The use of the Octal system

Question

I was teaching the Octal number system and a student raised the question "Why should we use the octal number system, if we have binary and hexadecimal?" I was very confused, trying to explain the importance of this topic.

For context, I am teaching a course about the relationship between hardware and software.

Answer 1

The value might not be in knowing octal itself (although as noted in the comments, there is inherent relevance to it). Rather, it might be what learning different bases teaches about abstraction and the representations of numbers in general.

I think of this Essential Knowledge statement from the AP CS Principles curriculum:

EK 2.1.1G Numbers can be converted from any base to any other base.

The transition from decimal to binary is one step, and the move to hexadecimal can be an even bigger jump. To then have students see that there can be base-8 or base-32 or base-60 or base-37 is at times mind-blowing. The design of a particular base system, while not entirely arbitrary, is a design choice that comes with advantages and trade-offs.

Moreover, this process of conversion to and from different base systems strengthens students understanding of the underlying algorithmic process. The example I share with students is that 100 in any base system is equal to the value of that base squared. (Indeed it verges upon tautology.) Part of that is breaking down the idea of the 1s place being followed by the 10s followed by the 100s, etc. and replacing it with 10^0 followed by 10^1, etc. Replacing 10 with 2, 8, 16, or any other integer becomes trivial.

Having students see that 100 is not so much "one hundred" but a combination of base numbers and exponents and why that is is powerful.

Answer 2

Sometimes you just want to group your bits 3 at a time instead of 4.

Octal groups 3 bits per digit. Hexadecimal groups 4 bits per digit.

For example Unix file permissions are written in octal because each group of 3 bits represents read, write and execute for either the user, group, or "other people". That is, the permissions 110110000 represent permissions rw-rw--- on the mask rwxrwxrwx where the group of 3 maps to (user, group, other). You can write it as 110110000, octal 660 or symbolically ug=rw. 6 always means read, write, but not execute no matter what position it is in. The position always means who is affected. However writing it in hexadecimal would be useless: 1B0 doesn't visually indicate to the reader what the permissions are.

In hardware systems grouping 3 bits at a time makes sense when many controllers each have 3 bits of value. An octal number like 0123 means send 001 to the first controller, 010 to the second controller, and 011 to the third controller.

Older computers were not 16-, 32-, or 64-bit but instead were often 9-bit or 36-bit. In that case, octal makes more sense because each group of 3 bits was a digit.

Reading older code often requires knowing octal because older code uses octal for many of the reasons listed above.

EDIT --- and even with recent code, you may encounter problems as

 january = 01;
 february = 02;
 ...               // disaster waiting to happen

https://softwareengineering.stackexchange.com/questions/221797/reasoning-behind-the-syntax-of-octal-notation-in-java

Answer 3

It's necessary to know about octal because some compilers and tools will interpret multi-digit numbers with leading zeros differently to how many people will read them. Tracking down this sort of bug is very hard (as in spoken language where two people interpret words differently - maybe a useful analogue to work into a conversation).

I found two other relevant questions on SE: https://softwareengineering.stackexchange.com/questions/98692/where-are-octals-useful

https://stackoverflow.com/questions/2609426/in-what-situations-is-octal-base-used

The first identifies quite a lot of genuine legacy applications, which seem to be driven in the main by resource constraints which are not likely to be so relevant any more. The second question doesn't add anything more that I can see.

About the only time where Octal could be useful is in packing more separate items into a 32 bit register, and I'm struggling to decide if this is just obvious or actually a useful concept. It's closely related to ideas like video encoding, where RGB can be packed as 5,5,6 bits - but I'm not aware of this leading to the use of octal in the representations. Video is of course one of the areas today where resource constraints are very real (8.3 Mp per frame@60 Hz).

Answer 4

An octal representation is shorter, in characters, than a binary representation for data with any number of bits greater than one.

Unlike hexadecimal, an octal representation can be displayed using only numeric characters, of which certain old technologies (such as Nixie tubes and 7-segment LEDs) were capable (back, depending on the students age, perhaps in ancient history).

Answer 5

3 bits per octal digit was convenient on old time machines which used words of 12, 18, 36 bits words. Easier to do mental calculations than with hexadecimal.

For example, you have the PDP family : PDP-1 & 4 are 18 bit machines, PDP-2 24 bits, PDP-3 and -6 36 bits, PDP-5 12 bits, etc.

PDP-11 was a 16 bit machine, but as far as I remember the documentation explained low level stuff with octal numbers.

At the time, characters were often coded on 6 bits (IBM 704 BCD, DEC SIXBIT, ...)

See https://en.wikipedia.org/wiki/Six-bit_character_code

So the fundamental question is: why are you teaching numeration systems at all? Did you ever have to explain why

int month = 09;

is refused by the C compiler (and lots of modern languages), where as 07 and 10 are ok? And 012 is october?

You'll have to talk about octal numbers, even if you don't want to! :-)

Answer 6

One lesson that hasn't been mentioned yet is that the basic (manual) algorithms that students use for addition, subtraction, etc. are independent of base. In Octal, you just "carry" when you get to "eight" rather than the "ten" they use with decimal numbers. This is a unifying concept that is useful to know.

Not that it is especially relevant, but it is said that if you have horse that can write and ask it to write out "ten" it will write "22". Similarly, if you ask it to stamp its foot 10 times, you hear "stamp, stamp, stamp, stamp".

While the above paragraph is intended as a joke, it provides a certain insight into number bases that might actually be helpful to students.

I also find it interesting that different European languages have names for the smaller numbers up to some limit, but the limit differs by language.

German, like English, has number names for 1 - 12, with 13 being a compound(-ish) name thir-teen (three and ten), or dreizehn. French, on the other hand has names for 1 - 16, with dix-sept (ten and seven) being 17. But the Latin-derived languages (Romance Languages) aren't consistent in this. Italian, for example, only has individual names for 1-10, with eleven being undici (one and ten) and Spanish has names for 1-15 with 16 being dieciséis (ten and six). German uses null for 0, and the Romance languages seem to use some form of zero, perhaps accented, zéro in French. I haven't done a comprehensive search, however.

It may be that the names of these numbers derive from commercial considerations, early marketplaces/marketdays, for example, and so depend more on local culture. This is a bit distinct from their use in mathematics, which tends to be more uniform.

And note that English is a compound language with both Germanic and French roots and we take 1-12 from German, but 0 from French.

Answer 7

One vestige of octal notation is in UNIX (includes macosx) file permissions. You can say ($ is your system prompt)

$ chmod 644 file.txt

and change all rings (user, group, and other) permissions in a quick and compact way. The first argument to chmod is a three-digit octal number. The first digit represents user permissions, the second group, and the third others. Read permission has value 4, write 2, and execute 1. The combinations work as follows

0  none
1  x
2  w
3  wx
4  r
5  rx
6  rw
7  rwx

So, in the example above, the user has read/write permission, groups read, and others read.

Answer 8

As a step toward learning binary or hexadecimal. Show that we use many number bases everyday: base 24 and 60 in a clock; base 7 and base 30ish in a calender; base 16 and 14 in weights (in US); base 12 and 3 in distance (US); etc.

Then introduce octal. Octal has an advantage that it is not too different to Denary (Decimal), and that it dose not introduce new symbols (like hexadecimal).

After octal, head toward binary, maybe brefly explore the idea of other bases, and ask how low can we go? Binary is the limit.

Then binary to octal, but look at how this does not fit will in to 8 bit bytes. What can we do about this. Base 4 or base 16. Base 4 does not help us much, but is used by DNA. Base 16 helps us reduce number of digits, but we need 6 more symbols (a-f).