Signed Numbers Up till now we've been concentrating on unsigned numbers. In real life we have to represent signed numbers ( like: -12, -45, 78). The difference between signed and unsigned numbers is the sign. A scheme is needed to represent the sign as part of the binary representation. There are a number of schemes for representing signed numbers in binary format. sign-magnitude representation the twos-complement representation.

Sign-Magnitude Representation In this representation, the leftmost bit of a binary code represents the sign of the value: 0 for positive, 1 for negative; the remaining bits represent the numeric value.

Sign-Magnitude Representation To Compute negative values using Sign/Magnitude (signmag) representation. Begin with the binary representation of the positive value, then flip the leftmost zero bit.

Sign-Magnitude Representation Ex 1. Find the signmag representation of Step1: find binary representation using 8 bits 6 10 = Step2: if the number is a negative number flip left most bit So: = (in 8-bit sign/magnitude form)

Sign-Magnitude Representation Ex 2. Find the signmag representation of Step 1: find binary representation using 8 bits = Step 2: if the number is a negative number flip left most bit So: = (in 8-bit sign/magnitude form)

Sign-Magnitude Representation Ex 3. Find the signmag representation of Step 1: find binary representation using 8 bits = Step 2: if the number is a negative number flip left most bit (no flipping, since it is +ve) So:70 10 = (in 8-bit sign/magnitude form)

Sign-Magnitude Representation What is this signmag number? The machine will think of it as - 0, which is a non valid value.

Two’s Complement Representation Another scheme to represent negative numbers The leftmost bit serves as a sign bit: 0 for positive numbers, 1 for negative numbers.

Two’s Complement Representation To Compute negative values using two’s Complement representation, begin with the binary representation of the positive value, complement (flip each bit if it is 0 make it 1 and visa versa) the entire positive number, and then add one.

Two’s Complement Representation Ex. Find the two’s complement representation of –6 10 Step1: find binary representation in 8 bits 6 10 =

Two’s Complement Representation Step 2: Complement the entire positive number, and then add one (complemented)-> (add one)-> So:-6 10 = (in 2's complement form, using any of above methods)

Two’s Complement Representation Alternative method for step 2 Scan binary representation from right too left, find first one bit, from low-order (right) end, and complement the remaining pattern to the left (left complemented)-->

Two’s Complement Representation Ex 2: Find the Two’s Complement of Step 1: Find the 8 bit binary representation of the positive value =

Two’s Complement Representation Step 2: Find first one bit, from low-order (right) end, and complement the pattern to the left (left complemented)-> So: = (in 2's complement form, using any of above methods)

Two’s Complement Representation Ex 3: Find the Two’s Complement of Step 1: Find the 8 bit binary representation of the positive value =

Two’s Complement Representation Step 2: Since number is positive do nothing. So:72 10 = (in 2's complement form, using any of above methods)

Two’s Complement Representation The most important characteristic of the two’s-complement system is that the binary codes can be added and subtracted as if they were unsigned binary numbers, without regard to the signs of the numbers they actually represent.

Two’s Complement Representation For example, to add +4 and -3, we simply add the corresponding binary codes, 0100 and 1101: 0100 (+4) (-3) 0001 (+1) A carry from the leftmost column has been ignored. The result, 0001, is the code for +1, the sum of +4 and -3.

Twos Complement Representation Likewise, to subtract +7 from +3, we add the code for -7, 1001, to that of +3, 0011: 0011 (+3) (-7) 1100 (-4) The result, 1100, is the code for -4, the result of subtracting +7 from +3.

Two’s Complement Representation Benefits of Twos Complements: addition and subtraction simplified in the two’s-complement system, In 8 bits, -0 has been eliminated, replaced by -128, for which there is no corresponding positive number.