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Binary Arithmetic and Binary Coding Schemes by Dr. Anita Goel, Associate Professor, Dyal Singh College, University of Delhi.......100001000001.......Binary Arithmetic and Binary Coding Schemes by Dr. Anita Goel, Associate Professor, Dyal Singh College, University of Delhi.......110000....Binary Arithmetic and Binary Coding Schemes by Dr. Anita Goel, Associate Professor, Dyal Singh College, University of Delhi.......
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  • Binary Arithmetic
    and
    Binary Coding Schemes


    1.
    Introduction
    2. Binary Arithmetic 3. Signed and Unsigned Numbers
    4. Binary Data Representation
    5. Binary Coding Schemes

    1. Introduction

    Since our school days, we have been using the decimal number system. The digits 0-9 are combined to get different numbers like 104, 4561 etc. We also perform decimal arithmetic, which involves addition, subtraction, multiplication and division of numbers. For example, a chocolate costs Rs 5/- and we want to buy two chocolates; the total cost of the two chocolates will be Rs. 10/-, i.e., (5*2) or (5+5).

    The binary number system is used in the computer systems. The digits 0 and 1 are combined to get different binary numbers like 1001, 11000110 etc. In a binary number, a digit 0 or 1 is called a bit. For example, 1001 is a 4-bit binary number, and, 11000110 is an 8-bit binary number. All kinds of data, be it alphabets, numbers, symbols, sound data or video data, are represented as combination of bits i.e. 0’s and 1’s. Each character is a unique combination of bits. We shall now discuss how to perform basic arithmetic operations in the binary number system.


    2. Binary Arithmetic

    The arithmetic operations - addition, subtraction, multiplication and division, performed on the binary numbers is called binary arithmetic . In computer systems, the basic arithmetic operations performed on the binary numbers is -

    • Binary Addition, and
    • Binary Subtraction,

    In the following subsections, we discuss the binary addition and the binary subtraction operations.


    2.1. Binary Addition

    Binary addition involves adding of two or more binary numbers. The binary addition rules are used while performing the binary addition. Table 1 shows the binary addition rules.

    Table 1. Binary Addition Rules for two inputs


    Binary addition of three inputs, when all the inputs are 1, follows the rule shown in Table 2.

    Table 2. Binary Addition Rules for three inputs


    Addition of the binary numbers involves the following steps-

    1. Start addition by adding the bits in unit column (the rightmost column). Use the rules of binary addition.
    2. The result of adding bits of a column is a sum with or without a carry.
    3. Write the sum in the result of that column.
    4. If carry is present, the carry is carried-over to the addition of the next left column.
    5. Repeat steps 2-4 for each column, i.e., the tens column, hundreds column and so on.

    Let us now understand binary addition with the help of some examples.

    Example 1. Add 10 and 01. Verify the answer with the help of decimal addition.

    When we add 0 and 1 in the unit column, sum is 1 and there is no carry. The sum 1 is written in the unit column of the result. In the tens column, we add 1 and 0 to get the sum 1. There is no carry. The sum 1 is written in the tens column of the result.



    Example 2. Add 01 and 11. Verify the answer with the help of decimal addition.

    When we add 1 and 1 in the unit column, sum is 0 and carry is 1. The sum 0 is written in the unit column of the result. The carry is carried-over to the next column, i.e. the tens column. In the tens column, we add 0, 1 and the carried-over 1, to get sum 0 and carry 1. The sum 0 is written in the tens column of the result. The carry 1 is carried-over to the hundreds column. In the hundreds column, the result is 1.

    Example 3. Add 11 and 11. Verify the answer with the help of decimal addition.

    When we add 1 and 1 in the unit column, sum is 0 and carry is 1. The sum 0 is written in the unit column of the result. The carry is carried-over to the tens column. In the tens column, we add 1, 1 and the carried-over 1. We use the rule for binary addition of three inputs shown in Table 2. We get sum 1 and carry 1. The sum 1 is written in the tens column of the result. The carry 1 is carried-over to the hundreds column. In the hundreds column, the result is 1.


    Example 4. Add 1101 and 1111. Verify the answer with the help of decimal addition.


    Example 5. Add 10111, 11100 and 11. Verify the answer with the help of decimal addition.



