# Numbering Systems. InThisAppendix...

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1 G InThisAppendix... Introduction Binary Numbering System Hexadecimal Numbering System Octal Numbering System Binary Coded Decimal (BCD) Numbering System Real (Floating Point) Numbering System BCD/Binary/Decimal/Hex/Octal - What is the Difference? DataTypeMismatch Signed vs. Unsigned Integers AutomationDirect.com Products and Data Types

2 G- Introduction As almost anyone who uses a computer is somewhat aware, the actual operations of a computer are done with a binary number system. Traditionally, the two possible states for a binary system are represented by the digits for zero (0) and one () although off and on or sometimes no and yes are closer to what is actually involved. Most of the time a typical PC user has no need to think about this aspect of computers, but every now and then one gets confronted with the underlying nature of the binary system. A PLC user should be more aware of the binary system specifically the PLC programmer. This appendix will provide an explanation of the numbering systems most commonly used by a PLC. Binary Numbering System Computers, including PLCs, use the Base numbering system, which is called Binary or Boolean. Like in a computer there are only two valid digits in Base a PLC relies on, zero and one, or off and on respectively. You would think that it would be hard to have a numbering system built on Base with only two possible values, but the secret is by encoding using several digits. Each digit in the base system when referenced by a computer is called a bit. When four bits are grouped together, they form what is known as a nibble. Eight bits or two nibbles would be a byte. Sixteen bits or two bytes would be a word as in. Thirty--two bits or two words is a double word as in Table. Word Double Word Word Byte Byte Byte Byte Nibble Nibble Nibble Nibble Nibble Nibble Nibble Nibble Table Binary is not natural for us to use since we have grown up using the base 0 system. Base 0 uses the numbers 0--9, as we are all well aware. From now on, the different bases will be shown as a subscripted number following the number. Example: 0 decimal would be 0 0. Table shows how base numbers relate to their decimal equivalents. A nibble of 00 would be equal to a decimal number 9 (* 3 +* 0 or ). A byte of 000 would be equal to 3 (* 7 +* +* 4 +* +* 0 or ). Table

3 G-3 Hexadecimal Numbering System The binary numbering system can be difficult and cumbersome to interpret for some users. Therefore, the hexadecimal numbering system was developed as a convenience for humans since the PLC (computer) only understands pure binary. The hexadecimal system is useful because it can represent every byte (8 bits) as two consecutive hexadecimal digits. It is easier for us to read hexadecimal numbers than binary numbers. The hexadecimal numbering system uses characters (base ) to represent values. The first ten characters are the same as our decimal system, 0--9, and the first six letters of the alphabet, A--F. Table 3 lists the first eighteen decimal numbers; 0--7 in the left column and the equivalent hexadecimal numbers are shown in the right column. Decimal Hex Decimal Hex A B 3 3 C D E 5 F Table 3 Note that 0 and in hex are not the same as 0 and in decimal. Only the first ten numbers 0--9 are the same in the two representations. For example, consider the hex number D8AF. To evaluate this hex number use the same method used to write decimal numbers. Each digit in a decimal number represents a multiple of a power of ten (base 0). Powers of ten increase from right to left. For example, the decimal number 35 means 3x0 + x In hex each digit represents a multiple of a power of sixteen (base ). Therefore, the hex number D8AF translated to decimal means 3x 3 +8x + 0x + 5 = However, going through the arithmetic for hex numbers in order to evaluate them is not really necessary. The easier way is to use the calculator that comes as an accessory in Windows. It can convert between decimal and hex when in Scientific view. Note that a hex number such as 35 is not the same as the decimal number 35. Its actual value in decimal terms is 3x x + 5 = 89. To avoid confusion, hex numbers are often labeled or tagged so that their meaning is clear. One method of tagging hex numbers is to append a lower case h at the end. Another method of labeling is to precede the number with 0x. Thus, the hex number D8AF can also be written D8AFh, where the lower case h at the end is just a label to make sure we know that it is a hex number. Also, D8AF can be written with a labeling prefix as 0xD8AF.

