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Session Two
Welcome to the second section of MTH140, Number bases. In this session you will learn about the place values in the octal system as well as how to convert octal numbers from standard notation to expanded notation.
 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 place values, 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.
Section One
Octal Numbers
The octal system that is a number system frequently used in computer applications. This system is based on 8 digits. The digits 0, 1, 2, 3, 4, 5, 6, and 7 are used in the octal system. When counting, once all of the digits have been used, a combination of digits must then be used. As digits are added, place value increases. For example, the "number" after 7 is 10. Two digits are used in the number 10; the digit placement in the number tells us the "size" of the number. The digits are used sequentially when we build numbers. The number after 17 is 20. This is because the digit on the far right has reached the maximum of value of 7 and we must begin to "reuse" values in this position. We begin counting in the octal system just a we would begin in the whole numbers with the number 0. We would count: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, and so on. Because it can be difficult to distinguish between a decimal number and an octal number, the octal numbers are usually written with a subscript to clearly identify the number as a member of the octal system. Thus, we could write: 18, 28, 38, 48, 58, 68, 78, and 108. We will continue to use the subscripts to identify the number as an octal numbers in this section.
2.1 ~ Converting from Standard Notation to Expanded Notation
The value of a number is determined by the position of each digit within the number. The "place" of the digit has a name. In the number 1458, we refer to the position where the 5 is as the "units place" or as "eight to the zero power place". The 4 is in the "eights" place or the "eight to the first power" place. The 5 is in the "sixty-fours" place or the "eight to the second power" place. Each additional place would be "eight" to the next successive power.
The number 1458 is in standard octal notation. If we looked at the number 1458 and wrote it in a different way, we might put the information in a table such as the one shown in Table 2.1.1.
Table 1 ~ 2.1
|
Digit |
x |
Power of Eight |
= |
Value |
|
5 |
x |
1 or 80 |
= |
58 |
|
4 |
x |
8 or 81 |
= |
408 |
|
1 |
x |
64 or 82 |
= |
1008 |
|
Total Value |
= |
12345 |
Notice that numbers in the value column are all multiples of eight. If we wrote this number without using the chart, it would be written as:
5 x 1 + 4 x 8 + 1 x 64
Another way of writing this would be:
5 x 80 + 4 x 81 + 1 x 82
Since the number 8 is represented in octal notation by 108, we could write the expanded notation in another way. It could be written as:
5 x 100 + 4 x 101 + 1 x 102
or putting in the subscript to identify the number as a base eight number, it could be written as:
5 x 1080 + 4 x 1081 + 1 x 1082
We call this expanded notation.
Notice the similarities between what has been written above and the manner in which decimal numbers are written in expanded notation. They are the same except for the subscript.
Figure 1 ~ 2.1
Place Values for Octal Number 1458
1 4 5
| | |__________1080
| | _________________1081
|________________________1082
Example: Convert 47158 to expanded notation.
Solution: First, identify the value of each digit. The first digit on the right, the digit 5, is in the units place or the eight to the zero power place; the digit 1 is in the eight to the first power place; the next digit 7 is in the eight to the second power place; the final digit 4 is in the eight to the third power place.
Table 2 ~ 1.2
|
Digit |
x |
Power Of Eight |
= |
Value |
|
5 |
x |
1 or 80 or 1080 |
= |
58 |
|
1 |
x |
8 or 81 or 1081 |
= |
108 |
|
7 |
x |
64 or 82 or 1082 |
= |
7008 |
|
4 |
x |
512 or 83 or 1083 |
= |
40008 |
|
Total Value |
= |
47158 |
If we wrote the number without using the chart, the number would be
5 x 80 + 1 x 81 + 7 x 82 + 4 x 83
The number is now in an expanded form. Since the number 8 in octal notation is represented by 108, we could write the expanded form in another way. It could be written as:
5 x 100 + 1 x 101 + 7 x 102 + 4 x 103
or putting in the subscript, it could be written as:
5 x 1080 + 1 x 1081 + 7 x 1082 + 4 x 1083 .
Now the number is in expanded notation.
Figure 2 ~ 1.2
Place Values for Octal Number 47158
4 7 1 5
| | | |_________________1080
| | | ________________________1081
| |________________________________1082
|_______________________________________1083
2.1~ Practice Problems
Directions: Turn off your VCR. Change the following numbers from standard notation to expanded notation. When you have finished working the problems, check your answers in the back of the book or click here to view the Answer Key. Turn on the VCR to view the solutions to the practice problems.
- 5638
- 720148
- 328
- 68
- 713054628
2.1~ Homework Problems
Directions: Turn off your VCR. Convert the following numbers from standard notation to expanded notation. Check your answers in the back of the book or click here to view the Answer Key. If you have any questions, contact your instructor.
- 57018
- 123658
- 1117648
- 758
- 1068
- 5778
- 725034618
- 51703218
- 58
- 777678
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 expanded notation to standard notation, 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.
