Download Binary Arithmetic: Multiplication, Two's-complement, Division, and Codes in EECC341 and more Lecture notes Number Theory in PDF only on Docsity!
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Binary Multiplication Binary Multiplication
- Multiplication is achieved by adding a list of shifted
multiplicands according to the digits of the multiplier.
11 1 0 1 1 multiplicand (4 bits)
X 13 X 1 1 0 1 multiplier (4 bits)
33 1 0 1 1
11 0 0 0 0
______ 1 0 1 1
143 1 0 1 1
1 0 0 0 1 1 1 1 Product (8 bits)
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Binary Multiplication (continued)
- Instead of listing all shifted multiplicands and then
adding, we can add each shifted multiplicand to a partial product. The previous un-signed example becomes:
11 1101 multiplicand
x 13 x 1101 multiplier
143 0000 partial product
1011 shifted multiplicand
01011 partial product
0000 shifted multiplicand
001011 partial product
1011 shifted multiplicand
0110111 partial product
1011 shifted multiplicand
10001111 product
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Binary Division Binary Division
Example:
19 10011 quotient
11 217 1011 11011001 dividend
11 1011 shifted divisor 107 0101 reduced dividend 99 0000 shifted divisor 8 1010 reduced dividend 0000 shifted divisor 10100 reduced dividend 1011 shifted divisor 10011 reduced dividend 1011 shifted divisor 1000 remainder
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Binary Codes Binary Codes
• Groups of binary bits are often organized in specific
ways or binary codes to:
- Represent decimal numbers or alphabetic characters:
- Binary Coded Decimal (BCD).
- American Standard Code for Information Interchange (ASCII)
- Detect errors:
- Even parity code.
- Odd parity code.
- Correct errors:
– Aid in transmission and storage of digital
information.
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BCD Addition BCD Addition
- Addition of BCD digits is similar to adding 4-bit
unsigned binary numbers except a correction must be made if a result exceeds 1001 by adding 6 to the digit.
0101
- 1001 1110
- 0110 Correction add 6 0001 0100
5
Example:
1 4
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Alphanumeric Binary Codes: ASCII Alphanumeric Binary Codes: ASCII
M S B s L S B s 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1
0 0 0 0 N U L^ D L E^ S P^0 @^ P^ `^ p 0 0 0 1 S O H D C 1! 1 A Q a q 0 0 1 0 S T X^ D C^2 “^2 B^ R^ b^ r 0 0 1 1 E T X D C 3 # 3 C S c s 0 1 0 0 E O T D C 4 $ 4 D T d t 0 1 0 1 E N Q N A K % 5 E U e u 0 1 1 0 A C K S Y N & 6 F V f v 0 1 1 1 B E L E T B ‘ 7 G W g w 1 0 0 0 B S^ C A N^ (^8 H^ X^ h^ x 1 0 0 1 H T^ E M^ )^9 I^ Y^ i^ y 1 0 1 0 L F S U B * : J Z j z 1 0 1 1 V T E S C + ; K [ k { 1 1 0 0 F F F S , < L \ l | 1 1 0 1 C R G S - = M ] m } 1 1 1 0 O R S. > N ^ n ~ 1 1 1 1 S I U S /? O _ o D E L
Seven bit codes are used to represent all upper and lower case letters, numbers, punctuation and control characters
