Binary Representation and Computer Arithmetic DECIMAL and BINARY

Binary Representation and Computer Arithmetic

DECIMAL and BINARY system n The Decimal System n This uses ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) to represent numbers n The Binary System n uses just two digits, 0 and 1 n easier for a computer to distinguish between two different voltage levels (than ten)

POSITIONAL NOTATION (in decimal) Consider the decimal number 5237. Positional thousands hundreds Name tens units Positional 103 Weight 102 101 100 Value 1 000 10 1 5 2 3 7 5 237 5 x 1 000 + 2 x 100 + 3 x 10 + 7 x 1

POSITIONAL NOTATION (in binary) Consider the byte 1100 0101 Positional Weight 27 26 25 24 23 22 21 20 Value 128 64 32 16 8 4 2 1 Number 1 0 0 0 1 Positional Notation 1100 0101 = 1 x 128 + 1 x 64 + 0 x 32 + 0 x 16 + 0 x 8 + 1 x 4 + 0 x 2 + 1 x 1 = 197 1 Note the LSD in 1100 0101 is the right most digit which is 1 and the MSD is the left most digit which is 1.

Representing Integers in Binary ASCII codes are pure or true binary numbers. By that we mean there is no positive or negative sign attached to these numbers. We are now going to look at representing integers in binary.

Representing Integers in Binary Here are some of the challenges we face when we move to representing integers in binary: 1. Integers can be positive or negative. 2. Must have a unique representation. 3. Should be able to add and subtract them 4. Operations must be efficient ( using the smallest number of circuits possible)

INTEGER REPRESENTATIONS There are 4 commonly known (1 not common) integer representations. All have been used at various times for various reasons. 1. unsigned 2. sign magnitude 3. one's complement 4. two's complement 5. biased (not commonly known)

UNSIGNED BINARY NUMBER SYSTEM The standard binary encoding already given n only positive values n range: 0 to 2**n - 1, for n bits example: 4 bits, values 0 to 15 n=4, 2**4 -1 is 15 n

Sign-Magnitude Binary Number System The sign and magnitude number system is a simple binary code system used to represent positive and negative integers. n In this system, the first bit (the MSB) in a binary representation is a sign bit (0 for positive and 1 for negative) followed by the magnitude bits. n

Sign-Magnitude Binary Number System Example 0101 represents the number +5. n The most significant bit (MSB) =0. The 0 indicates the number is positive. n The remaining 3 bits represent the magnitude of the number. The decimal value of binary 101 is 4 + 0 + 1 = 5. n Therefore, 0101, using a 4 -bit sign-magnitude convention is the integer +5

PROBLEM How many positive binary numbers can be represented in a nibble using the sign-magnitude notation? How many positive binary numbers can be represented in a byte using the sign-magnitude notation?

Representation of Negative Integers using Sign-Magnitude Notation Example n When an 8 -bit sign-magnitude convention for representing numbers is used, the number 1011 0111 represents the negative integer -55. n The first bit, the MSB, is 1 indicating the integer is negative. n The next 7 bits 0111 represent the magnitude of the integer. Binary 0111 has decimal value 32 + 16 + 4 + 2 + 1 = 55.

The 1's Complement form n The 1's complement representation in binary of a positive integer is same as the signmagnitude representation of that integer. Example Using a byte, the 1's complement in binary of +84 is 0101 0100.

The 1's Complement in Binary of a Negative Integer Rule: The 1's complement in binary of a negative integer is obtained by subtracting its magnitude from 2 n -1 where n is the number of bits used to store the integer in binary.

BINARY ARITHMETIC

BINARY ADDITION Rule 1 Rule 2 Rule 3 Rule 4 Rule 5 0 +0 0 0 +1 1 1 +0 1 1 + 1 1 0 1 1 +1 11

BINARY ADDITION View online lesson

BINARY ADDITION Example n Add these binary numbers: n 1100 (12)+ n + 1101 (13) n 1 1001 (25)

BINARY SUBTRACTION Rule 1 Rule 2 Rule 3 Rule 4 0 - 0 0 1 - 1 0 1 - 0 1 0 - 1 1

BINARY SUBTRACTION View online lesson

BINARY SUBTRACTION Example Subtract these binary numbers: n 1101 (13)n + 0110 ( 6) n 0111 (7) n

The 1's Complement in Binary of a Negative Integer Example Store the integer -36 in a byte in 1's complement form. Step 1: +3610 = 0010 01002 [convert the magnitude of the integer to binary] Step 2: 1111 - 0010 0100 1101 1011 [A byte contains 8 bits. Therefore, subtract 0010 0100 from 28 -1= 255. In binary 255 is 1111]

EASY WAY Determine the 1's complement of a negative integer n Determine the 1's complement of the corresponding positive integer n invert all bits.

EXAMPLE Using a byte, find the 1's complement of -57. n +57 stored in a byte in 1's complement form is 0011 1001. n Invert all bits to determine the 1's complement of -57 to be 1100 0110.

Two's Complement Notation The 2's Complement in Binary of a Positive Integer n same as sign-magnitude notation

Two's Complement Notation The 2's Complement of a Negative Integer 1. 2. 3. First, we write the positive value of the number in binary. Next, we reverse each bit to get the 1's complement. Last, we add 1 to the number.

Example Store -27 in a byte using 2's complement notation. n Step 1: +27 in binary is 0001 1011. n Step 2: Invert bits to yield 1110 0100. Then the 1's complement of -27 is 1110 0100. n Step 3: Add 1 to the 1's complement: 1110 0100 + 1 1110 0101 The 2's complement of -27 is 1110 0101.

Binary Arithmetic Subtraction with one’s complement n Subtraction with two’s complement n

Fractions Converting a fractional decimal value to binary

Floating Point Arithmetic Floating point arithmetic derives its name from something that happens when you use exponential notation. Consider the number 123: it can be written using exponential notation as: n 1. 23 * 10 2 n 12. 3 * 10 1 n 123 * 10 0 n. 123 * 10 3 n 1230 * 10 -1

Floating Point Arithmetic All of these representations of the number 123 are numerically equivalent. n They differ only in where the decimal point appears in the number. The decimal point "floats" within the number as the exponent is changed. This phenomenon gives floating point numbers their name. n

Floating Point Arithmetic n The first representation, 1. 23 * 10 2, is in a form called "scientific notation“ n In scientific notation, the significand is always a number greater than or equal to 1 and less than 10. (normalization)

Floating Point Number A floating-point number is often stored as three parts: n A significand or mantissa (indicating the digits that define the number's magnitude) n An exponent or scale (indicating the position of the radix point) n A sign (indicating whether the number is positive or negative)

Floating Point Arithmetic The exponent does not always indicate a power of ten; sometimes powers of sixteen are used instead, or, most commonly of all, powers of two The real number -0. 125 is equal to n -1. 25 x 10^-1 if powers of ten are used, n -2 x 16^-1 if powers of sixteen are used, n -1 x 2^-3 if powers of two are used. The absolute value of the mantissa is, however, always greater than or equal to 1 and less than the base of numeration.

Floating Point Formats n Over the years, floating point formats in computers have not exactly been standardized. While the IEEE (Institute of Electrical and Electronics Engineers) has developed standards in this area, they have not been universally adopted.

In the IEEE single-precision representation of a real number One bit is reserved for the sign, and it is set to 0 for a positive number and to 1 for a negative one. n A representation of the exponent is stored in the next eight bits. n The remaining twenty-three bits are occupied by a representation of the mantissa of the number. n

IEEE 754 Standard (32 bits) 31 30 29 28 27 26 25 24 23 22 21 2 Sign bit Exponent Mantissa (1 bit) (8 bits) (23 bits) 1 0

IEEE 754 Standard (32 bits) n n n Sign bit = 0 or 1 The exponent, which is a signed integer in the range from -126 to 127, is represented as a biased value. (127 is the bias used) Only the part of the mantissa that comes after the binary point is actually stored. This suppressed digit at the beginning of the mantissa is called the ``hidden bit. ''

Floating Point Arithmetic n In a binary computer, all numbers are stored in base 2 instead of base 10 n The absolute value of the mantissa is, however, always greater than or equal to 1 and less than 2.

EXAMPLE Find the IEEE single-precision representation of 5. 75. n The sign bit is 0, since the number is positive n (5) = (101) n To convert 0. 75 do repeated multiplication 10 2

EXAMPLE n . 75 X 2 = 1. 5 ---- 1 n . 5 X 2 = 1. 0 ----- 1 n 0 ------ (stop) (. 75) = (. 11) 10 2

EXAMPLE (5. 75)10 = (101. 11)2 n Normalizing 101. 11 --- 1. 0111 X 22 Hidden bit n n mantissa Adding bias to the exponent 127+2 = 129 (129)10 = ( 1000 0001 )2 Exponent

And the result…. Lining up, we get 0 10000001 01110000000000

More Examples Convert from Decimal to Binary