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Pertaining to a number system that has just two unique digits. For most purposes, we use the decimal number system, which has ten unique digits, 0 through 9. All other numbers are then formed by combining these ten digits. Computers are based on the binary numbering system, which consists of just two unique numbers, 0 and 1. All operations that are possible in the decimal system (addition, subtraction, multiplication, division) are equally possible in the binary system. We use the decimal system in everyday life because it seems more natural (we have ten fingers and ten toes). For the computer, the binary system is more natural because of its electrical nature (charged versus uncharged). In the decimal system, each digit position represents a value of 10 to the position's power. For example, the number 345 means: 3 three 100s (10 to the 2nd power) plus 4 four 10s (10 to the first power) plus 5 five 1s (10 to the zeroth power) In the binary system, each digit position represents a power of 2. For example, the binary number 1011 equals: 1 one 8 (2 to the 3rd power) plus 0 zero 4s (2 to the 2nd power) plus 1 one 2 (2 to the first power) plus 1 one 1 (2 to the zeroth power) So a binary 1011 equals a decimal 11. Because computers use the binary number system, powers of 2 play an important role. This is why everything in computers seems to come in 8s (2 to the 3rd power), 64s (2 to the 6th power), 128s (2 to the 7th power), and 256s (2 to the 8th power). Programmers also use the octal (8 numbers) and hexadecimal (16 numbers) number systems because they map nicely onto the binary system. Each octal digit represents exactly three binary digits, and each hexadecimal digit represents four binary digits.

Decimal  Binary 
1  1 
2  10 
3  11 
4  100 
5  101 
6  110 
7  111 
8  1000 
9  1001 
10  1010 
16  10000 
32  100000 
64  1000000 
100  1100100 
256  100000000 
512  1000000000 
1000  1111110100 
1024  10000000000 
binary format decimal 
hexadecimal octal 
Data Representation
This is Chapter 1 of Randall Hyde's book, "Art of Assembly Language." It describes the binary and hexadecimal numbering systems, binary data organization (bits, nibbles, bytes, words, and double words), signed and unsigned numbering systems, arithmetic, logical, shift, and rotate operations on binary values, bit fields and packed data, and the ASCII character set.
Updated on Aug 5, 1998
Binary primer
Introduction to binary numbers.
Connected Encyclopedia's binary arithmetic overview
Explains binary arithmetic, bits, and bytes for beginners.
Data Representation  The Binary Number System
Very clear and concise discussion of the binary system used by computers to represent data.
Updated on Apr 7, 1998
Learn to count in binary on your fingers
This page uses cartoon hands to demonstrate how to count on your fingers in binary. Though only the right hand is shown here, by "getting" the pattern, you can count to 1023 using both hands.
Lecture notes on binary numbers
These are notes on binary numbers and character sets from a computer architecture course.
Updated on Jul 17, 1998
Representing data
This page explains binary representation of numbers in computers.
Updated on Aug 4, 1998
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