8.1 Introduction

In everyday life, we use the decimal number system (base, or radix, 10), which allows the creation of numbers with digits in the set: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The ten possible digits are combined to create integer and real numbers. However, base 10 is not the only number system. Digital circuits and systems use the binary (base, or radix, 2) number system, which allows for the creation of numbers with digits in the set: 0, 1.

The 0 and 1 numbers are logic levels (0 = logic 0, 1 = logic 1), which are created by voltages in a circuit:

In this text, only positive logic will be used and will use the voltage levels shown in Table 8.1.

Table 8.1 Typical voltage levels representing positive logic

Decimal and binary number systems are only two of four number systems used in digital circuits and systems:

At some point in the design and analysis of a digital circuit, it will be necessary to convert between the different number systems to view and manipulate values propagating through the design. Such conversion is typically undertaken to aid the interpretation and understanding of the design operation.

In addition, a binary number can have different meanings as different binary coding can be chosen for different design requirement. The main binary coding schemes used are:

Unsigned binary numbers are used to represent positive numbers only. Signed binary numbers are used to represent positive and negative numbers that are coded to allow arithmetic using either 1s complement or 2’s complement notation. Twos complement notation is more commonly used and will be considered in this text. Gray code allows for a one-bit change when moving from one value to the next (or previous) value. BCD provides a simple conversion between binary and decimal numbers.

All four binary coding schemes are fully discussed in the following sections.

8.2 Decimal–Unsigned Binary Conversion

The conversion between decimal and binary involves identifying the particular decimal value for the particular binary code (or vice versa). Both decimal-to-binary and binary-to-decimal conversion is common and a binary number will be needed to represent each decimal number. If both the decimal and binary numbers are unrestricted in size, then an exact conversion is possible.

In unsigned (or straight) binary, the numbers represented by the binary code will be positive numbers only. Each digit in the binary number will contribute to the magnitude of the value. For example, consider the decimal value 8₁₀. In unsigned binary, this is represented by 1000₂. Each digit in the decimal number has a value in the set of (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Each digit in the binary number is in the set of (0, 1). A binary digit is referred to as a bit (binary digit).

The magnitude of the decimal number is the sum of the product of the value of each digit in the number (d) and its position (n). The position immediately to the left of the decimal point is position zero (0). The value of the digit has a weight of 2ⁿ where n is the position number. Moving left from position 0 (in the integer part of the number), the position increments by 1. Moving right from position zero (into the fractional part of the number), the position decrements by 1. Therefore, the magnitude of the number is given by:

Here, the decimal number is written as:

Some example decimal numbers are:

The binary number is a base 2 number whose magnitude is the sum of the product of the value of each digit in the number (b) and its position (n). Moving left from position 0 (in the integer part of the number), the position increments by 1. The value of the digit has a weight of 2ⁿ where n is the position number. Moving right from position zero (into the fractional part of the number), the position decrements by 1. This allows the creation of numbers with digits in the set: 0, 1. Therefore, in general the magnitude of the number (as a decimal number) is given by:

Here, the binary number is written as b_nb_n−1 b_n−2 … b₀.b ₋₁ …. b_−n. Some example binary numbers are:

The decimal number equivalent for a binary number can be created by taking the binary number and calculating its magnitude (as a decimal number):

Some example binary numbers are:

The binary number equivalent of a decimal number is created by dividing the decimal number by 2 until the result of the division is 0. The remainder of the total division forms the binary number digits, the remainder from the first division forms the least significant bit (LSB) of the binary number, and the remainder from the last division forms the most significant bit (MSB) of the binary number.

Consider the number 8₁₀. The conversion procedure is shown in Table 8.2.

Table 8.2 Conversion procedure for number 8₁₀

The binary number can be read as: 8₁₀ = (b₃b₂b₁b₀)₂ = 1000₂.

Consider now the number 218₁₀. The conversion procedure is shown in Table 8.3.

Table 8.3 Conversion procedure for number 218₁₀

8.3 Signed Binary Numbers

Unsigned (or straight) binary numbers are used when the operations use only positive numbers and the result of any operations is a positive number. However, in many cases, both the number and the result can be either positive or negative, and the unsigned binary number system cannot be used. The two coding schemes used to achieve this are the 1’s complement and 2’s complement.

The 1’s complement of a number is obtained by changing (or inverting) each of the bits in the binary number (0 becomes a 1 and a 1 becomes a 0):

The 2’s complement is formed by adding 1 to the 1’s complement:

The MSB of the binary number is used to represent the sign (0 = positive, 1 = negative) of the number, and the remainder of the number represents the magnitude. It is therefore essential that the number of bits used is sufficient to represent the required range, as shown in Table 8.4. Here, only integer numbers are considered.

