Number Systems

Skill Level: Beginner

by M-Short December 02, 2010 | 15 comments

When we write numbers, we have ten different digits that we can put in each place (0-9). After that we need to add a new column of digits and we keep going. This is called the decimal system (or base 10). What would happen if instead of using ten digits we used eight, or two, or even sixteen? Well, what happens is instead of a decimal system we would have what's called binary, octal, or hexadecimal systems (base 2, base 8, and base 16). We use subscripts to denote the system we are using: 9876₁₀ is decimal (if we don't use a subscript we assume we are using decimal), 7653₈ is octal, 1010₂ is binary, and BEEF₁₆ is yummy hexadecimal.

Binary – binary just has 1s and 0s. But much of our world is binary. Light switches are either up or down, your computer is either on or off, our servers are either currently up or down (hopefully up). Because computers are made up of billions of little switches (called transistors), they do calculations in binary. Everything in your computer is either on or off. And so we represent larger numbers in 1s and 0s. Each 1 or 0 is called a bit, and 8 bits is a byte.

Octal – at some point we end up with a lot of 1s and 0s and they are kind of hard to read. For example "010101000110100001100101" and that’s just to spell "The.”  But if we break that into 3 piece sections with 8 possible digits it becomes 25064145₈ which is a little easier to read. And thus we have octal.  Unfortunately for octal we rarely use it.

Hexadecimal – What happens if we break that down into 4 piece sections? Well, first of all we run out of digits because now we have 16 possible digits instead of our normal 10. So the powers that be just decided to start using letters so hexadecimal has 16 digits (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F). In other words if we wanted to write 10₁₀ we would write A₁₆ etc.  Now if we want to write out the word "the" it looks like 546865₁₆ which is a whole lot easier to read.

Conversion – In decimal, the least significant digit (the one on the right) keeps track of 10⁰. The next column is 10¹, then 10² etc.

So 382 is
(2 x 10⁰)+ (8 x 10¹) + (3 x 10²) = 2 + 80 + 300.

The same is true for other systems.

For example, 101111110₂ goes thusly:
(0 x 2⁰) + (1 x 2¹) + (1 x 2²) + (1 x 2³) + (1 x 2⁴) + (1 x 2⁵) + (1 x 2⁶) + (0 x 2⁷) + (1 x 2⁸) =
0 + 2 + 4 + 8 + 16 + 32 + 64 + 0 + 256 = 382₁₀

Now you try (highlight for answers):

BEED₁₆

D x 16⁰ = 13 x 1 = 13
E x 16¹ = 14 x 16 =224
E x 16² = 14 x 256 =3584
B x 16³ = 11 x 4096 =45056

Add that up and you get: 48877₁₀

And remember just because things have bytes does NOT mean you should bite them. None of our products should be ingested or inserted under the skin.