4 G-4 Octal Numbering System Many of the early computers used the octal numbering system for compiled printouts. Today, the PLC is about the only device that uses the Octal numbering system. The octal numbering system uses 8 values to represent numbers. The values are 0--7 being Base 8. Table 4 shows the first 3 decimal digits in octal. Note that the octal values are 0--7, 0--7, 0--7, and Octal Decimal Octal Decimal Table 4 This follows the DirectLOGIC PLCs. Refer to the bit maps in Chapter 4 and notice that the memory addresses are numbered in octal, as well as each bit. The octal system is much like counting in the decimal system without the digits 8 and 9 being available. The general format for four digits of the octal number is: (dx8 0 )+(dx8 )+(dx8 )+(dx8 3 ) where d means digit. This is the same format used in the binary, decimal, or hexadecimal systems except that the base number for octal is 8.

5 G-5 Using the powers of expansion, the example below shows octal 4730 converted to decimal. Binary Coded Decimal (BCD) Numbering System BCD is a numbering system where four bits are used to represent each decimal digit. The binary codes corresponding to the hexadecimal digits A--F are not used in the BCD system. For this reason numbers cannot be coded as efficiently using the BCD system. For example, a byte can represent a maximum of 5 different numbers (i.e ) using normal binary, whereas only 00 distinct numbers (i.e ) could be coded using BCD. Also, note that BCD is a subset of hexadecimal and neither one does negative numbers. BCD Bit Pattern Power Bit Value Max Value Table 5 One plus for BCD is that it reads like a decimal number, whereas 87 in BCD would mean 87 decimal. No conversion is needed.

6 G- Real (Floating Point) Numbering System The terms Real and floating--point both describe IEEE--754 floating point arithmetic. This standard specifies how single precision (3 bit) and double precision (4 bit) floating point numbers are to be represented as well as how arithmetic should be carried out on them. Most PLCs use the 3--bit format for floating point (or Real) numbers which will be discussed here. Real (Floating Point 3) Bit Pattern Bit # Sign Exponent Mantissa Mantissa (continues from above) Table Floating point numbers, which DirectLOGIC PLCs use, have three basic components: sign, exponent and mantissa. The 3 bit word required for the IEEE standard floating point numbers is shown in Table. It is represented as a number bits from 0 to 3, left to right. The first bit (3) is the sign bit, the next eight bits (30--3) are the exponent bits and the final 3 bits (--0) are the fraction bits. In summary: The sign bit is either 0 for positive or for negative; The exponent uses base ; The first bit of the mantissa is typically assumed to be.fff, where f is the field of fraction bits. The Internet can provide a more in--depth explanation of the floating point numbering system. One website to look at is:

7 G-7 BCD/Binary/Decimal/Hex/Octal - What is the Difference? Sometimes there is confusion about the differences between the data types used in DirectLOGIC PLCs. The PLC s native data format is BCD, while the I/O numbering system is octal. Other numbering formats used are binary, decimal and Real. Although data is stored in the same manner (0 s and s), there are differences in the way that the PLC interprets it. While all of the formats rely in the base numbering system and bit--coded data, the format of the data is dissimilar. The formats discussed are shown in Table 7 below. Binary/Decimal Bit Pattern Decimal Bit Value Max Value 5535 Hexidecimal Bit Pattern Decimal Bit Value Max Value F F F F BCD Bit Pattern Decimal Bit Value Max Value Octal Bit Pattern Bit Value Max Value Octal Bit Pattern Bit # Sign Exponent Mantissa Mantissa (continued from above) Table 7 As seen in Table 7, the BCD and hexadecimal formats are similar, although the maximum number for each grouping is different (9 for BCD and F for hexadecimal). This allows both formats to use the same display method. The unfortunate side effect is that unless the data type is documented, it s difficult to know what the data type is unless it contains the letters A--F.

