# What exactly is base 8?

222 views

I saw people making a joke about this in a comment section and I couldn’t seem to understand what they were talking about. Can someone fill me in?

In: Mathematics

we use base 10, which means our number system has 10 digits, 0-9

Base 8 has 8 digits, 0-7 (Base 2 uses 0-1, Base 16 uses 0-F, etc)

counting in base 8 goes 0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20…

11 in base 10 means (1 * 10^1 + 1*10^0) == 10 + 1 == 11

11 is base 8 means (1*8^1 + 1*8^0) == 8 + 1 == 9

Examples

Base 10 is:

0 1 2 3 4 5 6 7 8 9

Base 8 is:

0 1 2 3 4 5 6 7

There is no “8” or “9” in Base 8, but the rest of arithmetic is the same.

The number 8 in Base 8 is “10”. You run out of digits so you shift one over and start counting. The number 12 in Base 8 is “14”.

The regular decimal number system you use is base 10. This is because there are ten different digits that is used to represent numbers 0,1,2,3,4,5,6,7,8,9. If you want to count higher you have to start using two digits, 10,11,12,13, etc. Base 8 only use eight digits instead of ten. So you count 0,1,2,3,4,5,6,7 and then start using two digits 10,11,12, etc. This means that 10 in the decimal system is the same as 12 in base 8 and so on. Base 8 was used in some older computer systems because there is exactly three bits per digit. So it is very easy to convert between base 8 numbers and binary numbers. But base 8 lost out to base 16 which added another six digits by using letters from the alphabet and is exactly four bits per digit.

Let’s look at Base 10, which is the number system we normally use. It uses the digits 0-9. When we write a number like 234, what we actually means is 2 100’s, 3 10’s, and 4. because it’s base 10, the first digits go up to 9 and the next digits are 10, 10×10 (100), 10x10x10 (1000), and so on.

In Base 8, we use the digits 0-7. Instead of using the digit for 8, we use 10 (1 eight and 0 singles). In base 8, the number 234 would mean 2 64’s (8×8), 3 8’s, and 4.

If we were using, say, base 12, we would use the digits 0-B. 10 in base 12 would be A. In base 12, the number 234 would mean 2 144s (12×12), 3 12’s, and 4.

Base 8 is, essentially, what cartoon characters, who have three fingers and a thumb on each hand, would use.

Our system is base ten – both hands’ worth gets “chunked” into the next unit over. Ten ones is a ten, ten tens is a hundred, etc.

Theirs would use digits 0,1,2,3,4,5,6, and 7; two hands’ worth would be eight, and they’d write that as “one eight plus none” the way we write “one ten plus none”. It would look the same: “10”.

If they wrote “324”, it would mean 3 sixty-fours, 2 eights, and 4; in base ten those add up to 212.

There is a relevant Tom Lehrer song, [New Math](https://www.youtube.com/watch?v=W6OaYPVueW4).

–Dave, which actually covers it pretty well