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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, since digital systems operate with binary numbers that are represented in terms of "0's" and "1's". We have seen previously that an 8-bit byte can have a value from 0 to 255, 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 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 a sign also has only two values, a "+" or a "–". Then we can use a single bit to identify the sign of a signed binary number.

So to represent a positive (N) and a negative (-N) binary number we can use the binary numbers with sign. For signed binary numbers the most significant bit (MSB) is used as the sign. If the sign bit is "0", this means the number is positive. If the sign bit is "1", then the number is negative. The remaining bits are used to represent the magnitude of the binary number in the usual unsigned binary number format.

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.

Positive Signed Binary Numbers

Negative Signed Binary Numbers

The disadvantage here is that whereas before we had a n-bit unsigned binary number we now have a n-1 bit signed binary number giving a range of digits from:

-(2(n-1) - 1)  to  +(2(n-1) - 1)

So for example: if we have 4 bits to represent a signed binary number, (1-bit for the Sign bit and 3-bits for the Magnitude bits), then the actual range of numbers we can represent in sign-magnitude notation would be:

-(2(4-1) - 1)  to  +(2(4-1) - 1)

-2(3) - 1  to  +2(3) - 1

-7  to  +7

Whereas before, the range of an unsigned 4-bit binary number would have been from 0 to 15, or 0to F in hexadecimal. In other words, unsigned binary arithmetic does not have a sign-bit, and therefore can have a larger binary range as the most significant bit (MSB) is just an extra bit or digit rather than a sign bit.

Signed Binary Numbers Example No1

Convert the following decimal values into signed binary numbers using the sign-magnitude format:

Note that for a 4-bit, 6-bit, 8-bit, 16-bit or 32-bit signed binary number all the bits MUST have a value, therefore "0's" are used to fill the spaces between the leftmost sign bit and the first or highest value "1".

The sign-magnitude representation of a binary number is a simple method to use and understand for representing signed binary numbers, as we use this system all the time with normal decimal (base 10) numbers in mathematics. Adding a "1" to the front of it if the binary number is negative and a "0" if it is positive.

However, using this sign-magnitude method can result in the possibility of two different bit patterns having the same binary value. For example, +0 and -0 would be 0000 and 1000 respectively as a signed 4-bit binary number. So we can see that using this method there can be two representations for zero, a positive zero ( 00002 ) and also a negative zero ( 10002 ) which can cause big complications for computers and digital systems.

One's Complement of a Signed Binary Number

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 100101002 is simply 011010112 as all the 1's are changed to 0's and the 0's to 1's.

The easiest way to find the one's complement of a signed binary number when building digital arithmetic or logic decoder circuits is to use Inverters. The inverter is naturally a complement generator and can be used in parallel to find the 1's complement of any binary number as shown.

1's Complement Using Inverters

Then we can see that it is very easy to find the one's complement of a binary number N as all we need do is simply change the 1's to 0's and the 0's to 1's to give us a -N equivalent. Also just like the previous sign-magnitude representation, one's complement can also have n-bit notation to represent numbers in the range from: -2(n-1) - 1  and  +2(n-1) - 1. For example, a 4-bit representation in the one's complement format can be used to represent decimal numbers in the range from -7 to +7 with two representations of zero: 0000 (+0) and 1111 (-0) the same as before.

Addition and Subtraction Using One's Complement

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.

We saw in the Binary Adder tutorial that binary addition follows the same rules as for the normal addition except that in binary there are only two bits (digits) and the largest digit is a "1", (just as "9" is the largest decimal digit) thus the possible combinations for binary addition are as follows:

When the two numbers to be added are both positive, the sum A + B, they can be added together by means of the direct sum (including the number and bit sign), because when single bits are added together, "0 + 0", "0 + 1", or "1 + 0" results in a sum of "0" or "1". This is because when the two bits to be added together are odd ("0" + "1" or "1 + 0"), the result is "1". Likewise when the two bits to be added together are even ("0 + 0" or "1 + 1") the result is "0" until you get to "1 + 1" then the sum is equal to "0" plus a carry "1". Let's look at a simple example.

