Negative Binary Numbers

In mathematics, positive numbers (including zero) are represented as unsigned numbers. That is we do not put the +ve sign in front of them to show that they are positive numbers.

However, when dealing with negative numbers we do use a -ve sign in front of the number to show that the number is negative in value and different from a positive unsigned value, and the same is true with signed binary numbers.

However, in digital circuits there is no provision made to put a plus or even a minus sign to a number, since digital systems operate with binary numbers that are represented in terms of “0’s” and “1’s”. When used together in microelectronics, these “1’s” and “0’s”, called a bit (being a contraction of BInary digiT), fall into several range sizes of numbers which are referred to by common names, such as a byte or a word.

We have also seen previously that an 8-bit binary number (a byte) can have a value ranging from 0 (000000002) to 255 (111111112), that is 28 = 256 different combinations of bits forming a single 8-bit byte. So for example an unsigned binary number such as: 010011012 = 64 + 8 + 4 + 1 = 7710 in decimal. But Digital Systems and computers must also be able to use and to manipulate negative numbers as well as positive numbers.

Mathematical numbers are generally made up of a sign and a value (magnitude) in which the sign indicates whether the number is positive, ( + ) or negative, ( – ) with the value indicating the size of the number, for example 23, +156 or -274. Presenting numbers is this fashion is called “sign-magnitude” representation since the left most digit can be used to indicate the sign and the remaining digits the magnitude or value of the number.

Sign-Magnitude

Sign-magnitude notation is the simplest and one of the most common methods of representing positive and negative numbers either side of zero, (0). Thus negative numbers are obtained simply by changing the sign of the corresponding positive number as each positive or unsigned number will have a signed opposite, for example, +2 and -2, +10 and -10, etc.

But how do we represent signed binary numbers if all we have is a bunch of one’s and zero’s. We know that binary digits, or bits only have two values, either a “1” or a “0” and conveniently for us, a sign also has only two values, being a “+” or a “–“.

Then we can use a single bit to identify the sign of a signed binary number as being positive or negative in value. So to represent a positive binary number (+n) and a negative (-n) binary number, we can use them with the addition of a sign.

For signed binary numbers the most significant bit (MSB) is used as the sign bit. If the sign bit is “0”, this means the number is positive in value. If the sign bit is “1”, then the number is negative in value. The remaining bits in the number are used to represent the magnitude of the binary number in the usual unsigned binary number format way.

Then we can see that the Sign-and-Magnitude (SM) notation stores positive and negative values by dividing the “n” total bits into two parts: 1 bit for the sign and n–1 bits for the value which is a pure binary number. For example, the decimal number 53 can be expressed as an 8-bit signed binary number as follows.

There are disadvantages to this method. For example the range of numbers we can represent is now lower.

One's Complement

One’s Complement or 1’s Complement as it is also termed, is another method which we can use to represent negative binary numbers in a signed binary number system. In one’s complement, positive numbers (also known as non-complements) remain unchanged as before with the sign-magnitude numbers.

Negative numbers however, are represented by taking the one’s complement (inversion, negation) of the unsigned positive number. Since positive numbers always start with a “0”, the complement will always start with a “1” to indicate a negative number.

The one’s complement of a negative binary number is the complement of its positive counterpart, so to take the one’s complement of a binary number, all we need to do is change each bit in turn. Thus the one’s complement of “1” is “0” and vice versa, then the one’s complement of 10010100 is simply 01101011 as all the 1’s are changed to 0’s and the 0’s to 1’s.

One of the main advantages of One’s Complement is in the addition and subtraction of two binary numbers. In mathematics, subtraction can be implemented in a variety of different ways as A – B, is the same as saying A + (-B) or -B + A etc. Therefore, the complication of subtracting two binary numbers can be performed by simply using addition.

Two's Complement

Two’s Complement or 2’s Complement as it is also termed, is another method like the previous sign-magnitude and one’s complement form, which we can use to represent negative binary numbers in a signed binary number system. In two’s complement, the positive numbers are exactly the same as before for unsigned binary numbers. A negative number, however, is represented by a binary number, which when added to its corresponding positive equivalent results in zero.

In two’s complement form, a negative number is the 2’s complement of its positive number with the subtraction of two numbers being A – B = A + ( 2’s complement of B ) using much the same process as before as basically, two’s complement is one’s complement + 1.

The main advantage of two’s complement over the previous one’s complement is that there is no double-zero problem plus it is a lot easier to generate the two’s complement of a signed binary number. Therefore, arithmetic operations are relatively easier to perform when the numbers are represented in the two’s complement format.

Let’s look at the subtraction of two 8-bit numbers 115 and 27 from above using two’s complement, and we remember from above that the binary equivalents are:

115₁₀ in binary is: 01110011₂

27₁₀ in binary is: 00011011₂

Our numbers are 8-bits long, then there are 2⁸ digits available to represent our values and in binary this equals: 100000000₂ or 256₁₀. Then the two’s complement of 27₁₀ will be:

(2⁸)₂ – 00011011 = 100000000 – 00011011 = 11100101₂

The complementation of the second negative number means that the subtraction becomes a much easier addition of the two numbers so therefore the sum is: 115 + ( 2’s complement of 27 ) which is:

01110011 + 11100101 = 1 01011000₂

As previously, the 9th overflow bit is disregarded as we are only interested in the first 8-bits, so the result is: 01011000₂ or (64 + 16 + 8) = 88₁₀ in decimal the same as before.