    2.2. Binary Subtraction

    Binary subtraction involves subtracting of two binary numbers. The binary subtraction rules are used while performing the binary subtraction. The binary subtraction rules are shown in Table 3.


    The steps for performing subtraction of the binary numbers are as follows-

    1. Start subtraction by subtracting the bit in the lower row from the upper row, in the unit column.
    2. Use the binary subtraction rules. If the bit in the upper row is less than lower row, borrow 1 from the upper row of the next column (on the left side). The result of subtracting two bits is the difference.
    3. Write the difference in the result of that column.
    4. Repeat step 2-3 for each column, i.e., the tens column, hundreds column and so on.

    Let us now understand binary subtraction with the help of some examples.

    Example 1. Subtract 01 from 11. Verify the answer with the help of decimal subtraction. When we subtract 1 from 1 in the unit column, the difference is 0. Write the difference in the unit column of the result. In the tens column, subtract 0 from 1 to get the difference 1. Write the difference in the tens column of the result.


    Example 2. Subtract 01 from 10. Verify the answer with the help of decimal subtraction.

    When we subtract 1 from 0 in the unit column, we have to borrow 1 from the left column since 0 is less than 1. After borrowing from the left column, 0 in the unit column becomes 10, and, 1 in the left column becomes 0. We perform 10-1 to get the difference 1. We write the difference in the unit column of the result. In the tens column, subtract 0 from 0 to get the difference 0. We write the difference 0 in the tens column of the result.


    Example 3. Subtract 0111 from 1110. Verify the answer with the help of decimal subtraction.

    When we do 0-1 in the unit column, we have to borrow 1 from the left column since 0 is less than 1. After borrowing from the left column, 0 in the unit column becomes 10, and, 1 in the left column becomes 0. We perform 10-1 to get the difference 1. We write the difference in the unit column of the result. In the tens column, when we do 0-1, we again borrow 1 from the left column. We perform 10-1 to get the difference 1. We write the difference in the tens column of the result. In the hundreds column, when we do 0-1, we again borrow 1 from the left column. We perform 10-1 to get the difference 1. We write the difference in the hundreds column of the result. In the thousands column, 0-0 is 0. We write the difference 0 in the thousands column of the result.


    Example 4. Subtract 10010 from 10101. Verify the answer with the help of decimal subtraction.


    Example 5. Subtract 101111 from 110001. Verify the answer with the help of decimal subtraction.



    3. Signed and Unsigned Numbers

    A binary number may be positive or negative. Generally, we use the symbol “+” and “-” to represent positive and negative numbers, respectively. The sign of a binary number has to be represented using 0 and 1, in the computer. An n-bit signed binary number consists of two parts – sign bit and magnitude. The left most bit, also called the Most Significant Bit (MSB) is the sign bit. The remaining n-1 bits denote the magnitude of the number.

    In signed binary numbers, the sign bit is 0 for a positive number and 1 for a negative number. For example, 01100011 is a positive number since its sign bit is 0, and, 11001011 is a negative number since its sign bit is 1. An 8-bit signed number can represent data in the range -128 to +127 (-27 to +27-1). The leftmost bit is the sign bit.

    In an n-bit unsigned binary number, the magnitude of the number n is stored in n bits. An 8-bit unsigned number can represent data in the range 0 to 255 (28= 256).


    3.1. Complement of Binary Numbers

    Complements are used in computer for the simplification of the subtraction operation. We now see, how to find the complement of a binary number. There are two types of complements for the binary number system – 1’s complement and 2’s complement.

    • 1’s complement of Binary number is computed by changing the bits 1 to 0 and the bits 0 to 1. For example,
      • 1’s complement of 110 is 001
      • 1’s complement of 1011 is 0100
      • 1’s complement of 1101111 is 0010000
    • 2’s complement of Binary number is computed by adding 1 to the 1’s complement of the binary number. For example,
      • 2’s complement of 110 is 001 + 1 = 010
      • 2’s complement of 1011 is 0100 + 1 = 0101
      • 2’s complement of 1101111 is 0010000 + 1 = 0010001

    4. Binary Data Representation

    A binary number may also have a binary point, in addition to the sign. The binary point is used for representing fractions, integers and integer-fraction numbers. Registers are high-speed storage areas within the Central Processing Unit (CPU) of the computer. All data is brought into a register before it can be processed. For example, if two numbers are to be added, both the numbers are brought in registers, added, and the result is also placed in a register. There are two ways of representing the position of the binary point in the register - fixed point number representation and floating point number representation.