2.2 ~ Converting from Expanded Notation to Standard Notation
We can change numbers that have been written in expanded notation to standard notation by reversing the process. Suppose we have the notation
5 x 80 + 4 x 81 + 1 x 82
Recalling the information in Figure 2.1.1 and extending the idea, we could make a table for the powers of eight in octal notation.
Table 2 ~ 2.1
Table of Powers of 8 vs. octal Notation
|
Power of Eight |
Octal Notation |
|
80 |
18 |
|
81 |
108 |
|
82 |
1008 |
|
83 |
10008 |
|
84 |
100008 |
|
85 |
1000008 |
|
86 |
10000008 |
|
87 |
100000008 |
|
88 |
1000000008 |
|
89 |
10000000008 |
|
810 |
100000000008 |
We might think of our number in another way as
5 x 1080 + 4 x 1081 + 1 x 1082
Table 2 ~ 2.2
|
Digit |
x |
Power of Eight |
= |
Value |
|
5 |
x |
1080 (or 12) |
= |
48 |
|
4 |
x |
1081 (or 108) |
= |
508 |
|
1 |
x |
1082 (or 1008) |
= |
1008 |
|
Total Value |
= |
1458 |
Adding these values together, we get 1458 in standard notation.
Example: Convert 5 x 1080 + 1 x 1081 + 7 x 1082 + 4 x 1083 to standard notation.
Solution: If we follow the rules of order of operations, we must first evaluate the powers of eight.
5 x 18 + 1 x 108 + 7 x 1008 + 4 x 10008
Next, we perform the multiplication.
58 + 108 +7008 + 40008
Finally, we add the numbers together to obtain the sum:
47158
which is the number in standard form.
2.2 ~ Practice Problems
Directions: Turn off your VCR. Change the following numbers from expanded notation to standard notation. When you have finished working the problems, check your answers in the back of the book or click here to view the Answer Key. Turn on the VCR to see the solutions to the practice problems.
- 5 x 1080 + 6 x 1081 + 3 x 1082
- 4 x 1080 + 1 x 1081 + 0 x 1082 + 2 x 1083 + 7 x 1084
- 2 x 1080 + 3 x 1081
- 6 x 1080
- 2 x 1080 + 6 x 1081 + 4 x 1082 + 5 x 1083 + 3 x 1085 + 1 x 1086 + 7 x 1087
2.2 ~ Homework Problems
Directions: Turn off your VCR. Convert the following numbers from expanded notation to standard notation. Check your answers in the back of the book or click here to view the Answer Key. If you have any questions, contact your instructor.
- 7 x 1080 + 3 x 1081 + 0 x 1082 + 4 x 1083
- 1 x 1080 + 5 x 1081 + 2 x 1083 + 1 x 1085
- 7 x 1081
- 3 x 1080 + 7 x 1081 + 3 x 1082 + 6 x 1083 + 2 x 1084
- 3 x 1080
- 2 x 1080 + 4 x 1081 + 6 x 1082 + 0 x 1083 + 7 x 1084 + 5 x 1085 + 3 x 1086
- 1 x 1080 + 1 x 1081 + 1 x 1084
- 6 x 1082 + 4 x 1083 + 6 x 1084 + 5 x 1085
- 3 x 1083 + 6 x 1084
- 6 x 1087
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 numbers in the octal system, 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.
2.3 ~ Adding Numbers in Octal Notation
Addition in base eight is very similar to addition in base ten. The first step is learning the addition facts for base eight. Since there are only eight digits used in the octal system, there are only a handful of different mathematics facts that we must know. When we add any two digits together and their sum is less than the decimal number eight, the sum is the same in both bases. The first eight facts in the octal system are the same as in the decimal system, that is 0 + 0 = 0, 0 + 1 = 1, 0 + 2 = 2, 0 + 3 = 3, 0 + 4 = 4, 0 + 5 = 5, 0 + 6 = 6, 0 + 7 = 7. (Of course the reflexive property of addition holds so that 1 + 0 = 1, 2 + 0 = 2, and so on.) We know that the equivalent of the decimal number 8 is the octal value 108. Consequently, 7 + 1 = 10 in base eight. As with addition base ten, a table can provide easy reference to the addition facts in base eight. We can refer to the addition facts for base eight depicted in Table 2.3.1 as we work our examples.
Table 2.3.1 Addition Table Base Eight (Octal System)
|
+ |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
0 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
1 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
10 |
|
2 |
2 |
3 |
4 |
5 |
6 |
7 |
10 |
11 |
|
3 |
3 |
4 |
5 |
6 |
7 |
10 |
11 |
12 |
|
4 |
4 |
5 |
6 |
7 |
10 |
11 |
12 |
13 |
|
5 |
5 |
6 |
7 |
10 |
11 |
12 |
13 |
14 |
|
6 |
6 |
7 |
10 |
11 |
12 |
13 |
14 |
15 |
|
7 |
7 |
10 |
11 |
12 |
13 |
14 |
15 |
16 | |
Example 1: Add: 138 + 6548
Solution: The first step is to "line up" the digits, like we do for decimal numbers. We want to make sure that all of the digits in the same place value are beneath one another.
