Signed Binary | Digital Electronics - Electronics and Communication Engineering (ECE) PDF Download

Signed Binary Notation

All the binary arithmetic problems looked at in Module 1.3 used only POSITIVE numbers. The reason for this is that it is not possible in PURE binary to signify whether a number is positive or negative. This of course would present a problem in any but the simplest of arithmetic.

There are a number of ways in which binary numbers can represent both positive and negative values, 8 bit systems for example normally use one bit of the byte to represent either + or − and the remaining 7 bits to give the value. One of the simplest of these systems is SIGNED BINARY, also often called ‘Sign and Magnitude’, which exists in several similar versions, but is commonly an 8 bit system that uses the most significant bit (msb) to indicate a positive or a negative value. By convention, a 0 in this position indicates that the number given by the remaining 7 bits is positive, and a most significant bit of 1 indicates that the number is negative.

For example:

+4510 in signed binary is (0)01011012

-4510 in signed binary is (1)01011012

Note:

The brackets around the msb (the sign bit) are included here for clarity but brackets are not normally used. Because only 7 bits are used for the actual number, the values the system can represent range from is −12710 or 111111112, to +12710.

A comparison between signed binary, pure binary and decimal numbers is shown in Table 1.4.1. Notice that in the signed binary representation of positive numbers between +010 and +12710, all the positive values are just the same as in pure binary. However the pure binary values equivalents of +12810 to +25510 are now considered to represent negative values −0 to −127.

This also means that 010 can be represented by 000000002 (which is also 0 in pure binary and in decimal) and by 100000002 (which is equivalent to 128 in pure binary and in decimal).

 

Signed Binary Arithmetic

Signed Binary | Digital Electronics - Electronics and Communication Engineering (ECE)

Signed Binary | Digital Electronics - Electronics and Communication Engineering (ECE)

Because the signed binary system now contains both positive and negative values, calculation performed with signed binary arithmetic should be more flexible. Subtraction now becomes possible by adding a negative number to a positive number, without the problems of borrow and payback described in  Fig. 1.4.1 Adding Positive Number Systems Module 1.3. However there are still problems. Look at the two Numbers in Signed Binary examples illustrated in Fig. 1.4.1 and 1.4.2, using signed binary notation. Z

In Fig. 1.4.1 two positive (msb = 0) numbers are added and the correct answer is obtained. This is really no different to adding two numbers in pure binary.

In Fig. 1.4.2 however, the negative number −5 is added to +7, the same action in fact as SUBTRACTING 5 from 7, which means that subtraction should be possible by merely adding a negative number to a positive number. Although this principle works in the decimal version the result using signed binary is 100011002 or −1210 which of course is wrong, the result of 7 − 5 should be +2.   

                                                                                                                                                                     Signed Binary | Digital Electronics - Electronics and Communication Engineering (ECE)

 

 

 

            Fig. 1.4.2 Adding Positive & Negative Numbers in Signed Binary                                                           

Although signed binary can represent positive and negative numbers, if it is used for calculations, some special action would need to be taken, depending on the sign of the numbers used, and how the two values for 0 are handled, to obtain the correct result. Whilst signed binary does solve the problem of REPRESENTING positive and negative numbers in binary, and to some extent carrying out binary arithmetic, there are better sign and magnitude systems for performing binary arithmetic. These systems are the ONES COMPLEMENT and TWOS COMPLEMENT systems, which are described in Number Systems 

The document Signed Binary | Digital Electronics - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Digital Electronics.
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FAQs on Signed Binary - Digital Electronics - Electronics and Communication Engineering (ECE)

1. What is signed binary?
Ans. Signed binary is a method of representing both positive and negative numbers using binary digits. It includes a sign bit, where a 0 represents a positive number and a 1 represents a negative number. The remaining bits are used to represent the magnitude of the number.
2. How does signed binary work?
Ans. In signed binary, the leftmost bit (most significant bit) is used as the sign bit. If the sign bit is 0, the number is positive and the remaining bits represent its magnitude. If the sign bit is 1, the number is negative and the remaining bits represent the magnitude in two's complement form. This allows for the representation of both positive and negative numbers in binary.
3. What is the range of numbers that can be represented in signed binary?
Ans. The range of numbers that can be represented in signed binary depends on the number of bits used. For example, with 8 bits, a signed binary number can represent values from -128 to 127. The range is symmetric around zero, with an equal number of positive and negative values.
4. How is addition performed with signed binary numbers?
Ans. Addition with signed binary numbers follows the same rules as regular binary addition. The sign bit is treated as any other bit and is included in the addition. If there is a carry-out from the sign bit, it indicates an overflow and the result is not valid. Overflow occurs when the sum of two positive numbers is negative or the sum of two negative numbers is positive.
5. Can signed binary be converted to other number systems?
Ans. Yes, signed binary numbers can be converted to other number systems. To convert a signed binary number to decimal, the sign bit is first examined. If it is 0, the remaining bits are converted to decimal normally. If it is 1, the number is first converted from two's complement form to its magnitude, and then the magnitude is converted to decimal. Similarly, signed binary can be converted to octal or hexadecimal by grouping the bits and converting each group to the corresponding base.
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