Decimal System The radix or base of a number system determines the total number of different symbols or digits used by the system. The decimal system has a base of 10. In the decimal system, 10 unique numbers or digits ( 0 through 9) are used: the total number of symbols is the same as the base, and the symbol with the largest value is 1 less than the base.

Decimal System The decimal system can be summarized as follows: Ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Base: 10 Weights: 1, 10, 100, 1000, …(powers of base 10)

Decimal System Weighted value in the decimal system

Binary System The binary system has a base of 2. The only allowable digits are 0 and 1 Digital Signal Waveform: with digital circuits it is easy to distinguish between two voltage levels - +5 V and O V, which can be related to the binary digits 1 and 0. Time Volts +5 High (H) (1) Low (L) (0)

Binary System The binary system can be summarized as follows: Two digits: 0, 1 Base: 2 Weights: 1, 2, 4, 8, 16, 32, …(powers of base 2)

Binary System Since the binary system uses only Decimal Binary 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 Since the binary system uses only two digits, each position of a binary number can go through only two changes, and then a 1 is carried to the immediate left position. To express the number in the binary system requires many more digits than in the decimal system.

Converting For Binary To Decimal

Converting For Binary To Decimal Another Method In the binary number when you see a 1, multiply that 1 times the value that is directly over it. Where you see a 0 in the box, just ignore it. 128 64 32 16 8 4 2 1 If we add only those numbers which have a binary 1 in the box under them, we come up with 128+32+8+4+1 which equals 173.

Bits – Bytes - Words Each digit of a binary number is known as a bit. A group of 8 bits is known as a byte. A group of bits that occupies one or more storage locations and is treated as a unit is known as a word. A 16-bit word is made up of two bytes (Upper and Lower). The least significant bit (LSB) is the digit that represents the smallest value. The most significant bit (MSB) is the digit that represents the largest value. 16-Bit Word Bit Upper Byte MSB LSB

PLC Processor Memory Size The size of the programmable controller relates to the amount of user program that can be stored. The 1 K word memory size shown can store 1,024 words, or 16,380 (1,024 x 16) bits of information using 16-bit words or 32,768 (1,024 x 32) using 32 bit words.

Converting For Decimal To Binary

Binary Representation Of Data Even though the binary system has only two digits, it can be used to represent any quantity that can be represented in the decimal system. Computer memory is then a series of binary 1s and 0s. SLC 500 Modular Chassis Output Status File A word will be created in the table only if the processor finds an output module residing in a particular slot. Each bit represents the “on” or “off” state of one output point. These points are numbered 0 through15. One 16-bit output file word is reserved for each slot in the chassis. The column on the right lists the output module address. Made up of single bits grouped into 16-bit words

Read chapter 7-1 thru 7-3 p.241 - 250

Binary Addition When adding with binary numbers, there are only four conditions that can occur.

Binary Addition When adding larger binary numbers, the resulting 1’s are carried into higher-order columns.

Binary Subtraction To subtract from larger binary numbers, subtract column by column, borrowing from the adjacent column when necessary. Remember that when borrowing from the adjacent column, there are two digits, i. e., 0 borrow 1 gives 10.

Binary Subtraction To subtract using the 1’s complement: Change the subtrahend to 1’s complement Add the two numbers Remove the last carry and add it to the number 1’s complement

Binary Multiplication When multiplying binary numbers, there are only four conditions that can occur. 0 x 0 = 0 0 x 1 = 0 1 x 0 = 0 1 x 1 = 1

x 110 Binary Multiplication To multiply numbers with more than one digit, form partial products and add them together. 101 x 110 000 101 101 11110

Binary Division The process for dividing one binary number by another is the same for both binary and decimal numbers. 111 10 11 00 10 1110

1. The binary number system has a base of 8. (True/False) 2. The decimal number 7 would be written in binary as 1011. (True/False) 3. To express a number in decimal requires fewer digits than in the binary system. (True/False) 4. For a base 2 number system, the weight value associated with the 3rd digit would be 4. (True/False)

5. What is the decimal value of binary 110010 ? a. 48 c. 13 b. 26 d. 50 6. The decimal number 15 would be written in binary as: a. 1111 c. 4C b. 1000 d. 00011001 7. Data can be stored in one 16-bit word as two separate groups of 8-bit data. (True/False)

8. A group of 8 bits is known as a byte. (True/False) 9. The MSB is the digit that represents the smallest value. (True/False) 10. Since the binary system has only two digits, it is limited as far as representing very large quantities. (True/False)