Numeral Systems
There’s Tens and Then There’s Tens
There are 10 kinds of people in the world: those who understand binary, and those who don’t.
If that was a groaner for you, I’m sorry. If it doesn’t make sense at all, hopefully it will by the end of this article, and subsequently, I’m sorry. We are going to work our way down from decimal (10) until we get to binary (2).
10
decimal
Everybody gets ten. It’s solid, it makes sense, it’s just there. Right between nine and eleven. But ten is special, because it is the one we use. It’s our base. In this article I’m going to look at a bunch of different types of tens so it’s important we understand this one. It might seem self-explanatory, but this is the lens through which we will look at all the other numeral systems.
In base-10, there are ten different symbols we combine to make up all other numbers, which are the Arabic numerals you are hopefully already familiar with. What I want you to focus on is what happens between the number 9 and the number 10. Nothing special, really, it’s just the next number in an infinite line… except that instead of using a single symbol to represent it, there are two: 1 and 0. We have decided that ten is where we move up a place, and start counting over. Ten, in our system, is what’s called the radix.
thousands hundreds tens ones
(10 x 10 x 10) (10 x 10) (10) (1)
v v v v
1000 100 10 12,048 -> two thousands, zero hundreds, four tens, and eight ones 2000
0
40
+ 8
-----
2048
Every time we move one place to the left in base-10, we multiply by ten.
9
nonary
Nonary isn’t useful for very much, but it will help show the difference between decimal and other systems. It’s simple, just imagine that the number 9 doesn’t exist. One, two, three, four, five, six, seven, eight… then what? How would we represent that value? We do what we would normally do, just a little early. 9 is now 10, which comes directly after 8. How this compares to decimal:
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23...
(9) 1 2 3 4 5 6 7 8 10 11 12 13 14 15 16 17 18 20 21 22 23 24 25...(10) 100 101 102 103... (10) 1000 1001 1002 1003...
(9) 121 122 123 124... (9) 1331 1332 1333 1334...
And the same example from above:
seven-twenty-nines eighty-ones nines ones
(9 x 9 x 9) (9 x 9) (9) (1)
v v v v
1000 100 10 1this number is the same as 2,048 above:2,725 -> two seven-twenty-nines (1458), seven eighty-ones (567), two nines (18), and five ones (5) 2 x 729 = 1458
7 x 81 = 567
2 x 9 = 18
+ 5 x 1 = 5
------------------
2048
It’s hard! Our brains are so used to ten being the base, other systems are hard to decode. But the more we look at the more we understand how numeral systems are built up from their base.
8
octal
Languages in use: Pame (Mexico), Yuki (California — extinct)
Not everybody uses ten as their base. Around the globe (and throughout history) there are many numeral systems in use besides decimal.
Octal was also widely used in early digital computing, as a condensed way to represent binary digits. This will make more sense later when we talk about binary. Now there is no 8 and no 9!
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22...
(8) 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 20 21 22 23 24 25 26...2,048 in octal => 4,000 4 x 512 = 2048
0 x 64 = 0
0 x 8 = 0
+ 0 x 1 = 0
------------------
2048
8,000 in octal => 17,466 1 x 4096 = 4096
7 x 512 = 3594
4 x 64 = 256
6 x 8 = 46
+ 6 x 1 = 6
-------------------
17466
Huh, that’s odd, isn’t it? It was actually very easy to represent 2,048 in octal. 2,048 is a power of two (2¹¹), which should give you a hint as to why octal resonates with binary.
7
septenary
Not used for much, except for the timekeeping of weeks.
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22...
(7) 1 2 3 4 5 6 10 11 12 13 14 15 16 20 21 22 23 24 25 26 30 31...2,048 in septenary => 5,654 5 x 343 = 1715
6 x 49 = 294
5 x 7 = 35
+ 4 x 1 = 4
------------------
2048
7,000 in septenary = 26,260 2 x 2401 = 4802
6 x 343 = 2058
2 x 49 = 98
6 x 7 = 42
+ 0 x 1 = 0
-------------------
7000
6
senary
Languages in use: Ndom (extant in Indonesia), Yam language family (Indonesia and Papua New Guinea)
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22...
(6) 1 2 3 4 5 10 11 12 13 14 15 20 21 22 23 24 25 30 31 32 33 34...2,048 in senary => 13,252 1 x 1296 = 1296
3 x 216 = 648
2 x 36 = 72
5 x 6 = 30
+ 2 x 1 = 2
-------------------
2048360 in senary => 1,400 1 x 216 = 216
4 x 36 = 144
0 x 6 = 0
+ 0 x 1 = 0
-----------------
360
Senary is used in the study of prime numbers as well. All primes above 3 end in either 1 or 5:
(10) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61...
(6) 2 3 5 11 15 21 25 31 35 45 51 101 105 111 115 125 135 141...Diceware is also a neat cryptographic method for using dice to create passphrases from a word list.
5
quinary
Languages in use: Ateso (Uganda and Kenya), Dhuwal (Australia), Khmer (official language of Cambodia), Kuurn Kopan Noot (Australia), Luiseño (California), Nunggubuyu (Australia), Saraveca (Bolivia — extinct), Wolof (Senegal, the Gambia, Mauritania)
Likely a little bit more common as a base for counting in natural languages as it shares an important quality with decimal — it reflects how many fingers we have on each hand! Also when using tally marks these are also grouped by fives. We only have the digits from 0–4 now to work with:
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21...
(5) 1 2 3 4 10 11 12 13 14 20 21 22 23 24 30 31 32 33 34 40 41...2,048 in quinary => 31,143 3 x 625 = 1875
1 x 125 = 125
1 x 25 = 25
4 x 5 = 20
+ 3 x 1 = 3
-------------------
2048
2,500 in quinary => 40,000 4 x 625 = 2500
0 x 25 = 0
0 x 5 = 0
+ 0 x 1 = 0
------------------
2500
biquinary
This is the same as base-10 but uses 2 and 5 as sub-bases. Common implementations of this type of system are abacuses (which were used all over the world), Roman numerals, alongside Scandinavian runes, many units of currency, and even some early computers!
(10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17...
(RN) I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI XVII...OK, let’s take a breather, then we will resume with base-4.
PART 10 PART 11
