1、1,Binary Arithmetic,Binary addition Binary subtraction Binary multiplication Binary division,2,Complements of Binary Numbers,1s complements 2s complements,3,Complements of Binary Numbers,1s complement Change all 1s to 0s and all 0s to 1s,1 0 1 0 1 0 1 0,0 1 0 1 0 1 0 1,4,Complements of Binary Number
2、s,2s complement Find 1s complement and then add 1,1 0 1 0 1 0 1 0,0 1 0 1 0 1 0 1,Input bits Adder Output bits (sum),Carry In,(add 1),1,0 1 0 1 0 1 1 0,2s complement,1s complement,5,Signed Numbers,6,Topics for Signed Numbers,Signed-magnitude form 1s and 2s complement form Decimal value of signed num
3、bers (How to convert) Range of values (max and min) Floating-point numbers,7,Signed Numbers,Signed-magnitude form The sign bit is the left-most bit in a signed binary number A 0 sign bit indicates a positive magnitude A 1 sign bit indicates a negative magnitude,8,Signed Numbers,1s complement form A
4、negative value is the 1s complement of the corresponding positive value 2s complement form A negative value is the 2s complement of the corresponding positive value,9,Signed Numbers,Decimal value of signed numbers Sign-magnitude 1s complement 2s complement,10,Signed Numbers,Range of ValuesTotal comb
5、inations = 2n 2s complement form: (2n 1) to + (2n 1 1),Range for 8 bit number: n = 8 -(28-1) = -27 = -128 minimum +(28-1) 1 = +27 - 1 = +127 maximum Total combination of numbers is 28 = 256.,11,Signed Numbers,Range for 16 bit number: n = 16 -(216-1) = -215 = -32768 minimum +(216-1) - 1 = +215 = +327
6、67 maximum Total combinations is 216 = 65536 (64K),8 bit examples:10000000 =,-128,11111111 =,-1,10000001 =,-127,01111111 =,+127,12,Signed Numbers,Floating-point numbers Can represent very large or very small numbers based on scientific notation. Binary point “floats”. Two Parts Mantissa represents m
7、agnitude of number Exponent represents number of places that binary point is to be moved Three forms Single-precision (32 bits) float Double-precision (64 bits) double Extended-precision (80 bits) long double Also have Quadruple and Quadruple extended!,13,Single Precision,IEEE 754 standard Mantissa
8、(F) has hidden bit so actually has 24 bits. Gives 7 significant figures. 1st bit in mantissa is always a one Exponent (E) is biased by 127 called Excess-127 Notation Add 127 to exponent so easier to compare Range of exponents is -126 to +128 Sign (S) bit tells whether number is negative or positive,
9、S Exponent (E) Mantissa (fraction, F),32 bits,1 bit 8 bits 23 bits,14,Single Precision,Example: Convert 577710 to Floating Point 1st, convert to binary using divide by 2 method,577710 = 10110100100012 Positive number, so sign bit (S) equals 0.,2nd, count number of places to move binary point10110100
10、100012 = 1.011010010001 x 212Add 127 to 12 = 13910 = 100010112,Mantissa is fractional part, 011010010001 Finally, put everything together,15,Special Cases,Zero and infinity are special cases Can have +0 or -0 depending on sign bit Can also have + or - Not a Number (NaN) if underflow or overflow,16,E
11、xamples,17,Double Precision,Exponent has 11 bits so uses Excess-1023 Notation Mantissa has 53 bits (one hidden) 53 bits gives 16 significant figures,18,Arithmetic Operations with Signed Numbers,Addition Subtraction Multiplication Division,19,Arithmetic Operations with Signed Numbers,Addition of Sign
12、ed Numbers The parts of an addition function are: Augend - The first number Addend - The second number Sum - The result,Numbers are always added two at a time.,20,Arithmetic Operations with Signed Numbers,Four conditions for adding numbers:1. Both numbers are positive.2. A positive number that is la
13、rger than a negative number.3. A negative number that is larger than a positive number.4. Both numbers are negative.,21,Arithmetic Operations with Signed Numbers,Signs for Addition When both numbers are positive, the sum is positive. When the larger number is positive and the smaller is negative, th
14、e sum is positive. The carry is discarded.,22,Arithmetic Operations with Signed Numbers,Signs for Addition When the larger number is negative and the smaller is positive, the sum is negative (2s complement form). When both numbers are negative, the sum is negative (2s complement form). The carry bit