Table 8.4 Number range

Two’s complement number manipulation is as follows:

The 2’s complement number coding scheme is widely used in digital circuits and system design and so will be explained further. Table 8.5 shows the binary representations of decimal numbers for a four-bit binary number. In the unsigned binary number coding scheme, the binary number represents a positive decimal number from 0₁₀ to +15₁₀. In the 2’s complement number coding scheme, the decimal number range is −8₁₀ to +7₁₀.

Table 8.5 Decimal to binary conversion

In this, the most negative 2’s complement number is 110 greater in magnitude than the most positive 2’s complement number. The number range for an n-bit number is: −2^N to +(2^N – 1).

Addition and subtraction are undertaken by addition and if necessary inversion (creating a negative number from a positive number and vice versa). Table 8.6 shows the cases for addition and subtraction of two numbers (A and B). It is essential to ensure that the two numbers have the same number of bits, the MSB represents the sign of the binary number, and the number of bits used is sufficient to represent the range of possible inputs and the range of possible outputs.

Table 8.6 2’s complement addition and subtraction

Figure 8.1 shows an arrangement where two inputs are either added or subtracted, depending on the logic level of a control input. This arrangement requires an adder, a complement (a logical inversion of the inputs bits and add 1, disregarding any overflow), and a digital switch (multiplexer).

Input numbers in the range −8₁₀ to +7₁₀ are represented by four bits in binary. However, the range for the result of an addition is −16₁₀ to +14₁₀, and the range for the result of a subtraction is −15₁₀ to +15₁₀. The result requires five bits in binary to represent the number range (one bit more than the number of bits required to represent the inputs), so the number of bits to represent the inputs will be increased by one bit before the addition or subtraction:

Consider the addition of +2₁₀ to +3₁₀ using 2’s complement numbers. The result should be +5₁₀. The two input numbers can be represented by three bits, but if 3-bit addition is undertaken, the result will be in error:

If, however, the input word length is increased by one bit, then the addition is undertaken, the result becomes:

Consider the subtraction of +3₁₀ from −2₁₀. The result should be −5₁₀. The two input numbers can be represented by three bits, but if 3-bit addition is undertaken, then the result will be in error:

If, however, the input word length is increased by one bit, then the addition is undertaken, the result becomes:

8.4 Gray Code

The Gray code provides a binary code that changes by one bit only when it changes from one value to the next.

The Gray code and the decimal number equivalent of the binary number (in unsigned binary) are shown in Table 8.7. This is no longer a straight binary count sequence.

Table 8.7 Gray code

The Gray code is often used in position control systems which represent either a rotary position as in the output shaft of an electric motor or a linear position as in the position of a conveyor belt. Figure 8.2 shows the Gray code used on a sensor to identify the position of an object that can move left and right. Each code represents a point of position or span of distance in length. The Gray code removes the potential for errors when changing from sensing one position to the next position that could occur in a binary code when more than one bit could change. If there is a time delay in the circuitry that senses the individual bits, and the delay for sensing each bit is different, the result will be a short but finite time during which the position code would be wrong. If the circuitry that uses this position signal detects this wrong position code, it will react to a wrong position, and the result would be an erroneous operation of the circuit.

8.5 Binary-Coded Decimal

Binary-coded decimal (BCD) provides a simple conversion between a binary number and the decimal number. For a decimal number, each digit is represented by four bits. For example, the number 12₁₀ is represented by 00010010₂.

If the MSBs are 0, they might also be left out, so the BCD number could also be represented as 100102. This particular BCD code is referred to as 8421 BCD (or straight binary coding) because the binary number is a direct representation of the decimal value for decimal values 0₁₀ to 9₁₀. Decimal values 10₁₀ to 15₁₀ are not represented in the four bits. Other BCD codes can also be implemented.

It is important to understand that a BCD is not the same as a straight binary (unsigned binary) count. For example, consider the number 12₁₀:

8.6 Octal-Binary Conversion

The octal number is a number to the base (or radix) 8, and the magnitude of the number is the sum of the product of the value of each digit in the number (o) and its position (n). This allows the creation of numbers with digits in the set: 0, 1, 2, 3, 4, 5, 6, 7.

The position immediately to the left of the decimal point is zero (0). Moving left from position 0 (in the integer part of the number), the position increments by 1. The value of the digit has a weight of 8ⁿ where n is the position number. Moving right from position 0 (into the fractional part of the number), the position decrements by 1. The eight possible digits are combined to create integers and real numbers. Table 8.8 shows the conversion table.

