If you’re reading this, you probably learned how to count using your fingers going from one to ten. It is a convenient and intuitive system given that humans generally have 10 fingers. This is the decimal counting system that is also referred to as base ten because it is based on powers of ten (who would’ve thought?). It is the most widely used system today, but there are many other systems available that may be more appropriate for some tasks. If you’re not in a scientific field, you may not have encountered these other counting systems such as octal (base eight), binary (base two), or hexadecimal (base sixteen). There are an infinitesimal number of different counting systems that can be used, but I’ll focus on these three to explain the concepts after which you will be able to count in any base you like.

In the decimal (base ten) system, each digit in a number holds some significant position based on powers of ten. For example, the number 267 has a seven in the ones place, a six in the tens place, and a two in the hundreds place. You can find the total value of a number by multiplying the place-value by the digit in each respective place, then adding the products together. For example:

2 * 100 = 200
6 * 10 = 60
7 * 1 = 7

200 + 60 + 7 = 267

So we end up with 267, which is the value in base ten notation. To count in decimal, you add one to the next highest power place when the digit in a given place exceeds nine, then return that place to zero. We can count 7, 8, 9, and 10 to observe this. Once you exceed nine, you add one to the tens place and the ones place returns to zero. This is the basic idea behind counting in any number system, as you will see.

Octal is the number system based on powers of eight. It was used in some early computers and has been used by some Native Americans that counted using the eight spaces between fingers. Since it is based on powers of eight, the places (right to left) of a three digit number would be ones, eights, and sixty-fours. Let’s use the process from the last paragraph to discover the decimal value of the octal number 513.

5 * 64 = 320
1 * 8 = 8
3 * 1 = 3

320 + 8 + 3 = 331

We can see now that 331 is the value in base ten of octal number 513. Counting manually follows the same process described earlier. We can count 4, 5, 6, 7, 10. We reached 7 which is the highest value a single digit can represent, and added a one to the eights place. Continuing would reveal the same sequence: 11, 12, 13, 14, 15, 16, 17, 20, 21… and so on.

Counting in binary (base two) might seem a little trickier since there are only two digits to use, but the process is exactly the same. You have (right to left) a ones place, twos place, fours place, eights place, sixty-fours place, and so on. Let’s find the value of the four-bit number, 1101.

1 * 8 = 8
1 * 4 = 4
0 * 2 = 0
1 * 1 = 1

8 + 4 + 0 + 1 = 13

Counting in binary just involves a lot of place-shifting. It looks like this: 1, 10, 11, 100, 101, 110, 111. The binary value 111 is seven in decimal.

Until now, each number system we’ve studied uses standard Arabic digits 0-9. What happens if you want to use a base with digits above nine? Well, just get more digits! More specifically, symbols other than common decimal numbers can be used to represent values. The hexadecimal system (base sixteen) needs symbols to represent values zero through fifteen, so when a value is above nine letters are used. The letters A through F represent values ten through fifteen. Knowing this, we can apply the same process we’ve been using the find the value of hexadecimal number C4A.

12 * 256 = 3072
4 * 16 = 64
10 * 1 = 10

3072 * 64 + 10 = 3146

Counting in hexadecimal is just as straightforward as any other. A sequence looks like: 7, 8, 9, A, B, C… etc.

That’s all there is to it! For practice, try counting and finding the values of numbers in strange systems such as base seven or base nineteen.