Subtraction of Two Binary Numbers

An 8-bit digital system is required to subtract the following two numbers 115 and 27 from each other using one's complement. So in decimal this would be: 115 - 27 = 88.

First we need to convert the two decimal numbers into binary and make sure that each number has the same number of bits by adding leading zero's to produce an 8-bit number (byte). Therefore:

11510  in binary is:  011100112

2710   in binary is:  000110112

Now we need to find the complement of the second binary number, (00011011) while leaving the first number (01110011) unchanged.

By changing all the 1's to 0's and 0's to 1's, the one's complement of 00011011 is equal to 11100100.

Adding the first number and the complement of the second number gives:

Since the digital system is to work with 8-bits, only the first eight digits are used to provide the answer to the sum, and we simply ignore the last bit (bit 9). This bit is call an "overflow" bit. Overflow occurs when the sum of the most significant (left-most) column produces a carry forward. This overflow or carry bit can be ignored completely or passed to the next digital section for use in its calculations. Overflow indicates that the answer is positive. If there is no overflow then the answer is negative.

The 8-bit result from above is: 01010111 (the overflow "1" cancels out) and to convert it back from a one's complement answer to the real answer we now have to add "1" to the one's complement result, therefore:

So the result of subtracting 27 ( 000110112 ) from 115 ( 011100112 ) using 1's complement in binary gives the answer of: 010110002 or (64 + 16 + 8) = 8810 in decimal.

Then we can see that signed or unsigned binary numbers can be subtracted from each other using One's Complement and the process of addition. Binary adders such as the TTL 74LS83 or 74LS283 can be used to add or subtract two 4-bit signed binary numbers or cascaded together to produce 8-bit adders complete with carry-out.

Two's Complement of a Signed Binary Number

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 our 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.

Signed Binary Numbers Summary

We have seen that negative binary numbers can be represented by using the most significant bit (MSB) as a sign bit. If an n bit binary number is signed the leftmost bit is used to represent the sign leavingn-1 bits to represent the number.

For example, in a 4-bit binary number, this leaves only 3 bits to hold the actual number. If however, the binary number is unsigned then all the bits can be used to represent the number.

The representation of a signed binary number is commonly referred to as the sign-magnitude notation and if the sign bit is "0", the number is positive. If the sign bit is "1", then the number is negative. When dealing with binary arithmetic operations, it is more convenient to use the complement of the negative number.

Complementation is an alternative way of representing negative binary numbers. This alternative coding system allows for the subtraction of negative numbers by using simple addition.

Since positive sign-magnitude numbers always start with a zero (0), its complement will therefore always start with a one (1) to indicate a negative number as shown in the following table.

4-bit Signed Binary Number Comparison

Signed-complement forms of binary numbers can use either 1's complement or 2's complement. The 1's complement and the 2's complement of a binary number are important because they permit the representation of negative numbers.

The method of 2's complement arithmetic is commonly used in computers to handle negative numbers the only disadvantage is that if we want to represent negative binary numbers in the signed binary number format, we must give up some of the range of the positive number we had before.

Binary Coded Decimal or BCD as it is more commonly called, is another process of converting decimal numbers into its binary equivalent. As we have seen in this Binary Numbers section of tutorials, there are many different binary codes used in digital and electronic circuits, each with its own specific use.

As we naturally live in a decimal (base-10) world we need some way of converting these decimal numbers into a binary (base-2) environment that computers and digital electronic devices understand, and binary coded decimal code allows us to do that.

We have seen previously that an n-bit binary code is a group of "n" bits that assume up to 2ndistinct combinations of 1's and 0's. The advantage of the Binary Coded Decimal system is that each decimal digit is represented by a group of 4 binary digits or bits in much the same way as Hexadecimal. So for the 10 decimal digits (0-to-9) we need a 4-bit binary code.

