Signed numbers Assembly language programming By xorpd xorpd.net Objectives  You will recall the traditional ways to represent negative numbers  You will learn how to invoke “subtraction by addition” in base 10  You will learn about the “two’s complement” method to negate a binary number Motivation  We want to be able to represent negative numbers inside the computer  We want the usual arithmetic to work with the new representation for negative numbers Traditional negative decimals  In base 10, we traditionally use the ‘-’ symbol to represent a negative number  Examples:  −34510  −710  We can invoke arithmetic operations between positive numbers and negative numbers:  −510 + 710 = − = 210  −1110 − −3210 = 32 − 11 = 2110  It seems like everything works right Drawbacks of the traditional “-”  Before adding any two numbers, we have to check their signs, and act accordingly:  𝑎 + −𝑏 = 𝑎 − 𝑏 −𝑎 + 𝑏 = 𝑏 − 𝑎  −𝑎 + −𝑏 = − 𝑎 + 𝑏  𝑎+𝑏 =𝑎+𝑏   The usual subtraction is hard to invoke (We have to handle borrows from far away bits)  We want to only have only the addition operator  To calculate 𝑎 − 𝑏 We calculate instead 𝑎 + (−𝑏)  We have to keep the “−” sign before any signed number That means keeping one extra symbol before every number Subtraction by addition  We want to solve the following: 93210 − 15110 =?  We add and remove 100010 :  93210 − 15110 = 93210 + 100010 − 15110 − 100010  100010 − 15110 = 84910 (“Mechanical” operation)  We replace the subtraction with the result:  93210 − 15110 = 93210 + 84910 − 100010 = 178110 − 100010  178110 − 100010 = 78110 (“Mechanical” operation)  We conclude that 93210 − 15110 = 78110  We only used addition to solve it  All the subtraction operations were mechanical Subtraction by addition (Cont.)  We saw that:  93210 − 15110 = 78110  93210 + 84910 = 178110 Subtraction by addition (Cont.)  We saw that:  93210 − 15110 = 78110  93210 + 84910 = 178110 Subtraction by addition (Cont.)  We saw that:  93210 − 15110 = 78110  93210 + 84910 = 178110  Subtraction by Addition!  What is the relation between 15110 and 84910 ?  100010 − 15110 = 84910  Mechanically: Every digit 𝑑 is replaced by (9 − 𝑑), and finally we add to the result  84910 is virtually the negation of 15110 , in the world of decimal digits  We call this method of turning 15110 into 84910 : The ten’s complement The ten’s complement  Summary of the method:  We want to calculate 63710 − 29110 =?  We confine ourselves to the world of decimal digits  We find the ten’s complement of 29110 by finding − 𝑑 for every digit, and finally adding 1:  29110 → 70810 + 110 = 70910  We add: 63710 + 70910 = 134610 , and consider only the lowest digits  We get that 63710 − 29110 = 34610 Take a break  And come back when you are ready for the Binary version of ten’s complement The two’s complement  We apply the same method to binary numbers  We want to calculate 101102 − 1112 =?  We confine ourselves to the world of bits  We find two’s complement of 1112 by calculating (1 − 𝑏) for every bit 𝑏 (“Flip” every bit), and finally adding 12  1112 = 001112 → 110002 + 12 = 110012  We add 101102 + 110012 = 1011112 and consider only the lowest digits  We get that 101102 − 1112 = 011112 = 11112 Observations  Using the two’s complement imposes a limit on the amount of bits used  We have to fix the amount of bits used before finding the two’s complement of a number  Example: 1012 → 0112 in the world of bits, however 1012 → 1110112 in the world of bits  Adding a number with his complement gives  Example: 1012 + 1110112 = 10000002 Signed binary numbers  Let us look at all the binary numbers with bits  Also called Byte  We call numbers that begin with the bit “1” negative  To change the sign of a number we use the two’s complement method  Example: The number 101100012 is a negative number 101100012 → 010011102 + 12 = 010011112 = 7910 Hence 101100012 represents −7910  We call this representation Signed Binary Numbers of size 8, as opposed to the simple representation that is called Unsigned Binary Numbers of size  We can add signed numbers “in the usual way”, and the sign of the numbers is taken into account automatically! Examples of signed addition  Example: 4610 − 1710 = 2910  In binary:  4610 = 001011102  1710 = 000100012  000100012 → 111011102 + 12 = 111011112 = −1710  4610 − 1710 = 4610 + −1710 = 001011102 + 111011112 = 1000111012  We consider only the lowest bits  000111012 = 2910  The “+” operator works well with two’s complement Examples of signed addition (Cont.)  Example:−1510 − 10110 = −11610  In binary:  1510 = 000011112  −1510 = 111100002 + 12 = 111100012  10110 = 011001012  −10110 = 100110102 + 12 = 100110112  −1510 − 10110 = −1510 + −10110 = 111100012 + 100110112 = 1100011002  We consider only the lower bits  100011002 begins with 1, it is a negative number  100011002 → 011100112 + 12 = 011101002 = 11610 Exceptions  The two’s complement takes positive numbers into negative numbers and vice versa  The highest bit is usually flipped after invoking two’s complement  Two Exceptions:  complements himself   02 → 111111112 + 12 = 100000002 Begins with 0, therefore it is formally positive  “The most negative number” 100000002 complements himself:   100000002 → 011111112 + 12 = 100000002 Also called the “weird number” Graphical view Graphical view Numbers that complement themselves Some Philosophy of representation  You now know about at least two interpretations for every binary number you see  How could you decide which interpretation is the right one?  The bits don’t know what they represent, and they don’t care  100011002 could mean −11610 in two’s complement representation, or 14010 in simple representation  The meaning of a number is obtained from your thoughts about it  And the actions you perform on it, accordingly Some philosophy (Cont.)  Example:  010010112 + 111001112 = 1001100102  This could mean that:  7510 + 23110 = 30610  Because 7510 = 010010112 ; 23110 = 111001112 ; 1001100102 = 30610 in the unsigned binary interpretation  This could also mean that:  7510 − 2510 = 5010  Because 7510 = 010010112 ; 111001112 = −2510 ; 001100102 = 5010 in the signed two’s complement interpretation  Amazingly, both are correct Exercises  Basic ten’s and two’s complement calculations  Use a pen and a paper to solve Use a calculator to check your results  Some more interesting exercises  ... 10 0010 − 15110 − 10 0010  10 0010 − 15110 = 84910 (“Mechanical” operation)  We replace the subtraction with the result:  93210 − 15110 = 93210 + 84910 − 10 0010 = 178110 − 10 0010  178110 − 10 0010. .. interpretation  This could also mean that:  7510 − 2510 = 5010  Because 7510 = 0100 10112 ; 111001112 = −2510 ; 0011 00102 = 5010 in the signed two’s complement interpretation  Amazingly, both... is called Unsigned Binary Numbers of size  We can add signed numbers “in the usual way”, and the sign of the numbers is taken into account automatically! Examples of signed addition  Example: