Number systems and Conversion

Binary and hexadecimal numbers are a complete mystery for many of us. Often we don't find it really interesting because on the internet there are plenty of "subnet" or "binary" calculators where you can easily calculate from decimal to binary to hexadecimal or the other way around, without knowing how the exact calculation works.
This is no problem when you are not configuring or designing networks on a daily basis, but it will be a problem as soon as you take a networking examn, so it's best to know how to do these calculations off the top of your head.
Another advantage you will have is once you have mastered the art of binary calculations you can immediately see how big a network is and what the subnet mask is when people start throwing numbers at you.
So, let's start:

1. Decimal and Binary Numbers
When we write decimal (base 10) numbers, we use a positional notation system. Each digit is multiplied by an appropriate power of 10 depending on its position in the number:
For example:

843 = 8*10^2 + 4*10^1 + 3*10^0 = 8*100 + 4*10 + 3*1 = 800 + 40 + 3 = 843

For whole numbers, the rightmost digit position is the one's position (10^0 = 1). The numeral in that position indicates how many ones are present in the number. The next position to the left is ten's, then hundred's, thousand's, and so on. Each digit position has a weight that is ten times weight of the position to its right.
In the decimal number system, there are ten possible values that can appear in each digit position, and so there are ten numerals required to represent the quantity in each digit position. The decimal numerals are the familiar zero through nine (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).
In a positional notation system, the number base is called the radix. Thus, the base ten system that we normally use has a radix of 10. The term radix and base can be used interchangeably. When writing numbers in a radix other than ten, or where the radix isn't clear from the context, it is customary to specify the radix using a subscript. Thus, in a case where the radix isn't understood, decimal numbers would be written like this:

Generally, the radix will be understood from the context and the radix specification is left off.

The binary number system is also a positional notation numbering system, but in this case, the base is not ten, but is instead two. Each digit position in a binary number represents a power of two. So, when we write a binary number, each binary digit is multiplied by an appropriate power of 2 based on the position in the number:
For example:

101101 = 1*2^5 + 0*2^4 + 1*2^3 + 1*2^2 + 0*2^1 + 1*2^0 = 1*32 + 0*16 + 1*8 + 1*4 + 0*2 + 1*1 = 32 + 8 + 4 + 1 = 45

In the binary number system, there are only two possible values that can appear in each digit position rather than the ten that can appear in a decimal number. Only the numerals 0 and 1 are used in binary numbers. The term "bit" is a contraction of the words "binary" and "digit", and when talking about binary numbers the terms bit and digit can be used interchangeably. When talking about binary numbers, it is often necessary to talk of the number of bits used to store or represent the number. This merely describes the number of binary digits that would be required to write the number. The number in the above example is a 6 bit number.
The following are some additional examples of binary numbers:

2. Conversion between Decimal and Binary
Converting a number from binary to decimal is quite easy. All that is required is to find the decimal value of each binary digit position containing a 1 and add them up.

The method for converting a decimal number to binary is one that can be used to convert from decimal to any number base. It involves successive division by the radix until the dividend reaches 0. At each division, the remainder provides a digit of the converted number, starting with the least significant digit.

3. Hexadecimal Numbers
In addition to binary, another number base that is commonly used in digital systems is base 16. This number system is called hexadecimal, and each digit position represents a power of 16. For any number base greater than ten, a problem occurs because there are more than ten symbols needed to represent the numerals for that number base. It is customary in these cases to use the ten decimal numerals followed by the letters of the alphabet beginning with A to provide the needed numerals. Since the hexadecimal system is base 16, there are sixteen numerals required. The following are the hexadecimal numerals:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

The reason for the common use of hexadecimal numbers is the relationship between the numbers 2 and 16. Sixteen is a power for 2 (16 = 2^4). Because of this relationshipp, four digits in a binary number can be represented with a single hexadecimal digit. This makes conversion between binary and hexadecimal numbers very easy, and hexadecimal can be used to write large binary numbers with much fewer digits. When working with large digital systems such as computers, it is common to find binary numbers with 8, 16 and even 32 digits. Writing a 16 or 32 bit binary number would be quite tedious and error prone. By using hexadecimal, the numbers can be written with fewer digits and much less likelihood of error.
To convert a binary number to hexadecimal, divide it into groups of four digits starting with the rightmost digit. If the number of digits isn't a multiple of 4, prefix the number with 0's so that each group contain 4 digits. For each four digit group, convert the 4 bit binary number into an equivalent hexadecimal digit.
There are several ways in common use to specify that a given number is in hexadecimal representation rather than some other radix. In cases where the context makes it absolutely clear that numbers are represented in hexadecimal, no indicator is used. In much wirtten material where the context doesn't make it clear what the radix is, the numeric subscript 16 following the hexadecimal number is used. In most programming languages, this method isn't really feasible, so there are several conventions used depending on the language. In the C and C++ languages, hexadecimal constants are represented with a "0x" preceding the number, as in: 0x317F, or 0x1234, or 0xAF. In assembler programming languages that follow the Intel style, a hexadecimal constant begins with a numeric character (so that the assembler can distinguish it from a variable name), a leading "0" being used if necessary. The letter "h" is then suffixed onto the number to inform the assembler that it is a hexadecimal constant. In Intel style assembler format: 371Fh and 0FABCh are valid hexadecimal constants.

