Korea University of Technology and Education

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1 MEC52 디지털공학 Binary Systems Jee-Hwan Ryu School of Mechanical Engineering Binary Numbers a 5 a 4 a 3 a 2 a a.a - a -2 a -3 base or radix = a n r n a n- r n-...a 2 r 2 a ra a - r - a -2 r -2...a -m r -m 7392 = (.) 2 = = (26.75) = Kilo = Mega = Giga = Tera

2 Binary Number Augend Minuend: Multiplicand: Addend Subtrahend: - Multiplier: * sum Difference: Product: Number Base Conversions Ex -) Convert decimal 4 to binary. Integer Quotient Remainder Coefficient Integer 4 Remainder 4/2 = 2/2 = /2 = 2 5 ½ a = a = a 2 = 2 5/2 = 2/2 = /2 = 2 ½ ½ a 3 = a 4 = a 5 = 5 2 answer : (4) = (a 5 a 4 a 3 a 2 a a ) 2 = () 2

3 Number Base Conversions Ex -2) Convert decimal 53 to octal = (23) 8 Ex -3) Convert (.6875) to binary Integer Fraction Coefficient.6875*2 =.375 a - =.375*2 =.75 a -2 =.75*2 =.5 a -3 =.5*2 =. a -4 = Answer:(.6875) = (.a - a -2 a -3 a -4 ) 2 = (.) 2 Number Base Conversions Ex -4) Convert (.53) to octal..53 *8 =.4 *8 =.832 *8 =.656 *8 =.248 * 8 =.984 * 8 = Coefficie nt a - = 4 a -2 = a -3 = 6 a -4 = 5 a -5 = a -6 = 7 Answer:(.53) = (.a - a -2 a -3 a -4. ) 2 = ( ) 8

4 Octal and Hexadecimal Numbers (. ) 2 = ( ) (. ) 2 = (2C6B.F2) 6 2 C 6 B F 2 Examples. Convert decimal to binary 2. Convert the following numbers with the indicated bases to decimal: (43)5 and (98)2 3. Convert the hexadecimal number 68BE to binary and then from binary convert it to octal

5 Complements Complements are used in digital computers for simplifying the subtraction operation and for logical manipulation. Two types of complements for each base-r system - radix complement or r s complement - Diminished radix complement or r- s complement For binary numbers, r = 2, 2 s complement or s complement For decimal numbers, r =, s complement or 9 s complement Complements (r-) s complements of N is (r n -)-N r=, r-=9, 9 complements of N is ( n -)-N Ex) the 9 s complements of 5467 is = the 9 s complements of 2398 is = 9876 For binary number, r=2, r-= complements of N is (2 n -)-N Ex) the s complements of is the s complements of is

6 Complements The r s complements of an n-digit number N is r n -N, N, N= r n -N=[(r n -)-N] => The r s complements is obtained by adding to the (r-) s complements Ex) The s complements of 2398 is The s complements of 2467 is 7533 The 2 s complements of is The 2 s complements of is Examples. Find the 9 s and s-complement of the following decimal numbers: (a) (b) ( c) (d) 2. Find the 6 s- complement of AF3B 3. Obtain the s and 2 s complements of the following binary numbers: (a) (b) (c)

7 Complements Subtraction with Complements Ex-5) using s complement, subtract M = s complement of N = Sum = Discard end carry 5 = Answer = Ex-6) Using s complement, subtract M = s complement of N = Sum = 378 There is no end carry Therefore, the answer is ( s complement of 378)= Complements Subtraction with Complements Ex-7) X=, Y=, (a) X-Y, (b) Y-X (a) X-Y (b) Y-X X = 2 s complement of Y = Sum = Discard end carry 2 7 = Answer: X-Y = Y = 2 s complement of X = Sum = - There is no carry. The answer is Y-X = -(2 s complement of )=-

8 Complements Subtraction with Complements Ex-8) Repeat Example -7 using complement. (a) X-Y = - X = s complement of Y = Sum = End-around carry = Answer: X-Y = (b) Y-X = - Y = s complement of X = Sum = There is no carry. The answer is Y-X = -( s complement of )=- Complement of a number N contains radix point EX.) Complement of. and

9 Signed Binary Numbers Ex) The number 9 represented in binary with eight bit 9 : -9 : (signed-magnitude representation) (signed- s-complement representation) (signed-2 s-complement representation) Signed Binary Numbers Arithmetic Addition - signed-magnitude system follows the rules of ordinary arithmetic. - signed-complement system requires only addition. 2 s complement If the sum obtained after the addition is negative, it is 2 scomplement form

10 Problems of Signed-magnitude System (25) (-37) = -(37 25) = -2. If the signs are the same, sign comparison operation - add the two magnitudes addition operation - give the sum the common sign 2. If the signs are different, sign comparison operation - subtract the smaller magnitude from the larger subtraction - give the result the sign of the larger magnitude sign 과 magnitude 비교, addition or subtraction 필요 Signed Binary Numbers If 8 bit signed-complement system (?) -29 (?) 8-bit signed-complement system 의 Range: -28 ~ 27 Overflow - If two n-bit numbers, the sum occupies n bits, we say that an overflow occurs. - Overflow is a problem in computers.

11 Arithmetic Subtraction. Take the 2 s complement of the subtrahend (including the sign bit) 2. Add it to the minuend(including the sign bit) 3. A carry out of the sign-bit position is discarded Arithmetic Subtraction (±A)-(B) = (±A)(-B) (±A)-(-B) = (±A)(B) -> A subtraction operation can be changed to an addition operation if the sign of the subtrahend is changed. Computers need only one common hardware circuit to handle both types of arithmetic. Binary Code-BCD code(binary Code for Decimal) - the 4-bit code for one decimal (85) = ( ) BCD = () 2 BCD Addition if the binary sum is greater or equal to, we add to obtain the correct BCD (This is because the difference between msb of the sum and a decimal carry differ by 6- = 6.)

12 Binary Code-Other Decimal Codes Conversion Flow Chart

13 Binary Code-Gray Code Gray Code 의장점과용도는? Binary Code- ASCII Character Code

14 Binary Code-Error-Detecting Code Error-Detecting Code: To detect errors in data communication and processing, and eighth bit is sometimes added to the ASCII character. ASCII A = ASCII T = With even parity With odd parity Examples. Decode the following ASCII code: Answer: Jane Doe 2. Represent decimal number 627 in (a) BCD, (b) excess-3 code, (c) 242 code Answer: a) b) Excess-3: c) 242:

15 Binary Storage and Registers Registers A register with n cells can store any discrete quantity of information that contains n bits. Register Transfer Binary Logic Definition of Binary Logic Logic Gates

16 Binary Logic