    The fixed point number representation assumes that the binary point is fixed at one position. The binary point is not actually present in the register, but its presence is assumed based on whether the number which is stored is a fraction or an integer.

    The floating point number representation uses two registers. The first register stores the number without the binary point. The second register stores a number that indicates the position of the binary point in the first register.

    We shall now discuss representation of data in the fixed point number representation and floating point number representation.


    4.1. Fixed Point Number Representation

    The integer binary signed number is represented as follows –

    • For a positive integer binary number, the sign bit is 0 and the magnitude is a positive binary number.
    • For a negative integer binary number, the sign bit is 1. The magnitude is represented in any one of the three ways-
      • Signed Magnitude representation - The magnitude is the positive binary number itself.
      • Signed 1’s complement representation - The magnitude is the 1’s complement of the positive binary number.
      • Signed 2’s complement representation - The magnitude is the 2’s complement of the positive binary number.
    Table 4 shows the representation of the signed number 18.


    Signed magnitude and signed 1’s complement representation are seldom used in computer arithmetic. Let’s now perform arithmetic operations on the signed binary numbers. We use the signed 2’s complement representation to represent the negative numbers.

    • Addition of signed binary numbers – The addition of any two signed binary numbers is performed as follows -
      • Represent the positive number in binary form.(For e.g., +5 is 0000 0101 and +10 is 0000 1010)
      • Represent the negative number in 2’s complement form. (For e.g., -5 is 1111 1011 and -10 is 1111 0110)
      • Add the bits of the two signed binary numbers.
      • Ignore any carry out from the sign bit position.
      • Please note that the negative output is automatically in the 2’s complement form. We get the decimal equivalent of the negative output number, by finding its 2’s complement, and attaching a negative sign to the obtained result. Let’s understand the addition of two signed binary numbers with the help of some examples.

        Example 1. Add +5 and +10.

        We represent +5 in binary form, i.e., 0000 0101. We represent +10 in binary form, i.e., 0000 1010. Add the two numbers. The result is 0000 1111 i.e. +15.




        Example 2. Add -5 and +10.

        We represent -5 in 2’s complement form, i.e., 1111 1011. We represent +10 in binary form, i.e., 0000 1010. Add the two numbers. The result is 0000 0101 i.e. +5.


        Example 3. Add +5 and -10.

        We represent +5 in binary form, i.e., 0000 0101. We represent -10 in 2’s complement form, i.e., 1111 0110. Add the two numbers. The result is 1111 1011. The result is in 2’s complement form.



        To find the decimal equivalent of the result 1111 1011 – Find the 2’s complement of 1111 1011, i.e., 0000 0100 + 1 = 0000 0101. This is binary equivalent of +5. Attaching a negative sign to the obtained result gives us -5.

        Example 4. Add -5 and -10.

        We represent -5 in 2’s complement form, i.e., 1111 1011. We represent -10 in 2’s complement form, i.e., 1111 0110. Add the two numbers. The result is 1111 0001. The result is in 2’s complement form.


        To find the decimal equivalent of the result 1111 0001- Find the 2’s complement of 1111 0001, i.e., 0000 1110 + 1 = 0000 1111. This is binary equivalent of +15. Attaching a negative sign to the obtained result gives us -15.

    • Subtraction of signed binary numbers – The subtraction of signed binary numbers is changed to the addition of two signed numbers. For this, the sign of the second number is changed before performing the addition operation.

    • (-A) – (+B) = (-A) + (-B) (+B in subtraction is changed to –B in addition)

      (+A) – (+B) = (+A) + (-B) (+B in subtraction is changed to –B in addition)

      (-A) – (-B) = (-A) + (+B) (-B in subtraction is changed to +B in addition)

      (+A) – (-B) = (+A) + (+B) (-B in subtraction is changed to +B in addition)

      We see that the subtraction of signed binary numbers is performed using the addition operation.

      The hardware logic for the fixed point number representation is simple, when we use 2’s complement for addition and subtraction of the signed binary numbers.