Table 8.8 Octal-decimal-unsigned 4-bit binary conversion

The magnitude of the number (as a decimal number) is given by:

Here, the octal number is written as (using the decimal equivalent of the octal number).

Some example octal numbers are:

For binary numbers, each octal number represents three bits. Therefore a 6-bit binary number is represented by two octal numbers, an 8-bit binary number is represented by three octal numbers, a 9-bit binary number is also represented by three octal numbers, a 16-bit binary is represented by six octal numbers, and so on. For example, 7₈ is 111₂ and 17₈ is 001111₂:

Some example octal numbers are:

The decimal number equivalent for an octal number is created by calculating the magnitude of the octal number as a decimal number:

Converting from decimal to octal is accomplished in a similar manner as converting from decimal to binary, except now dividing by 8 rather than 2. Consider the number 7₁₀. The conversion procedure is shown in Table 8.9.

Table 8.9 Conversion procedure for number 7₁₀

The octal number can be read as: 7₁₀ = (o₀)₈ = 7₈.

Consider the number 100₁₀. The conversion procedure is shown in Table 8.10.

Table 8.10 Conversion procedure for number 100₁₀

The octal number can be read as: 100₁₀ = (o₂o₁o₀)₈ = 144₈.

8.7 Hexadecimal-Binary Conversion

The hexadecimal number is a number to the base (or radix) 16, and its magnitude is the sum of the product of the value of each digit in the number (h) and its position (n). This allows the creation of numbers with digits in the set: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

The position immediately to the left of the decimal point is zero (0). Moving left from position 0 (in the integer part of the number), the position increments by 1. The value of the digit has a weight of 16ⁿ where n is the position number. Moving right from position zero (into the fractional part of the number), the position decrements by 1. The sixteen possible digits are combined to create integers and real numbers. In a decimal equivalent number, the hexadecimal digits A16 to F16 are the numbers 10₁₀ to 15₁₀. Table 8.11 shows the conversion table.

Table 8.11 Hexadecimal-decimal-unsigned four-bit binary conversion

The magnitude of the number (as a decimal number) is given by:

Here, the hexadecimal number is written as h_nh_n-1h_n-2…h₀.h₋₁….h_−n (using the decimal equivalent of the hexadecimal number).

Some example hexadecimal numbers are:

For binary numbers, each hexadecimal number represents four bits. Therefore, an 8-bit binary number is represented by two hexadecimal numbers, a 16-bit binary is represented by four hexadecimal numbers, and so on. For example, 8₁₆ is 1000₂ and A8₁₆ is 10101000₂.

Some example hexadecimal numbers are:

The decimal number equivalent for a hexadecimal number is created by calculating the magnitude of the hexadecimal number, using the decimal equivalent for hexadecimal numbers A to F, as a decimal number:

Converting from decimal to hexadecimal is accomplished in a similar manner to converting from decimal to binary, except now dividing by 16 rather than 2, and using the letters A to F for decimal remainder values of 10 to 15. Consider the number 7₁₀. The conversion procedure is shown in Table 8.12.

Table 8.12 Conversion procedure for number 7₁₀

The hexadecimal number can be read as: 7₁₀ = (h₀)₁₆ = 7₁₆.

Consider the number 100₁₀. The conversion procedure is shown in Table 8.13.

Table 8.13 Conversion procedure for number 100₁₀

The hexadecimal number can be read as: 100₁₀ = (h₁h₀)₁₆ = 64₁₆.

Consider the number 255₁₀. The conversion procedure is shown in Table 8.14.

Table 8.14 Conversion procedure for number 255₁₀

The hexadecimal number can be read as: 255₁₀ = (h₁h₀)₁₆ = FF₁₆.

Converting from hexadecimal to octal, or vice-versa, is accomplished by converting the number to either a binary or decimal equivalent and from that to the octal to hexadecimal number.

A summary table for the number systems is shown in Table 8.15. Here, unsigned decimal numbers from 0₁₀ to 15₁₀ are considered.

Table 8.15 Number systems summary

Both binary and decimal numbers can only be integers or real numbers. Table 8.16 shows the binary and decimal numbers for a real number represented by 40 bits in binary, with 24 bits representing the integer part of the number and 16 bits representing the fractional part of the number.

Table 8.16 Decimal-binary conversion table, with the positive position to the left of the decimal point and the negative position to the right of the decimal point and the negative