Today is February 211, in the year 2202222 AD. Well that certainly looks like a computer glitch, it can’t be right!
Guess what, it is correct, but the problem is that you are likely looking at it with decimal numbers in mind. The number system that I used, which is base-3 (ternary), is scarcely utilized. You likely don’t encounter numbers in bases other than base-10 (decimal) if you don’t write any code on a computer or select a hexadecimal value for a color in RGB, but I’m sure you have heard of binary numbers, which are base-2. Maybe you even know that hexadecimal is base-16, but have you heard about other number bases that are practically unused, like base-4, base-19, or base-76?
Number Bases
Here is a list of several number bases along with their names, but keep in mind that most of these number bases have more than one name; I have put decimal in bold because it is the most common base.
- Base-1 (Unary)
- Base-2 (Binary)
- Base-3 (Ternary)
- Base-4 (Quaternary)
- Base-5 (Quinary)
- Base-6 (Senary)
- Base-7 (Septenary)
- Base-8 (Octal)
- Base-9 (Nonary)
- Base-10 (Decimal)
- Base-11 (Undecimal)
- Base-12 (Duodecimal)
- Base-13 (Tredecimal)
- Base-14 (Tetradecimal)
- Base-15 (Pentadecimal)
- Base-16 (Hexadecimal)
- Base-17 (Heptadecimal)
- Base-18 (Octodecimal)
- Base-19 (Nonadecimal)
- Base-20 (Vigesimal)
Well, how many number bases are there, twenty? Thirty-six? Maybe one hundred?
To the best of my understanding, there theoretically should be as many number bases as there are numbers: infinite. Sure, we can only name a finite amount of number bases, just like numbers, but as long as you have a consistent symbol to represent each digit, you can use a number base as high as you desire.
Digits in Number Bases
Each number base only has as many digits as the number of the base, starting with 0, which means binary (meaning two) has two digits: 0 and 1; decimal (meaning ten) has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9; and hexadecimal (meaning sixteen) has sixteen digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. This is the pattern for all number bases.
Here is a list of several digits along with their values (in decimal, of course). I’ll also indicate which number bases use each given digit.
- 0 (0), used in all number bases except base-1
- 1 (1), used in all number bases
- 2 (2), used in base-3 and higher
- 3 (3), used in base-4 and higher
- 4 (4), used in base-5 and higher
- 5 (5), used in base-6 and higher
- 6 (6), used in base-7 and higher
- 7 (7), used in base-8 and higher
- 8 (8), used in base-9 and higher
- 9 (9), used in base-10 and higher
- A (10), used in base-11 and higher
- B (11), used in base-12 and higher
- C (12), used in base-13 and higher
- D (13), used in base-14 and higher
- E (14), used in base-15 and higher
- F (15), used in base-16 and higher
- G (16), used in base-17 and higher
- H (17), used in base-18 and higher
- I (18), used in base-19 and higher
- J (19), used in base-20 and higher
- K (20), used in base-21 and higher
- L (21), used in base-22 and higher
- M (22), used in base-23 and higher
- N (23), used in base-24 and higher
- O (24), used in base-25 and higher
- P (25), used in base-26 and higher
- Q (26), used in base-27 and higher
- R (27), used in base-28 and higher
- S (28), used in base-29 and higher
- T (29), used in base-30 and higher
- U (30), used in base-31 and higher
- V (31), used in base-32 and higher
- W (32), used in base-33 and higher
- X (33), used in base-34 and higher
- Y (34), used in base-35 and higher
- Z (35), used in base-36 and higher
| Base-2 | Base-3 | Base-4 | Base-5 | Base-6 | Base-7 | Base-8 | Base-9 | Base-10 | Base-11 | Base-12 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| 11 | 10 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| 100 | 11 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| 101 | 12 | 11 | 10 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |
| 110 | 20 | 12 | 11 | 10 | 6 | 6 | 6 | 6 | 6 | 6 |
| 111 | 21 | 13 | 12 | 11 | 10 | 7 | 7 | 7 | 7 | 7 |
| 1000 | 22 | 20 | 13 | 12 | 11 | 10 | 8 | 8 | 8 | 8 |
| 1001 | 100 | 21 | 14 | 13 | 12 | 11 | 10 | 9 | 9 | 9 |
| 1010 | 101 | 22 | 20 | 14 | 13 | 12 | 11 | 10 | A | A |
| 1011 | 102 | 23 | 21 | 15 | 14 | 13 | 12 | 11 | 10 | B |
| 1100 | 110 | 30 | 22 | 20 | 15 | 14 | 13 | 12 | 11 | 10 |
| 1101 | 111 | 31 | 23 | 21 | 16 | 15 | 14 | 13 | 12 | 11 |
| 1110 | 112 | 32 | 24 | 22 | 20 | 16 | 15 | 14 | 13 | 12 |
| 1111 | 120 | 33 | 30 | 23 | 21 | 17 | 16 | 15 | 14 | 13 |
As you might have noticed, Unary (Base-1) is the only number base that doesn’t entirely follow the expected pattern. This is because unary would only have the digit 0 if it followed the pattern, which would suggest that it is impossible to have any value in unary. Instead, unary only has the digit 1, which means 2 is written 11 in unary, 5 is 11111, and 7 is represented as 1111111, and so on. You could think of unary like counting in tally marks.