8 G-8 Data Type Mismatch Data type mismatching is a common problem when using an operator interface. Diagnosing it can be a challenge until you identify the symptoms. Since the PLC uses BCD as the native format, many people tend to think it is interchangeable with binary (unsigned integer) format. This is true to some extent, but not in this case. Table 8 shows how BCD and binary numbers differ. Data Type Mismatch Decimal BCD Binary Table 8 As the table shows, BCD and binary share the same bit pattern up until you get to the decimal number 0. Once you get past 0, the bit pattern changes. The BCD bit pattern for the decimal 0 is actually equal to a value of in binary, causing the number to jump six digits by when viewing it as the BCD. With larger numbers, the error multiplies. Binary values from 0 to 5 Decimal are actually invalid for the BCD data type. Looking at a larger number, such as the value shown in Table 9, both the BCD bit pattern and the decimal bit pattern correspond to a base 0 value of If bit patterns are read, or interpreted, in a different format than what is used to write them, the data will not be correct. For instance, if the BCD bit pattern is interpreted as a decimal (binary) bit pattern, the result is a base 0 value of Similarly, if you try to view the decimal (binary) bit pattern as a BCD value, it is not a valid BCD value at all, but could be represented in hexadecimal as 0xFFF. Base 0 Value BCD Bit Pattern Binary Bit Pattern Table 9 Look at the following examples and note the same value represented by the different numbering systems. The decimal values of 7 and 40 are used. 7 Decimal 4 0 Decimal 00 0 BCD BCD Binary Binary 4 3 Hex 3 4 Hex 0 3 Octal 0 4 Octal

9 G-9 Signed vs. Unsigned Integers So far, we have dealt with unsigned data types only. Now we will deal with signed data types (negative numbers). The BCD and hexadecimal numbering systems do not use signed data types. In order to signify that a number is negative or positive, we must assign a bit to it. Usually, this is the Most Significant Bit (MSB) as shown in Table 0. For a --bit number, this is bit 5. This means that for --bit numbers we have a range of to 377. Table 0 There are two ways to encode a negative number: two s complement and Magnitude Plus sign. The two methods are not compatible. The simplest method to represent a negative number is to use one bit of the PLC word as the sign of a number while the remainder of the word gives its magnitude. It is general convention to use the most significant bit (MSB) as the sign bit: a will indicate a negative, anda0apositivenumber.thus, a bit word allows numbers in the range 377. The following tables show a representation of 00 and a representation of --00 in this format. Decimal Magnitude Plus Sign Binary Table Two s complement is a bit more complicated. Without getting involved with a full explanation, a simple formula for two s complement is to invert the binary and add one (see Table ). Basically, s are being changed to 0 s and all 0 s are being changed to, then aisadded. Decimal Two s Compliment Binary Table More information about s complement can be found on the Internet, however most of the websites deal with 8--bit examples. Two fairly good websites are listed below, you can also do a search and checkout more websites.

10 G-0 AutomationDirect.com Products and Data Types DirectLOGIC PLCs C -more/c -more Micro -Graphic Panels The DirectLOGIC PLC family uses the octal numbering system for all addressing which includes: inputs, outputs, internal V--memory locations, timers, counters, internal control relays (bits), etc. Most data in the PLC, including timer and counter current values, is in BCD format by default. User data in V--memory locations may be stored in other data types if it is changed by the programmer, or comes from some external source, such as an operator interface. Any manipulation of data must use instructions appropriate for that data type which includes: Load instructions, Math instructions, Out box instructions, comparison instructions, etc. In many cases, the data can be changed from one data type to another, but be aware of the limitations of the various data types when doing so. For example, to change a value from BCD to decimal (binary), use a BIN instruction box. To change from BCD to a real number, use a BIN and a BTOR instruction box. When using Math instructions, the data types must match. For example, a BCD or decimal (binary) number cannot be added to a real number, and a BCD number cannot be added to a decimal (binary) number. If the data types are mismatched, the results of any math operation will be meaningless. When using the Data View in DirectSOFT, be certain that the proper format is selected for the element to be viewed. The data type and length is selected using the drop--down boxes at the top of the Data View window. Also notice that BCD is called BCD/Hex. Remember that BCD is a subset of hexadecimal so they share a display format even though the values may be different. This is where good documentation of the data type stored in memory is crucial. In the C--more and C--more Micro--Graphic HMI operator panels, the --bit BCD format is listed as BCD int. Binary format is either Unsigned int or Signed int depending on whether or not the value can be negative. Real number format is Floating PT 3. More available formats are, BCD int 3, Unsigned int 3 and Signed int 3.

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