LESSON 01 – COUNTING IN ANY BASE

LESSON NOTE

BASE

A base is a number system that is defined by the number of symbols used in it

The number system used throughout our world uses ten different symbols (0,1,2,3,4,5,6,7,8,9) and is therefore known as base 10.

The number system that has only have 5 symbols in it would be called base 5.

COMMON NAMES

As opposed to naming commonly known bases by their numbers, we often use a common names.

Base #	Symbols	Common Name
Base 2	0,1	Binary
Base 3	0,1,2	Tertiary
Base 8	0,1,2,3,4,5,6,7	Octal
Base 10	0,1,2,3,4,5,6,7,8,9	Decimal
Base 16	0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F	Hexadecimal (or Hex)

COUNTING IN DECIMAL

Recall that the symbols we use are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

It is important to realize that counting is simply adding one of a number over and over.

To add one to a number, we simply replace its right-most digit’s symbol by the next symbol available. For example, if we have 823, the right-most digit’s symbol, 3, is replaced by the next symbol which is 4. So, the next number is 824.

However, a small complication arises when we are dealing with the last of our symbols, number 9. We can’t simply replace it by the next symbol. What we do is restart that digit to zero and instead add one in column on the left.

COUNTING

Counting in any base is the same as in decimal.  When we reach the greatest symbol on the right, we add one by changing it to zero and carrying over one to the next column.

Base-2 (Symbols: 0, 1)

0, 1, 10, 11, 100, 101, 110, 111, 1000, …

Base-8 (Symbols: 0, 1, 2, 3, 4, 5, 6, 7)

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, …

Base-10 (Symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, …

Base-16 (Symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F)

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12, 13, …