**What are number bases?**

A base is a counting system. Base 10, or "decimal"
uses the digits 0,1,2,3,4,5,6,7,8, and 9.

Each number further left represents a power of 10.

For example, 987 is 9 100s, 8 10s, and 7 1s

or 9*10^2 + 8*10^1 + 7*10^0

If we count in hexadecimal, which is base 16

("hex" meaning 6 and "decimal" meaning 10),

we have digits 0,1,2,3,4,5,6,7,8,9,a,b,c,d,e, and f.

Notice that 10 is a, 11 is b... and we go up to 15 (f), which is 16-1,

just like in base 10 we go up to 9, which is 10-1.

**Examples**

For example, for the number 3db in hex,

instead of the 1s, 10s, and 100s places like decimal,

we have the 1s, 16s, and 256s places, for powers of 16 instead of powers of 10.

So 3db is 3*16^2 + d*16^1 + b*16^0 where d is 13 and b is 11.

So we have 3*256 + 13*16 + 11*1 = 768 + 208 + 11 = 987.

So 3db in hex is 987 in decimal.

Similarly, we can convert decimal back to base 16:

987/16 is 61 remainder 11.

61/16 is 3 remainder 13.

and 3 doesn't go into 16, so we finish with remainder 3.

We take the remainders and go back upwards:
so we have 3, then 13, then 11, which is 3db

**About Bases**

If we count in binary, which is base 2 ("bi" meaning 2) we have digits 0 and 1.

The prefix "bi" is used in "bicycle", "bipetal", and "bilingual".

In octal (base 8, "oct" meaning 8) we have digits 0,1,2,3,4,5,6, and 7.

The prefix "oct" is used in "octopus" and "octagon".

Base 2,8, and 16 are often used in programming because they're all powers of 2,

data is stored in binary, and conversion between them is easy.

We probably count in decimal because we have ten fingers.

**Using the Table as a Shortcut**
Converting between base 2, 8, and 16 is easy.

You can just look at the quick reference table

and substitute every 4 binary digits for a hex digit, or 3 for an oct digit.

If the amount isn't divisable by 3 or 4 (whichever you need), pad the beginning with 0s.

To convert from hex or oct, do the opposite.

**For example:**
3db in hex: 3 is 0011, d is 1101 and b is 1011, so 3db is 001111011011

getting rid of the leading 0s we have 1111011011

converting 1111011011 back, we add 0s to make the number of digits divisable by 4

this is because 2^4 is 16.

001111011011 split into groups of 4 is 0011 1101 1011 which in the table is 3, d, and b

**Misc. Facts:**
Sexigesimal (base 60) was used by ancient sumerians.

That is why there are 60 seconds in a minute and 360 degrees in a circle.

Sometimes bases above 10 use lowercase letters, but most of the time it's capital.

Bases above 36 often use both capital and lowercase.

Sometimes base 26 starts at A (where Z is 0).

We often use hexadecimal to abbreviate RGB colors because each color is between 0-255,

so each color is 2 hexadecimal digits.

Sometimes people call hexadecimal "hex".

Most of the time you can assume "hex" refers to base 16 not base 6.

Binary is sometimes called "bin".

RGB hexadecimal codes often start with a "#",

and sometimes are only 3 digits meaing they store only the most significant digit of each color.