15、 is discarded.,23,Examples (8 bit numbers),Add 7 and 4 (both positive)Add 15 and -6 (positive negative)Add 16 and -24 (negative positive)Add -5 and -9 (both negative),00000111 7 +00000100 + 4 00001011 11,24,Overflow,Overflow occurs when number of bits in sum exceeds number of bits in addend or augen
16、d. Overflow is indicated by the wrong sign. Occurs only when both numbers are positive or both numbers are negative,01111101 126+ 00111010 + 58 _ _10110111 183,Sign Incorrect Magnitude Incorrect,25,Arithmetic Operations with Signed Numbers,Subtraction of Signed Numbers The parts of a subtraction fun
17、ction are: Minuend - The first number Subtrahend - The second number Difference - The result,Subtraction is addition with the sign of the subtrahend changed.,26,Arithmetic Operations with Signed Numbers,Subtraction The sign of a positive or negative binary number is changed by taking its 2s compleme
18、nt To subtract two signed numbers, take the 2s complement of the subtrahend and add. Discard any final carry bit.,27,Subtraction Examples,Find 8 minus 3.Find 12 minus -9.Find -25 minus 19.Find -120 minus -30.,00001100 12 +00001001 - -9 00010101 21,10001000 -120 +00011110 - -30 10100110 -90,28,Arithm
19、etic Operations with Signed Numbers,Multiplication of Signed Numbers The parts of a multiplication function are: Multiplicand - First number Multiplier - Second number Product - Result,Multiplication is equivalent to adding a number to itself a number of times equal to the multiplier.,29,Arithmetic
20、Operations with Signed Numbers,There are two methods for multiplication: Direct addition add multiplicand multiple times equal to the multiplier Can take a long time if multiplier is large Partial products Similar to long hand multiplication,The method of partial products is the most commonly used.,
21、30,Arithmetic Operations with Signed Numbers,Multiplication of Signed Numbers If the signs are the same, the product is positive. (+ X + = + or - X - = +) If the signs are different, the product is negative. (+ X - = - or - X + = -),31,Multiplication Example,Both numbers must be in uncomplemented fo
22、rm Multiply 3 by -5.,Opposite signs, so product will be negative.310 = 000000112 -510 = 111110112 2s complement of -5 00000101,00000011 Multiplicand X 00000101 Multiplier 00000011 First partial product + 0000000 Second partial product 00000011 Sum of 1st and 2nd,+ 000011 Third partial product 000011
23、11 Sum and Final Product,Final result is negative, so take 2s complement. 11110001 is the result which in decimal is -15.,32,Arithmetic Operations with Signed Numbers,Division of Signed Numbers The parts of a division operation are: Dividend Divisor Quotient,Division is equivalent to subtracting the
24、 divisor from the dividend a number of times equal to the quotient.,33,Arithmetic Operations with Signed Numbers,Division of Signed Numbers If the signs are the same, the quotient is positive. (+ + = + or - - = +) If the signs are different, the quotient is negative. (+ - = - or - + = -),34,Division
25、 Example,Both numbers must be in uncomplemented form Divide 01100100 by 00110010.,Both numbers are positive so quotient will be positive. Set the quotient to zero initially.,01100100 Dividend + 11001110 2s complement of Divisor 1 00110010 First partial remainder Add 1 to quotient: 00000000 + 1 = 000
26、00001,quotient: 00000000,Subtract the divisor from the dividend by using 2s complement addition. (11001110) Ignore the carry bit.,00110010 First partial remainder + 11001110 2s complement of Divisor 1 00000000 zero remainder Add 1 to quotient: 00000001 + 1 = 00000010,Subtract the divisor from the 1s
27、t partial remainder using 2s complement addition.,So final quotient is 00000010 and final remainder is 00000000,35,Hexadecimal Numbers,36,Hexadecimal Numbers,Decimal, binary, and hexadecimal numbers 4 bits is a nibble FF16 = 25510,37,Hexadecimal Numbers,Binary-to-hexadecimal conversion Hexadecimal-t
28、o-decimal conversion Decimal-to-hexadecimal conversion,38,Hexadecimal Numbers,Binary-to-hexadecimal conversion Break the binary number into 4-bit groups Replace each group with the hexadecimal equivalent,Convert 1100101001010111 to Hex,Convert 10A416 to binary,39,Hexadecimal Numbers,Hexadecimal-to-decimal conversion Convert the hexadecimal to groups of 4-bit binary Convert the binary to decimal,