4. Binary Coded Decimal Numbers
Another number system that is encountered occasionally is Binary Coded Decimal. In this system, numbers are represented in a decimal form, however each decimal digit is encoded using a four bit binary number.

136 = 0001 0011 0110

Conversion of numbers between decimal and BCD is quite simple. To convert from decimal to BCD, simply write down the four bit binary pattern for each decimal digit. To convert from BCD to decimal, divide the number into groups of 4 bits and write down the corresponding decimal digit for each 4 bit group.
There are a couple of variations on the BCD representation, namely packed and unpacked. An unpacked BCD number has only a single decimal digit stored in each data byte. In this case, the decimal digit will be in the low four bits and the upper 4 bits of the byte will be 0. In the packed BCD representation, two decimal digits are placed in each byte. Generally, the high order bits of the data byte contain the more significant decimal digit.
Example:

01010110 10010011
This is converted to a decimal number as follows:
0101 0110 1001 0011
 5     6   9     3

The use of BCD to represent numbers isn't as common as binary in most computer systems, as it is not as space efficient. In packed BCD, only 10 of the 16 possible bit patterns in each 4 bit unit are used. In unpacked BCD, only 10 of the 256 possible bit patterns in each byte are used. A 16 bit quantify can represent the range 0 - 65535 in binary, 0 - 9999 in packed BCD and only 0-99 in unpacked BCD.

5. Fixed Precision and Overflow
So far, in talking about binary numbers, we haven't considered the maximum size of the number. We have assumed that as many bits are available as needed to represent the number. In most computer systems, this isn't the case. Numbers in computers are typically represented using a fixed number of bits. These sizes are typically 8 bits, 16 bits, 32 bits, 64 bits and 80 bits. These sizes are generally a multiple of 8, as most computer memories are organized on an 8 bit byte basis. Numbers in which a specific number of bits are used to represent the value are called fixed precision numbers. When a specific number of bits are used to represent a number, that determines the range of possible values that can be represented. For example, there are 256 possible combinations of 8 bits, therefore an 8 bit number can represent 256 distinct numeric values and the range is typically considered to be 0-255. Any number larger than 255 can't be represented using 8 bits. Similarly, 16 bits allows a range of 0-65535.
When fixed precision numbers are used, the concept of overflow must be considered. An overflow occurs when the result of a calculation can't be represented with the number of bits available. For example when adding the two eight bit quantities: 150 + 170 = 320. This is outside the range 0-255, and so the result can't be represented using 8 bits. The result has overflowed the available range. When overflow occurs, the low order bits of the result will remain valid, but the high order bits will be lost. This result in a value that is significantly smaller than the correct result.

6. Signed and Unsigned Numbers
So far, we have only considered positive values for binary numbers. When a fixed precision binary number is used to hold only positive values, it is said to be unsigned. In this case, the range of positive values that can be represented is 0 --2^n - 1, where n is the number of bits used. It is also possible to represent signed (negative as well as positive) numbers in binary. In this case, part of the total range of values is used to represent positive values, and the rest of the range is used to represent negative values.
There are several ways that signed numbers can be represented in binary, but the most common representation used today is called two's complement. The term two's complement is somewhat ambiguous, in that it is used in two different ways. First, as a representation, two's complement is a way of interpreting and assigning meaning to a bit pattern contained in a fixed precision binary quantify. Second, the term two's complement is also used to refer to an operation that can be performed on the bits of a binary quantity. As an operation, the two's complement of a number is formed by inverting all of the bits and adding 1. In a binary number being interpreted using the two's complement representation, the high order bit of the number indicates the sign. If the sign bit is 0, the number is positive, and if the sign bit is 1, the number is negative. For positive numbers, the rest of the bits hold the true magnitude of the number. For negative numbers, the lower order bits hold the complement (or bitwise inverse) of the magnitude of the number. It is important to note that two's complement representation can only be applied to fixed precision quantities, that is, quantities where there are a set number of bits.
Two's complement representation is used because it reduces the complexity of hardware in the arithmetic-logic unit of a computer's CPU. Using two's complement representation, all of the arithmetic operations can be performed by the same hardware whether the numbers are considered to be unsigned or signed. The bit operations performed are identical, the difference comes from the interpretation of the bits. The interpretation of the value will be different depending on whether the value is considered to be unsigned or signed.

Example: Find the two's complement of the following 8 bit number: 00101001
 11010110 (first invert the bits)
+00000001 (then add 1)
=11010111

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