But do not get confused, binary coded decimal is not the same as hexadecimal. Whereas a 4-bit hexadecimal number is valid up to F16 representing binary 11112, (decimal 15), binary coded decimal numbers stop at 9 binary 10012. This means that although 16 numbers (24) can be represented using four binary digits, in the BCD numbering system the six binary code combinations of: 1010 (decimal 10), 1011 (decimal 11), 1100 (decimal 12), 1101 (decimal 13), 1110(decimal 14), and 1111 (decimal 15) are classed as forbidden numbers and can not be used.

The main advantage of binary coded decimal is that it allows easy conversion between decimal (base-10) and binary (base-2) form. However, the disadvantage is that BCD code is wasteful as the states between 1010 (decimal 10), and 1111 (decimal 15) are not used. Nevertheless, binary coded decimal has many important applications especially using digital displays.

In the BCD numbering system, a decimal number is separated into four bits for each decimal digit within the number. Each decimal digit is represented by its weighted binary value performing a direct translation of the number. So a 4-bit group represents each displayed decimal digit from 0000 for a zero to 1001 for a nine.

So for example, 35710 (Three Hundred and Fifty Seven)  in decimal would be presented in Binary Coded Decimal as:

35710 = 0011 0101 0111 (BCD)

Then we can see that BCD uses weighted codification, because the binary bit of each 4-bit group represents a given weight of the final value. In other words, the BCD is a weighted code and the weights used in binary coded decimal code are 8, 4, 2, 1, commonly called the 8421 code as it forms the 4-bit binary representation of the relevant decimal digit.

Binary Coded Decimal Representation of a Decimal Number

The decimal weight of each decimal digit to the left increases by a factor of 10. In the BCD number system, the binary weight of each digit increases by a factor of  2 as shown. Then the first digit has a weight of  1 ( 20 ), the second digit has a weight of  2 ( 21 ), the third a weight of  4 ( 22 ), the fourth a weight of  8 ( 23 ).

Then the relationship between decimal (denary) numbers and weighted binary coded decimal digits is given below.

Truth Table for Binary Coded Decimal

Then we can see that 8421 BCD code is nothing more than the weights of each binary digit, with each decimal (denary) number expressed as its four-bit pure binary equivalent.

Decimal-to-BCD Conversion

As we have seen above, the conversion of decimal to binary coded decimal is very similar to the conversion of hexadecimal to binary. Firstly, separate the decimal number into its weighted digits and then write down the equivalent 4-bit 8421 BCD code representing each decimal digit as shown.

Binary Coded Decimal Example No1

Using the above table, convert the following decimal (denary) numbers: 8510, 57210 and 857910 into their 8421 BCD equivalents.

8510 = 1000 0101 (BCD)

57210 = 0101 0111 0010 (BCD)

857910 = 1000 0101 0111 1001 (BCD)

Note that the resulting binary number after the conversion will be a true binary translation of decimal digits. This is because the binary code translates as a true binary count.

BCD-to-Decimal Conversion

The conversion from binary coded decimal to decimal is the exact opposite of the above. Simply divide the binary number into groups of four digits, starting with the least significant digit and then write the decimal digit represented by each 4-bit group. Add additional zero's at the end if required to produce a complete 4-bit grouping. So for example, 1101012 would become: 0011 01012 or 3510in decimal.

Binary Coded Decimal Example No2

Convert the following binary numbers: 10012, 10102, 10001112 and 10100111000.1012 into their decimal equivalents.

10012 = 1001BCD = 910

10102 = this will produce an error as it is decimal 1010 and not a valid BCD number

10001112 = 0100 0111BCD = 4710

10100111000.1012 = 0101 0011 0001.1010BCD = 538.62510

The conversion of BCD-to-decimal or decimal-to-BCD is a relatively straight forward task but we need to remember that BCD numbers are decimal numbers and not binary numbers, even though they are represented using bits. The BCD representation of a decimal number is important to understand, because microprocessor based systems used by most people needs to be in the decimal system.