Place Values in Number Bases
Another thing that separates number bases is the value of each place. The place furthest right, assuming there is no decimal point, is always 1 times the digit in that place, the next place left is the base number times the digit, the next place left is the base number squared times the digit, the next place left is the base number cubed times the digit, and so on. This could be represented in the following manner.
Remember that any number to the power of 0 equals 1, and a number to the power of 2 is the number times itself, and a number to the power of 3 is the number times itself times itself, and so on. You could remember this by saying that the exponent number is telling you how many instances of a number to multiply together: 22 means 2 x 2, 23 means 2 x 2 x 2, and 24 means 2 x 2 x 2 x 2.
Converting into Decimal
To convert the date I provided in ternary into decimal, you would need to find the sum of each digit times the digit’s place value. The number 211 is converted to decimal by rewriting it in expanded notation and solving it: 2 1 1 = (2 x 32) + (1 x 31) + (1 x 30) = (2 x 9) + (1 x 3) + (1 x 1) = 18 + 3 + 1 = 22. So, I was simply saying that it is February 22 today, which is correct. Well what about the year? Let’s pretend for a second that we don’t know the current year. I said the year was 2202222, which you rewrite using the same pattern, but increasing the exponent by 1 each time you move 1 place left: 2 2 0 2 2 2 2 = (2 x 36) + (2 x 35) + (0 x 34) + (2 x 33) + (2 x 32) + (2 x 31) + (2 x 30) = (2 x 729) + (2 x 243) + (0 x 81) + (2 x 27) + (2 x 9) + (2 x 3) + (2 x 1) = 1,458 + 486 + 0 + 54 + 18 + 6 + 2 = 2,024, which turns out to be the current year. That is one way to say February 22, 2024, in code.
Believe it or not, this is also how our regular decimal numbers work, even though we take them at face value. This is because base-10 is super easy, each place leftward is ten times greater than the last, but we normally don’t write our number in expanded notation, since we have an intuitive comprehension of the value of each place. For example, the number 100 really means (1 x 102) + (0 x 101) + (0 x 100), but we simply know that the 1 in the hundreds place is 100 (or 102) times greater than if it was in the units place.
Does this look familiar to another notation? The x 10x is also used to express numbers in scientific notation, for example, 1.776 x 103. The reason a number is always multiplied by an exponent of 10 and not any other number in scientific notation is because decimal numbers are being used. If this number was in binary, I believe it would look something like this: 1.11000110101001111111 x 101011. Other number bases can have a x 10x, but the number with the exponent needs to be the number of the base, for example, the expression 3.5 x 24 would look like 11.1 x 10100 in binary, and the expression 5.25 x 416 would also look like 11.1 x 10100 in quaternary, but the topic of fractions and decimals in other number bases is a little complex for this introduction.
Fun fact: number bases are labeled using standard decimal numbers, such as base-2, base-3, and base-4, but because each number base only has as many digits as the number of its base and because the first digit is always 0, no bases can represent their base number with a single digit; they will always have to add one to the place left of the units place, therefore base-2 would be written base-10 in base-2, base-3 is base-10 in base-3, base-10 is base-10 in base-10, et cetera. This means base-10 is only base-10 in, well, base-10. Decimal is base-1010 in binary, base-101 in ternary, but it is base-A in base-11 and everything higher; this is just an interesting concept, but all bases will be represented by a decimal number.
How to Indicate a Number Base
If you are using a number base other than decimal, you could choose to specify the base number in subscript immediately following the number, which could look like 112 or 315. This means that it would have been less deceiving if I had said that the year was 22022223, but that wouldn’t have been nearly as fun, would it?
Now, would you believe me if I told you that the United States is F7 years old? Perhaps it would be more appropriate to say that the United States is F716 years old, as of July 4, 2023. Am I telling the truth, or is this a false statement? I have given you all the tools to figure this out, now it’s your turn.
What is your favorite number base?
Onward American 