18-06-2012, 12:05 PM
Number Systems and Binary Arithmetic
Binary Arithmetic.ppt (Size: 160 KB / Downloads: 54)
Introduction to Numbering Systems
We are all familiar with the decimal number system (Base 10). Some other number systems that we will work with are:
Characteristics of Numbering Systems
The digits are consecutive.
The number of digits is equal to the size of the base.
Zero is always the first digit.
The base number is never a digit.
When 1 is added to the largest digit, a sum of zero and a carry of one results.
Numeric values determined by the have implicit positional values of the digits.
Binary Number System
Also called the “Base 2 system”
The binary number system is used to model the series of electrical signals computers use to represent information
0 represents the no voltage or an off state
1 represents the presence of voltage or an
on state
Decimal to Binary Conversion
The easiest way to convert a decimal number to its binary equivalent is to use the Division Algorithm
This method repeatedly divides a decimal number by 2 and records the quotient and remainder
The remainder digits (a sequence of zeros and ones) form the binary equivalent in least significant to most significant digit sequence
Binary to Decimal Conversion
The easiest method for converting a binary number to its decimal equivalent is to use the Multiplication Algorithm
Multiply the binary digits by increasing powers of two, starting from the right
Then, to find the decimal number equivalent, sum those products
Octal Number System
Also known as the Base 8 System
Uses digits 0 - 7
Readily converts to binary
Groups of three (binary) digits can be used to represent each octal digit
Also uses multiplication and division algorithms for conversion to and from base 10
Bias Notation
The exponent field (8 bits) can be used to represent integers from 0-255
Because of the need for negative exponents to be represented as well, the range is offset or biased from – 128 to + 127
In this way, both very large and very small numbers can be represented
Computational Errors
When converting base 10 fractions to binary, only those fractions whose values can be expressed as a sum of base 2 fractions will convert evenly
All other base 10 fractions feature a least significant bit that is either rounded or truncated – an approximation
When two such numbers are multiplied, the rounding error is compounded .