.  Check your answers in the Answer Key.  If you have any questions, contact your instructor.

1. 101₂ +110₂
2. 1110₂ + 1101₂
3. 1110₂ + 1111₂
4. 10₂ + 1011₂
5. 1111₂ + 10111₂
6. 111₂ +11₂
7. 110₂ + 10001₂
8. 11110₂ + 10111₂
9. 10₂ + 111110₂
10. 1111₂ + 1111111₂

 Turn on your VCR and continue with this section.  If you have purchased a textbook, follow along in your book. If you are using the Internet version, follow along on the webpage. If you have any difficulties, contact your instructor.  After the explanation of adding hexadecimal numbers, turn off the video player and try to work the practice problems.  Once you have worked the problems, check your answers in the answer key listed in the back of your book or click on the Answer Key links at the end of every section.

 9.3 ~ Adding Hexadecimal Numbers

Working with hexadecimal numbers when the digits used in the numbers are 0 through 9 is similar to working with octal numbers on the calculator. To work with hexadecimal numbers, we must access the Hex function on the calculator. After accessing the MODE screen (2^nd MORE) do this by using the following keystrokes:

¯ ¯ ¯ ¯ ¯ ® ® ®

The screen appears as follows:

Note that the numbers that are used in the hexadecimal system are 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. The letters A-F cannot be accessed using the ALPHA keys since A-F in this case represent NUMBERS, not letters. The numbers A-F are accessed through the F1 key in the BASE sub-menu. To access the BASE sub-menu use the following keystrokes:

 

 

The screen appears as follows:

Once this screen is shown on the calculator, the letters A-F can be accessed by pressing the F1 key.

 

The screen appears as follows:

By using the function keys, the hexadecimal values A-F can be accessed. To access A press 2^nd F1, to access B press F1, to access C press F2, to access D press F3, to access E press F4, to access F press F5.

Example: Find the sum of the hexadecimal numbers A14 and BD2.

Solution: The first step is to access the mode screen and highlight "Hex" so that the calculator is ready to accept hexadecimal numbers and calculate an answer in the hexadecimal system. The next step is to access the special hexadecimal numbers A-F. Do this by first accessing the BASE submenu by pressing 2^nd 1 and then F1. The calculator is now ready to accept hexadecimal numbers for addition.

The next step is to enter the addends with the plus sign between them by using the following keystrokes:

2nd F1 1 4 + F1 F3 2

 

The screen appears as follows:

The solution to the problem A14₁₆ + BD2₁₆ is 15E6₁₆. Note the h immediately following the answer 15E6 on the screen. This h signifies that the answer is a hexadecimal number.

The mode screen must always be set to the number system in which the answer is required. Any operation (addition, subtraction, multiplication, division, roots, and powers) can be performed by the calculator. Always remember to set the calculator back to decimal notation when you are finished working in another number system.

9.3 ~ Practice Problems

Directions: Turn off your VCR.  Add the following numbers in the hexadecimal system using your calculator.  Check your answers in the Answer Key.  Turn your VCR back on to view the solutions to the practice problems.

1. 169₁₆ + 157₁₆
2. 191₁₆ + 199₁₆
3. ABC₁₆ + DEF1₁₆
4. A2D2₁₆ + A1B2₁₆
5. FFFF₁₆ + 1111₁₆

9.3 ~ Homework Problems

Directions: Turn off your VCR. Add the following numbers in the hexadecimal system using your calculator.  Check your answers in the Answer Key.  If you have any questions, contact your instructor.

1. 987₁₆ +110₁₆
2. A2B4₁₆ + 1A01₁₆
3. 1110₁₆ + DEF₁₆
4. 19₁₆ + 1AC71₁₆
5. DE₁₆ + 1A711₁₆
6. DDD₁₆ +EE11₁₆
7. DAD₁₆ + CAB₁₆
8. BABE₁₆ + BAF₁₆
9. 99₁₆ + 19785₁₆
10. 97865₁₆ + 879432₁₆

 Turn on your VCR and continue with this section.  If you have purchased a textbook, follow along in your book. If you are using the Internet version, follow along on the webpage. If you have any difficulties, contact your instructor.  After the explanation of converting from one number base to another, turn off the video player and try to work the practice problems.  Once you have worked the problems, check your answers in the answer key listed in the back of your book or click on the Answer Key links at the end of every section.

9.4 ~ Converting From One Number Base to Another

The CONV sub-menu will convert any number in any base to an equivalent number in another base. There are two different methods that you can use to do this. The easiest way to perform the conversion is to make sure that the MODE screen is set to the system of the given number. For example, if you wish to convert a binary number to a decimal number, make sure the MODE screen is set to bin.

Example 1: Determine the decimal equivalent of 1001 base two.

Solution 1: Make sure that the MODE screen is set to the binary number system, EXIT the mode screen, then use the following keystrokes 2nd 1 F3 to access the CONV submenu. Enter in the binary value to be expressed as a decimal number and obtain the solution by using the following keystrokes:

1 0 0 1 F4

The screen appears as follows:

Therefore, 9 in the decimal system is the same as 1001 in the binary system.