    4.2. Floating Point Number Representation

    The floating point representation of a number has two parts – mantissa and exponent. The mantissa is a signed fixed point number. The exponent shows the position of the binary point in the mantissa.
    For example, the binary number +11001.11 with an 8-bit mantissa and 6-bit exponent is represented as follows -

    • Mantissa is 01100111. The left most 0 indicates that the number is positive.
    • Exponent is 000101. This is the binary equivalent of decimal number +5.
    • The floating point number is Mantissa x 2exponent i.e. + (.1100111) x 2+5.

    The arithmetic operation with the floating point numbers is complicated, and uses complex hardware as compared to the fixed point representation. However, floating point calculations are required in scientific calculations, so, computers have a built-in hardware for performing floating point arithmetic operations.


    5. Binary Coding Schemes

    The alphabetic data, numeric data, alphanumeric data, symbols, sound data and video data, all are represented as combination of bits in the computer. The bits are grouped in a fixed size, such as 8 bits, 6 bits or 4 bits. A code is made by combining bits of definite size. Binary Coding schemes represent the data such as alphabets, digits 0-9, and symbols in a standard code. A combination of bits represents a unique symbol in the data. The standard code enables any programmer to use the same combination of bits to represent a symbol in the data.
    The binary coding schemes that are most commonly used are -

    • Extended Binary Coded Decimal Interchange Code (EBCDIC),
    • American Standard Code for Information Interchange (ASCII),and
    • Unicode

    5.1. EBCDIC

    • The Extended Binary Coded Decimal Interchange Code (EBCDIC) uses 8 bits (4 bits for zone, 4 bits for digit) to represent a symbol in the data.
    • EBCDIC allows 2 8 = 256 combinations of bits.
    • 256 unique symbols are represented using EBCDIC code. It represents decimal numbers (0-9), lower case letters (a-z), uppercase letters (A-Z), Special characters, and Control characters (printable and non-printable e.g. for cursor movement, printer vertical spacing etc.).
    • EBCDIC codes are used, mainly, in the mainframe computers.

    5.2. ASCII

    • The American Standard Code for Information Interchange (ASCII) is widely used in computers of all types.
    • ASCII codes are of two types –ASCII-7 and ASCII-8.
    • ASCII-7 is a 7-bit standard ASCII code. In ASCII-7, the first 3 bits are the zone bits and the next 4 bits are for the digits. ASCII-7 allows 27 = 128 combinations. 128 unique symbols are represented using ASCII-7. ASCII-7 has been modified by IBM to ASCII-8.
    • ASCII-8 is an extended version of ASCII-7. ASCII-8 is an 8-bit code having 4 bits for zone and 4 bits for the digit. ASCII-8 allows 28= 256 combinations. ASCII-8 represents 256 unique symbols. ASCII is used widely to represent data in computers.
    • The ASCII-8 code represents 256 symbols.
      • Codes 0 to 31 represent control characters (non-printable), because they are used for actions like, Carriage return (CR), Bell (BEL) etc.
      • Codes 48 to 57 stand for numeric 0-9.
      • Codes 65 to 90 stand for uppercase letters A-Z.
      • Codes 97 to 122 stand for lowercase letters a-z.
      • Codes 128-255 are the extended ASCII codes.

    5.3. Unicode

    • Unicode is a universal character encoding standard for the representation of text which includes letters, numbers and symbols in multi-lingual environments. The Unicode Consortium based in California develops the Unicode standard.
    • Unicode uses 32 bits to represent a symbol in the data.
    • Unicode allows 232 = 4164895296 (~ 4 billion) combinations.
    • Unicode codes can uniquely represent any character or symbol present in any language like Chinese, Japanese etc. In addition to the letters; mathematical and scientific symbols are also represented in Unicode codes.
    • An advantage of Unicode is that it is compatible with the ASCII-8 codes.
    • The first 256 codes in Unicode are identical to the ASCII-8 codes.
    • Unicode is implemented by different character encodings. UTF-8 is the most commonly used encoding scheme. UTF stands for Unicode Transformation Format. UTF-8 uses 8 bits to 32 bits per code.
    If you wish to see the Unicode character encoding in MS-Word 2007, do as follows-
    < >. A Symbol dialog box will appear which displays the symbols, and the character codes in a coding scheme, as shown in Figure 1.


    Figure 1 Unicode Coding

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