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Electronics Catlog
DIC Catlog

Number System
Conversions Between Number System
Arithematic Operations
1's & 2's Complement
Gray Codes
Arithmetic Circuits
Logical Gates and Truth Table Funtions
Boolean Expressions
Boolean Algebra
Karnaugh Map
Encoder & Decoder
TTL Circuits
555 Timer
Flip Flops
RS Flip - Flop
JK Flip - Flop
D Flip - Flop
Shift Register
Schmitt Trigger
Asynchronous Counters
Synchronous Counters
Digital - Analog Conversion
Data Flow
Memory Drives
Electronics Equation
Resistor Color Codes

Conversion Between Different Number Systems

Positional number systems

    Our decimal number system is known as a positional number system, because the value of the number depends on the position of the digits. For example, the number 123 has a very different value than the number 321, although the same digits are used in both numbers.

    (Although we are accustomed to our decimal number system, which is positional, other ancient number systems, such as the Egyptian number system were not positional, but rather used many additional symbols to represent larger values.)

    In a positional number system, the value of each digit is determined by which place it appears in the full number. The lowest place value is the rightmost position, and each successive position to the left has a higher place value.

    In our decimal number system, the rightmost position represents the "ones" column, the next position represents the "tens" column, the next position represents "hundreds", etc. Therefore, the number 123 represents 1 hundred and 2 tens and 3 ones, whereas the number 321 represents 3 hundreds and 2 tens and 1 one.

    The values of each position correspond to powers of the base of the number system. So for our decimal number system, which uses base 10, the place values correspond to powers of 10:

Position Number System

Converting from other number bases to decimal

    Other number systems use different bases. The binary number system uses base 2, so the place values of the digits of a binary number correspond to powers of 2. For example, the value of the binary number 10011 is determined by computing the place value of each of the digits of the number:

Decimal number representation

So the binary number 10011 represents the value

Binary to Decimal

The same principle applies to any number base. Example, the number 2132 base 5 corresponds to

Base5 Number

So the value of the number is

Base5 to Decimal Number

Converting from decimal to other number bases

    In order to convert a decimal number into its representation in a different number base, we have to be able to express the number in terms of powers of the other base. For example, if we wish to convert the decimal number 100 to base 4, we must figure out how to express 100 as the sum of powers of 4.

Decimal to Base4

Then we use the coefficients of the powers of 4 to form the number as represented in base 4:

Base4 equivalent value

    One way to do this is to repeatedly divide the decimal number by the base in which it is to be converted, until the quotient becomes zero. As the number is divided, the remainders - in reverse order - form the digits of the number in the other base.

Example: Convert the decimal number 82 to base 6:

Decimal to Base6

The answer is formed by taking the remainders in reverse order: 2 1 4 base 6

The base conversion table is as follows:

Base Conversion Table