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Introduction to Numbering Systems. Decimal (base-10)

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Decimal (base-10)

The numberidg system we use is called the decimal system.  It consists of 10 numbers: 0123456789. Some ancient civilizations used other numbering systems. For example, the Babylonians used the sexagesimal system, based on the number 60. The Babylonians divided the circle into 360 degrees (a multiple of 60). The Mayas used the vigesimal system, based on the number 20.


Unary (base-1)

A base-1 numbering system was used by some people in ancient times before more advanced systems were developed.  In a base-1 system, you simply put a mark (or a 1) for each item you are counting. For example, bones have been found with 29 and 30 notches, possibly corresponding to the cycle of the moon:

Placing a mark for every item you are counting would be quite cumbersome if you need to count larger quantities.  Slightly more advanced are numbering systems that have special symbols to represent larger numbers.  For example, Roman numerals use I, V, X, etc.  The ancient Egyptians also used symbols to represent larger quantities.  The number below is 2,382,358.


Binary (base-2)

The electronic circuitry within computers is composed of millions of transistors.  Each of these circuits has two states – on and off. On and off is viewed as representing 1 and 0, respectively. Because of this, a base-2 numbering system was devised for computers.


Counting in binary is often a tricky concept to understand for newcomers. One method that might help you count in binary would be to use the decimal system, but only keep the numbers that contain 0's and 1's.  The following table shows these numbers in bold:

One binary digit is called a bit. A bit can be either 0 or 1. Eight binary digits is call a byte. One byte of data can store a decimal number between 0 and 255.

Smallest number a byte can be: 00000000 = 0 in decimalLargest number a byte can be: 11111111 = 255 in decimal


Converting a Binary Number to DecimalThe easiest way to convert a binary number to decimal is to use the Windows calculator. To do it manually:(1) write the binary number (e.g. 10101110)(2) on the rightmost number, put a 1 on top of it. On the next number to the left, put a 2 on top of it. Keep doubling the numbers until you have a number on top of each binary digit.(3) add up all the numbers that have a 1 below them.Example: Convert binary 10101110 to decimal

128 + 32 + 8 + 4 + 2 = 174

doing calculations with bytes
Doing Calculations with Bytes

Example 1: You have some video files that are 2 GB each. How many ofthese files will fit on a 1 TB external hard drive?

Solution: 1 TB 1,000,000,000,000 1,000 ------- = ------------------------ = -------- = 500 video files

2 GB 2,000,000,000 2

Example 2: How many bits are in 8 KB?

Solution: 8 KB = 8,000 bytes = 64,000 bits (multiply by 8 to get bits)


ASCII - In most computers, each character (including letters, numbers, punctuation, spaces) of data is stored using one byte (8 bits) of memory.  Since each byte represents a number between 0 and 255, ASCII is way of coding data so that each character has a corresponding number between 0 and 255.

The word “Hi” is stored as ASCII numbers 72 and 105.


Unicode – Since the ASCII chart only has the characters for Latin derived languages, Unicode is a newer coding system used to store the characters for almost all writing systems (see table on right).

Each Unicode character requires 2 bytes of storage giving a total of 65,536 different characters.

Here is a complete Unicode chart: