Unsigned and Signed fixed point Addition and subtraction
- 1. Computer Architecture & organization ALU S.CIYAMALA KUSHBU ASSISTANT PROFESSOR/ECE
- 2. ALU
- 3. NUMBER SYSTEMS Human beings use decimal (base 10) and duodecimal (base 12) number systems for counting and measurements. Computers use binary (base 2) number system, as they are made from binary digital components (known as transistors) operating in two states - on and off. In computing, we also use hexadecimal (base 16) or octal (base 8) number systems, as a compact form for representing binary numbers. Example :10101010 AA(H) and 252(O)
- 5. WHAT IS FIXED AND FLOATING POINT? A fixed point number just means that there are a fixed number of digits after the decimal point. EX:123.45, 1234.56, 12345.67 A floating point number allows for a varying number of digits after the decimal point. EX:1.234567, 0.00001234567
- 6. FIXED POINT NUMBERS Computers use a fixed number of bits to represent an integer. The commonly-used bit-lengths for integers are 8-bit, 16-bit, 32-bit or 64-bit. There are two representation schemes for integers: Unsigned Integers: can represent zero and positive integers. Signed Integers: can represent zero, positive and negative integers. Three representation schemes had been proposed for signed integers: Sign-Magnitude representation 1's Complement representation 2's Complement representation
- 7. CONVERTION OF DECIMAL NUMBER 30 INTO 8 BIT BINARY NUMBER 8 7 6 5 4 3 2 1 8-bits 2^7 2^6 2^5 2^4 2^3 2^2 2^1 2^0 2^n representation 128 64 32 16 8 4 2 1 value 0 0 0 1 1 1 1 0 Eg: for value 30 in 8 bit
- 9. UNSIGNED NUMBERS N bits -Magnitude (absolute value) of the integer EXAMPLE: UNSIGNED ADDITION AND SUBTRACTION 8 - 4 in 4 bit representation 8 1000 4 0100 (-) ------------------ 4 0100 ---------------------- 8 + 4 in 4 bit representation 8 1000 4 0100 (+) ------------------ 12 1100 ----------------------
- 10. SIGNED NUMBERS
- 11. SIGN MAGINTUDE 1’S COMPLEMENT 2’S COMPLEMENT Nth bit or MSB N-1bits Sign bit FOR POSITIVE INTEGER Magnitude (absolute value) of the integer FOR NEGATIVE INTEGER Magnitude (absolute value) of the integer Sign bit FOR POSITIVE INTEGER Magnitude (absolute value) of the integer. FOR NEGATIVE INTEGER Magnitude of the complement (inverse) of the (n- 1)-bit binary pattern Sign bit FOR POSITIVE INTEGER Magnitude (absolute value) of the integer. FOR NEGATIVE INTEGER Magnitude of the complement of the (n-1)-bit binary pattern plus one Example: Taken 8 bit representati on +10000 0001 -11000 0001 +00000 0000 -01000 0000 +10000 0001 -11111 1110 +00000 0000 -01111 1111 +10000 0001 -11111 1111 +00000 0000 -00000 0000 Drawback 1. There are two representations for the number zero. 2. Positive and negative integers need to be processed separately. 1. There are two representations for the number zero. 2. Positive and negative integers need to be processed separately. 1.There is only one representation for the number zero 2.Positive and negative integers can be treated together in addition and subtraction. Subtraction can be carried out using the "addition logic".
- 12. SIGN MAGINTUDE 1’S COMPLEMENT 2’S COMPLEMENT Usual Subtraction in 5 bit representation +8 01000 +4 00100 (-) --------------- +4 00100 ---------------- +8 01000 +4 00100 (-) --------------- +4 00100 --------------- +8 01000 +4 00100 (-) --------------- +4 00100 --------------- Using addition to perform subtraction +8 01000 -4 10100 (+) --------------- -12 11100 ----------------- +8 01000 -4 11011 (+) --------------- +3 00011 --------------- with carry 1 which should be omitted +8 01000 -4 11100 (+) --------------- +4 00100 --------------- with carry 1 which should be omitted
- 13. OVERFLOW RULE FOR ADDITION If two numbers which are both positive or negative are added, then overflow occurs if and only if the result has the opposite sign OVERFLOW RULE FOR SUBTRACTION If two numbers with opposite sign are subtracted, then overflow occurs if and only if the result has the opposite sign