However, while BCD is easy to code and decode, it is not an efficient way to store numbers. In the standard 8421 BCD encoding of decimal numbers, the number of individual data bits needed to represent a given decimal number will always be greater than the number of bits required for an equivalent binary encoding.

For example, in binary a three digit decimal number from 0-to-999 requires only 10-bits (11111001112), whereas in binary coded decimal, the same number requires a minimum of 12-bits (0011 1110 0111BCD) for the same representation.

Also, performing arithmetic tasks using binary coded decimal numbers can be a bit awkward since each digit can not exceed 9. The addition of two decimal digits in BCD, will create a possible carry bit of 1 which needs to be added to the next group of 4-bits.

If the binary sum with the added carry bit is equal to or less than 9 (1001), the corresponding BCD digit is correct. But when the binary sum is greater than 9 the result is an invalid BCD digit. Therefore it is better to convert BCD numbers into pure binary, perform the required addition, and then convert the back to BCD before displaying the results.

Nevertheless, the use of a BCD coding system in both microelectronics and computer systems is particularly useful in situations where the binary coded decimal is intended to be displayed on one or more 7-segment LED or LCD displays and there are many popular integrated circuits available that are configured to give a BCD output or outputs.

One common IC is the 74LS90 asynchronous counter/divider that contains independent divide-by-2 and divide-by-5 counters that can be used together to produce a divide-by-10 decade counter with BCD outputs. Another is the 74LS390 which is a dual version of the basic 74LS90, and can also be configured to produce a BCD output.

But the most commonly used BCD encoded IC's are the 74LS47 and the 74LS48 BCD to 7-segment decoder/driver, which converts a 4-bit BCD code of a counter, etc. and converts it into the required display code to drive the individual segments of a 7-segment LED display. While both IC's are functionally the same, the 74LS47 has active-low outputs for driving common-anode displays, while the 74LS48 has active-high outputs for driving common-cathode displays.

Binary Coded Decimal Decoder IC

Binary Coded Decimal Summary

We have seen here that Binary Coded Decimal or BCD is simply the 4-bit binary code representation of a decimal digit with each decimal digit replaced in the integer and fractional parts with its binary equivalent. BCD Code uses four bits to represent the 10 decimal digits of 0 to 9.

So for example, if we wanted to display decimal numbers in the range of 0-to-9, (one digit) we would need 4 data bits (a nibble), decimal numbers in the range of 0-to-99, (two digits) we would need 8 bits (one byte), decimal numbers in the range of 0-to-999, (three digits) we would need 12 bits, and so on. The use of a single byte (8-bits) to store or display two BCD digits, allowing a byte to hold a BCD number in the range of 00 – 99, is known as packed BCD.

Standard binary coded decimal code is commonly known as a weighted 8421 BCD code, with 8, 4, 2 and 1 representing the weights of the different bits starting from the most significant bit (MSB) and proceeding towards the least significant bit (LSB). The weights of the individual positions of the bits of a BCD code are: 23=8, 22=4, 21=2, 20=1.

The main advantage of the Binary Coded Decimal system is that it is a fast and efficient system to convert the decimal numbers into binary numbers as compared to the pure binary system. But the BCD code is wasteful as many of the 4-bit states (10-to-16) are not used but decimal displays have important applications

Binary Numbers Tutorial Summary

There are different numbering systems used in digital electronic circuits and computers. However, the numbering system used in one type of circuit may be different to that of another type of circuit, for example, the memory of a computer would use hexadecimal numbers while the keyboard uses decimal numbers.

Then the conversion from one number system to another is very important with the four main forms of arithmetic being.

