Farewell to Mars. Aside from lots of iron oxide and some terrific a capella groups, they haven't much to offer us ten-fingered folk. There are some similarly odd number bases in use here, and I'd like to take a quick detour through one that occupies a separate world right here on Earth: The world of Digital Equipment Corporation, better known as DEC.
Back in the '60s, DEC invented the minicomputer as a challenger to the massive
mainframes pioneered by IBM. To ensure that no software could possibly be moved from an IBM mainframe to a DEC minicomputer, DEC designed its machines to understand only numbers expressed in base 8.
Let's think about that for a moment, given our experience with the Mar-tians. In base 8, there must be eight digits. DEC was considerate enough not to invent their own digits, so what they used were the traditional digits from 0 through 7. There is no digit 8 in base 8!
That always takes a little getting used to, but it's part of the definition of a number base.
DEC gave a name to its base 8 system: octal.
A columnar number in octal follows the rule we encountered in thinking about the Martian system: Each column has a value 8 multiplied by that of the column to its right.
Who Stole 8 and 9?
Counting in octal starts out in a very familiar fashion: One, two, three, four, five, six, seven...ten.
This is where the trouble starts. In octal, ten comes after seven. What happened to eight and nine? Did the Grinch steal them? (Or the Martians?) Hardly. They're still there—but they have different names. In octal, when you say "ten" you mean "eight." Worse, when you say "eleven" you mean "nine."
Unfortunately, what DEC did not do was invent clever names for the column values. The first column is, of course, the units column. The next column to the left of the units
column is the tens column, just as it is in our own decimal system. But here's the rub, and the reason I dragged Mars into this: Octal's "tens" column actually has a value of 8.
A counting table will help. Table 1.3 counts up to thirty octal, which has a value of 24 decimal. I dislike the use of the terms eleven, twelve, and so on in bases other than ten, but the convention in octal has always been to pronounce the numbers as we would in
decimal, only with the word "octal" after them.
Remember, each column in a given number base has a value base multi-plied by the
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column to its right, so the tens column in octal is actually the eights column. (They call it the tens column because it is written 10, and pronounced "ten.") Similarly, the column to the left of the tens column is the hundreds
Table 1.3.
Counting
in octal, base 8
Octal Numerals Pronunciation Decimal Equivalent
0 Zero 0
1 One 1
2 Two 2
3 Three 3
4 Four 4
5 Five 5
6 Six 6
7 Seven 7
10 Ten 8
11 Eleven 9
12 Twelve 10
13 Thirteen 11
14 Fourteen 12
15 Fifteen 13
16 Sixteen 14
17 Seventeen 15
20 Twenty 16
21 Twenty-one 17
22 Twenty-two 18
23 Twenty-three 19
24 Twenty-four 20
25 Twenty-five 21
26 Twenty-six 22
27 Twenty-seven 23
30 Thirty 24
column (because it is written 100) but the hundreds column actually has a value of 8
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multiplied by 8, or 64. The next column over has a value of 64 multiplied by 8, or 512, and the column left of that has a value of 512 multiplied by 8, or 4096.
This is why if someone talks about a value of "ten octal" they mean 8; "one hundred octal"
they mean 64, and so on. Table 1.4 summarizes the octal column values and their decimal equivalents.
A digit in the first column (the units, or 1's column) tells how many units are contained in the octal number. A digit in the next column to the left, the tens column, tells how many 8's are contained in the octal number. A digit in the third column, the hundreds column, tells how many 64's are in the number, and so on. For example, 400 octal means that the number contains 4 64's; that is, 256 in decimal.
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It works the same way it does in Martian, or decimal, or any other number base. In general: Each column has a value consisting of the number base raised to the power
represented by the ordinal position of the column minus one. That is, the value of the first column is the number base raised to the 1-1, or 0, power. Since any number raised to the zero power is one, the first column in any number base always has the value of one and is called the units column. The second column has the value of the number based raised to the 2—1, or 1st power, which is the value of the number base itself. In octal this is 8; in decimal, 10; in Martian base fooby, fooby. The third column has a value consisting of the number base raised to the 3-1, or 2nd power, and so on.
Within each column, the digit holding that column tells how many in-stances of that column's value is contained in the number as a whole. Here, the 6 in 76225 octal tells us that there are six instances of its column's value in the total value 76225 octal. The six occupies the fourth column, which has a value of 84-1, which is 83, or 512. This tells us that six 512 values are in the number as a whole.
You can convert the value of a number in any base to decimal (our base 10) by
determining the value of each column in the alien base, then multiplying the value of each column by the digit contained in that column, (to create the decimal equivalent of each digit) and then finally taking the sum of the decimal equivalent of each column. This is done in Figure 1.2, and the octal number and its decimal equivalent are both shown.
Now that we've looked at columnar notation from both a Martian and an octal perspective, make sure you understand how columnar notation works in any arbitrary base before we go on.