138
+ 6548
We start the addition process from the right, just as if we were adding decimal numbers. We know that three plus four equals seven in the octal number system. We can write down the seven in the units place; we have no numbers to "carry."
138
+ 6548
7
The next step is to add the one and the five. The sum of one and five is six in the octal number base. We write down the six in the eights place and have no numbers to carry.
138
+ 6548
67
The final step is to look at the eight to the second power place. Since we have not carried any numbers and there is no value in the eight to the second power place in the first addend, all we need to do is bring down the six to obtain the answer.
138
+ 6548
6678
This shows that the sum of 138 plus 6548 equals 6678 .
Example 2: Add: 4028 + 6438
Solution: The first step is to "line up" the digits, like we do for decimal numbers. We want to make sure that all of the digits in the same place value are beneath one another.
4028
+ 6438
We start the addition process from the right, just as if we were adding decimal numbers. We know that two plus three equals five in the octal number system. We can write down the five in the units place; we have no numbers to "carry."
4028
+ 6438
5
The next step is to add the four and the zero. We know that their sum is four. We write down the four in the eights place and have no numbers to carry.
4028
+ 6438
45
The final step is to look at the eight to the second power place. Since we have not carried any numbers we must just add four and six. Looking at the Table 2.3.1, we see that in the octal system 4 + 6 = 12. So, we can write 12 to obtain the answer.
4028
+ 6438
12458
This shows that the sum of 4028 plus 6438 equals 12458 .
Example 3: Add: 22458 + 5338
Solution: The first step is to "line up" the digits, like we do for decimal numbers. We want to make sure that all of the digits in the same place value are beneath one another.
22458
+ 5338
We start the addition process from the right, just as if we were adding decimal numbers. We know that five plus three equals 10 from Table 2.3.1. Since there is room for only one digit in the sum line, we must begin to carry numbers. When we carry, we use the same process as the decimal system, that is, we write down the digit on the right, in the units place, and carry the digit on the left (in this case, in the eights place.) We can write down the zero in the units place and carry the one.
1 "carried one"
22458
+ 5338
0
The next step is to add the four and the three. We know that their sum is seven from Table 2.3.1. Now we must add the sum that we have just obtained with the "carried" one. Seven plus one is 10. We write down the zero in the eights place and carry the one to the eight to the second place.
1 "carried one"
22458
+ 5338
00
The next step is to add the two and the five. We know that their sum is seven. Now we must add the sum that we have just obtained with the "carried" one. Seven plus one is 10. We write down the zero in the eight to the second power place and carry the one to the eight to the third place.
1 "carried one"
22458
+ 5338
000
The final step is to look at the eight to the third power place. We must add one and two to get 3. We write down the 3 to obtain the answer.
1 "carried one"
22458
+ 5338
30008
This shows that the sum of 22458 plus 5338 equals 30008 .
Example 4: Add: 62458 + 7478
Solution: The first step is to "line up" the digits.
62458
+ 7478
We start the addition process from the right. Five plus seven is 14. We write down the four and carry the one.
1 "carried one"
62458
+ 7478
4
The next step is to add the four and the four in the eights place. We know that their sum is 10. At this point, you may wish to carry the one to the next place.
1 1 "carried ones"
62458
+ 7478
4
Now we must add the remaining zero with the carried one. Since 0 + 1 = 1, we write down the one in the eights place.
1 "carried one"
62458
+ 7478
14
The next step is to add the two and the seven. We know that their sum is 11 from Table 2.3.1. Carry the one.
1 1 "carried ones"
62458
+ 7478
14
Now add the one left with the carried one to obtain a value of 2 to write in the eight to the second power place.
1 "carried one"
62458
+ 7448
214
The final step is to look at the eight to the third power place. We must add six in the first addend with the carried one to get 7. We write down the 7 to obtain the answer.
1 "carried one"
62458
+ 7478
72148
This shows that the sum of 62458 plus 7478 equals 72148 .
2.3 ~ Practice Problems
Directions: Turn off your VCR. Add the following numbers in octal notation. Check your answers using the Answer Key. Turn your VCR back on to view solutions.
- 4028 +508
- 7158 + 2068
- 37708 + 1358
- 328 + 6578
- 77778 + 437408
2.3 ~ Homework Problems
Directions: Turn off your VCR. Add the following numbers in octal notation. Check your answers in the Answer Key. If you have any questions, contact your instructor.
- 6118 + 7058
- 45278 + 50228
- 52748 + 63428
- 478 + 31258
- 27718 + 436128
- 7678 + 268
- 7348 + 434458
- 434308 + 236058
- 408 + 1635248
- 35278 + 47405458
THIS CONCLUDES YOUR WORK ON SESSION TWO.
Last Updated January 2004
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