Digital Electronics/Number Base System

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Introduction[edit]

In digital electronics how information is represented is key and there are different radices, i.e. number bases, that a numbering system can use, with the most common ones being: binary (base-2), octal (base-8), decimal (base 10) and hexadecimal (base 16). When a reference is made to a numbering system's base, i.e. its' radix, the base number represents the number of digits used in that numbering system. As an example, we are all familiar with the decimal numbering system and the digits used it are: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The numbering systems that have a base/radix less than 10 recycle the "digits" used to be common with the decimal numbering system. Therefore the digits for binary are: 0 and 1, the digits for octal are: 0, 1, 2, 3, 4, 5, 6 and 7. For the hexadecimal numbering system, base 16, the digits are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

The decimal numbering is obviously linked to the fact that us humans have 10 fingers...10 digits. However, the other "common" numbering systems in digital electronics are linked to logic, in particularly boolean logic and boolean algebra as developed by George Boole. In logic, including the philosophical kind, there can only be two answers: TRUE or FALSE. In digital electronics, we can represent TRUE as a high voltage (historically 5 Volts, but it has changed to save power) which we denote as a 1 and we represent FALSE as low voltage (0 Volts, but lower voltages can also be used too) which we denote as a 0. Therefore with two values to work with we then have the binary numbering system. The octal numbering is simply a more compact group of three binary digits. So instead of having to type "101" we can just type "5"; assuming that the numbering system is known to be octal based. In a similar manner, hexadecimal based numbering system is based on a grouping of four binary digits, so "D" is easier to type instead of "1101"

Beyond Digits, Representing Numbers[edit]

Now that we understand a numbering system's radix, the next step is to use these digits to present numbers, ideally large numbers. Independent of the numbering system's radix, the order of digits as they are listed in the representation of a number is of GREAT "significance", pardon the pun, and needs to be fully understood to be able to use any numbering system properly; even the decimal numbering system. The digit furthest to the right has a numerical weight of (radix)^0, which is always 1 given that any number raised to the zero power is one. The next digit moving to the left has a numerical weight of (radix)^1, which is the first digit where the radix value begins to being a factor. The weighting for each subsequent digit continues to increase by one with respect to the "power" that the radix is raised. And like in our more traditional decimal numbering systems there are ALWAYS zero digits in front of the highest weighted digit, but we simply do not bother writing all those zeros as such zeroes are not significant. The same is true with other radices, hence the order of the digits is representing a number in any radix is significant.

The next sections provide illustrations on how numbers are represented in the differing radices.

Number of Base Two[edit]

Base two numbers only use the two digits 0 and 1 . Any number greater than 1 is represented by a series of 0 and 1 digits

1110 = 1 * 2^3 + 1 * 2^2 + 1 * 2^1 + 0 * 2^0 = 14

Numbers that contain only two digit 0 and 1 are called Binary Numbers. Each 0 or 1 is called a Bit, from binary digit. A binary number of 4 bits is called a Nibble. A binary number of 8 bits is called a Byte. A binary number of 16 bits is called a Word on some systems, on others a 32-bit number is called a Word while a 16-bit number is called a Halfword.

Using 2 bit 0 and 1 to form

a binary number of 1 bit, there are 2 such numbers 0 and 1
a binary number of 2 bit, there are 4 such numbers 00, 01, 10, 11
a binary number of 3 bit, there are 8 such numbers 000, 001, 010, 011, 100, 101, 110, 111
a binary number of 4 bit, there are 16 such numbers 0000, 0010, 0100, 0110, 1000, 1010, 1100, ..., 1111

Therefore , using n bits there are 2n binary numbers of n bits

Numbers of Base Ten[edit]

Base ten numbers use ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Any number greater than 9 is represented by a series of digits in the 0 through 9 range.

14 = 1 * 10^1 + 4 * 10^0

Numbers of Base sixteen(hexa decimal)[edit]

Base sixteen numbers use sixteen digits 0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f. any number greater than 15 is represented by digits in the 0 through f range.

1A = 1 * 16^1 + 10 * 16^0 = 26

Communication (Conversion) between numbers of Different base[edit]

The same number 14 expressed

in base ten is 14
in base two is 1110 .

Hence , any number in our base ten system can be represented by a binary number of certain bit