The same problem can be done with the calculator set in the decimal system mode. If the calculator is set in the decimal mode, the binary number must be entered with the letter b directly after the numbers (accessed through the TYPE submenu, NOT the alpha character b). Do this with the following keystrokes:

2nd 1 F2 1 0 0 1 F1 EXIT F3 F4 ENTER

The screen appears as follows:

This again shows that the binary number 1001 is the same as the decimal number 9. Any number in any number system can be changed from one number system to another in a similar fashion.

Example: Determine the binary equivalent of 1A2 base sixteen.

Solution: Make sure that the MODE screen is set to the hexadecimal number system, EXIT the mode screen, then use the following keystrokes 2nd 1 to access the BASE menu. Enter in the hexadecimal value to be expressed as a binary number and obtain the solution by using the following keystrokes:

1 F1 2^nd F1 2 EXIT F3 F1 ENTER

The screen appears as follows:

Therefore, 1A2 in the hexadecimal system is the same as 110100010 in the binary system.

There is an alternate way of performing this same addition when the MODE screen is set to the decimal system. If this is the case, the computer must be told that the number entered is not a decimal number and that the equivalent value for the number is not to be found in the decimal system. Using the same example of converting the hexadecimal number 1A2 to a binary number can be done by entering the following keystrokes when the calculator is in the decimal mode:

2^nd 1 F1 1 2^nd F1 2 EXIT F2 F2 EXIT F3 F1 ENTER

The screen appears as follows:

This method shows that the hexadecimal number 1A2 has a binary equivalent of 110100010.

Any problem can be done in any number system if the TYPE submenu is used along with the CONV menu. The answer will always have a b, d, h or o after it if the CONV sub-menu is used to convert any number to a number that is in a different number system than that of the mode. Remember, when in doubt, put in the extra characters so that the answer will always be correct.

9.4~ Practice Problem

Directions: Turn off your VCR.  Perform the following conversions using your calculator.  Check your answers in the Answer Key.  Turn your VCR back on to view the solutions to the practice problems.

1. Convert 1111₂ to a decimal number
2. Convert 100011₂ to an octal number
3. Convert 1745₈ to a hexadecimal number
4. Convert ABC₁₆ to a decimal number
5. Convert the decimal number 875 to a binary number.

9.4 ~Homework Problems

Directions: Turn off your VCR.  Perform the following conversions using your calculator. Either method can be used.  Check your answers in the Answer Key.  Contact your instructor if you are having difficulty working these problems.

1. Convert 1101₂ to a decimal number
2. Convert 111110₂ to an octal number
3. Convert 111011110₂ to a hexadecimal number
4. Convert 43510₈ to a binary number
5. Convert 753241₈ to a hexadecimal number
6. Convert 437777₈ to a decimal number
7. Convert 11AB0₁₆ to a decimal number
8. Convert CAB₁₆ to a binary number
9. Convert AB4CD₁₆ to an octal number
10. Convert 100111₂ to a hexadecimal number

 Turn on your VCR and continue with this section.  If you have purchased a textbook, follow along in your book. If you are using the Internet version, follow along on the webpage. If you have any difficulties, contact your instructor.  After the explanation of Boolean Operators, turn off the video player and try to work the practice problems.  Once you have worked the problems, check your answers in the answer key listed in the back of your book or click on the Answer Key button to the left.

9.5 ~ Boolean Operators

The BOOL sub-menu accesses the Boolean operators and, or, xor, and not.

When the calculator is in the binary mode, the operators can be used to determine the truth value of a statement. The BOOL submenu is found under the BASE menu. First access the base menu by using the keystrokes 2^nd 1. Then hit F4.

The screen appears as follows:

Example 1: Perform the indicated operation: 11 and 10

Solution: Once the BOOL submenu is accessed, use the following keystrokes:

1 1 F1 1 1 1 ENTER

The screen appears as follows:

This means that 11 and 10 yields 10.

Example 2: Perform the indicated operation:

Solution: Once the BOOL submenu is accessed, use the following keystrokes:

1 1 F2 1 0 ENTER

The screen appears as follows:

This means that 11 or 10 results in an answer of 11.

Example 3: Perform the indicated operation 111 xor 001

Solution: Once the BOOL submenu is accessed, use the following keystrokes:

1 1 1 F3 0 0 1 ENTER

The screen appears as follows:

This means that 111 xor 001 results in an answer of 110. Of note is the fact that if the answer results in an initial zero in the answer, the zero will not appear on the screen.

9.5 ~ Practice Problems

Directions: Turn off your VCR.  Perform the indicated operations using the BOOL submenu on the calculator.  Check your answers in the Answer Key. Turn your VCR back on to view the solutions to the problems.

1. 101 or 101
2. 111 or 111
3. 101 and 101
4. 111 and 111
5. 101 xor 101
6. 111 xor 111

9.5 ~ Homework Problems

Directions: Turn off your VCR.  Perform the indicated operations using the BOOL submenu on the calculator.  Check your answers in the Answer Key. If you have any questions, contact your instructor.

1. 11 or 10
2. 1010 or 1110
3. 1111 or 1001
4. 1011 and 1111
5. 1110 and 1100
6. 10101 and 11100
7. 10101 xor 11000
8. 11111 xor 11100
9. 101010 xor 100100
10. 1110001 xor 1100110

This Concludes Session Nine