You are correct. Just as 778 is as high as we can get in the octal system with 2 digits, 7778 is as high as we can get with 3 digits, so we must go on to the next power of 8 to go beyond 7778. By expanding 7778 and10008 you will find that 7778 = 51110 and 10008 = 51210. This indicates that 7778 and 10008 are consecutive numbers, just as 51110 and 51210 are.
We have learned that the octal number system is built on the same principle as the decimal system, except that powers of 8 are used as the underlying value for each digit instead of powers of 10. The position of the coefficients with relation to the octal point tells us to which power of 8 they refer. These coefficients are written with the subscript 8 to indicate the value of the octal number.
5, 3608 = 5(83) + 3(82) + 6(81) + 0(80).
Now, what does 53608 mean? We’re used to thinking in values of 10’s and 100’s and 1,000’s. How can we assign a value to a number based on 8’s and 64’s and 512’s? Converting octal numbers to decimal numbers is done by expansion, remembering that the digits of an octal number are simply coefficients of powers of 8. Following this method, 1238 is 1(82) + 2(81) + 3(80), or 53110.
In octal fractions, the digits are related to negative powers of 8. The first digit to the right of the octal point refers to 8 − 1, the next digit to the right refers to 8 − 2, and so on.
If we wish to convert
.3218 to a decimal fraction by expansion, we proceed in this way:
.3218
= 3(8 − 1) + 2(8 − 2) + 1(8 − 3)
= 3⎛⎝(1)/(8)⎞⎠ + 2⎛⎝(1)/(64)⎞⎠ + 1⎛⎝(1)/(512)⎞⎠.
Now,
(1)/(8) = .125;
(1)/(64) = .015625;
(1)/(512) = .001953125.
3 × .125
=
.375
2 × .015625
=
.031250
1 × .001953125
=
.001953125
Total
.408203125
or
.3218 = .40820312510.
Counting in the octal system is similar to counting in the decimal system.
When we count in either octal or decimal, we have a limited number of digits to work with. the ten digits used in the decimal system are, as you know, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The eight digits used in the octal system are 0, 1, 2, 3, 4, 5, 6, and 7.
When we want to count beyond 9 in the decimal system we start using combinations of digits in a very straightforward way. We have to use as additional number-place to the left, and start using the same symbols over again. Thus the number after 9 is 10.
Mathematics is an orderly business. Just as 7 is greater than 6, so 17 is greater than 16. Because you know how to count in the decimal system, you can inspect any two numbers and tell which one is larger, although you may never have seen those exact numbers before. If this seems ridiculously obvious, it’s a tribute to the elegance of the number system and the thoroughness with which we are trained to use numbers. Historically, things weren’t always so clear.
The octal system works just like the decimal system, except that there are just eight digits instead of ten. The highest single digit in each system is one less than the base of the system. When we count above 7 in the octal system, we add a number-place to the left. There is no single digit meaning “eight” in the octal system, so the number after 7 is 10.
When we see the quantity 108, we know it means eight octal units: 18 + 18 + 18 + 18 + 18 + 18 + 18 + 18.
In octal, 778 is followed by 1008, just as 99 is followed by 100 in decimal.
You can always check your counting accuracy in octal by converting to the decimal system:
178
= 1(81) + 7(80) = 1(8) + 7(1) = 8 + 7 = 1510;
208
= 2(81) + 0(80) = 2(8) + 0(1) = 16 + 0 = 1610.
Since 16 follows 15 in decimal counting, the expansion above demonstrates that 208 follows 178 in octal counting.
In the next lesson, we shall take up a particularly neat method of converting decimal numbers to octal numbers. But first, go to the Self-Test Questions on Lesson 6. You’ll find them in the Appendix.