• Decimal – The decimal numbering system has a base of 10 (MOD-10) and uses the digits from 0 through 9 to represent a decimal number value.
• Binary – The binary numbering system has a base of 2 (MOD-2) and uses only two digits a "0" and a "1" to represent a binary number value.
• Octal – The octal numbering system has a base of 8 (MOD-8) and uses 8 digits between 0and 7 to represent an octal number value.
• Hexadecimal – The Hexadecimal numbering system has a base of 16 (MOD-16) and uses a total of 16 numeric and alphabetic characters to represent a number value. Hexadecimal numbers consist of digits 0 through 9 and letters A to F.

Long binary numbers are difficult to both read or write and are generally converted into a system more easily understood or user friendly. The two most common derivatives based on binary numbers are the Octal and the Hexadecimal numbering systems, with both of these limited in length to a byte (8-bits) or a word (16-bits).

Octal numbers can be represented by groups of 3-bits and hexadecimal numbers by groups of 4-bits together, with this grouping of the bits being used in electronic or computer systems in displays or printouts. The grouping together of binary numbers can also be used to represent Machine Code used for programming instructions and control such as an Assembly Language.

Comparisons between the various Decimal, Binary, Hexadecimal and Octal numbers are given in the following table.

Comparison Table

We can see from the table above that the Hexadecimal numbering system uses only four digits to express a single 16-bit word length, and as a result it is the most commonly used Base Numbering System for digital, micro-electronic and computer systems.

The Octal Numbering System

The Octal Number System is another type of computer and digital base number system. TheOctal Numbering System is very similar in principle to the previous hexadecimal numbering system except that in Octal, a binary number is divided up into groups of only 3 bits, with each group or set of bits having a distinct value of between 000 (0) and 111 ( 4+2+1 = 7 ).

Octal numbers therefore have a range of just "8" digits, (0, 1, 2, 3, 4, 5, 6, 7) making them a Base-8 numbering system and therefore, q is equal to "8".

Then the main characteristics of an Octal Numbering System is that there are only 8 distinct counting digits from 0 to 7 with each digit having a weight or value of just 8 starting from the least significant bit (LSB). In the earlier days of computing, octal numbers and the octal numbering system was very popular for counting inputs and outputs because as it works in counts of eight, inputs and outputs were in counts of eight, a byte at a time.

As the base of an Octal Numbers system is 8 (base-8), which also represents the number of individual numbers used in the system, the subscript 8 is used to identify a number expressed in octal. For example, an octal number is expressed as:  2378

Just like the hexadecimal system, the "octal number system" provides a convenient way of converting large binary numbers into more compact and smaller groups. However, these days the octal numbering system is used less frequently than the more popular hexadecimal numbering system and has almost disappeared as a digital base number system.

Representation of an Octal Number

As the octal number system uses only eight digits (0 through 7) there are no numbers or letters used above 8, but the conversion from decimal to octal and binary to octal follows the same pattern as we have seen previously for hexadecimal.

To count above 7 in octal we need to add another column and start over again in a similar way to hexadecimal.

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21….etc

Again do not get confused, 10 or 20 is NOT ten or twenty it is 1 + 0 and 2 + 0 in octal exactly the same as for hexadecimal. The relationship between binary and octal numbers is given below.

Octal Numbers

Then we can see that 1 octal number or digit is equivalent to 3 bits, and with two octal number, 778we can count up to 63 in decimal, with three octal numbers, 7778 up to 511 in decimal and with four octal numbers, 77778 up to 4095 in decimal and so on.

Octal Numbers Example No1

Using our previous binary number of 11010101110011112 convert this binary number to its octal equivalent, (base-2 to base-8).

Thus, 0011010101110011112 in its Binary form is equivalent to 1527178 in Octal form or 54,735 in denary.

Octal Numbers Example No2

Convert the octal number 23228 to its decimal number equivalent, (base-8 to base-10).

Then, converting octal to decimal shows that  23228 in its Octal form is equivalent to 123410 in its Decimal form.

While Octal is another type of digital numbering system, it is little used these days instead the more commonly used Hexadecimal Numbering System is used as it is more flexible.