Computer Arithmetic and Binary Math.pptx
- 1. Computer Arithmetic Nazmun Nessa Moon Assistant Professor Department of CSE Daffodil International University
- 2. Learning Objectives In this lecture you will learn about: Reasons for using binary instead of decimal numbers Basic arithmetic operations using binary numbers Addition (+) Subtraction (-) Multiplication (*) Division (/)
- 3. Binary over Decimal Information is handled in a computer by electronic/ electrical components Electronic components operate in binary mode (can only indicate two states – ON (1) or OFF (0) Binary number system has only two digits (0 and 1), and is suitable for expressing two possible states In binary system, computer circuits only have to handle two binary digits rather than ten decimal digits causing: Simpler internal circuit design Less expensive More reliable circuits Arithmetic rules/processes possible with binary numbers
- 4. Examples of a Few Devices that work in Binary Mode Binary
- 5. Binary Number System System Digits: 0 and 1 Bit (short for binary digit): A single binary digit LSB (least significant bit): The rightmost bit MSB (most significant bit): The leftmost bit Upper Byte (or nybble): The right-hand byte (or nybble) of a pair Lower Byte (or nybble): The left-hand byte (or nybble) of a pair The term nibble used for 4 bits being a subset of byte.
- 7. Binary Equivalents 1 Nybble (or nibble) = 4 bits 1 Byte = 2 nybbles = 8 bits 1 Kilobyte (KB) = 1024 bytes 1 Megabyte (MB) = 1024 kilobytes = 1,048,576 bytes 1 Gigabyte (GB) = 1024 megabytes = 1,073,741,824 bytes
- 8. Binary Arithmetic Binary arithmetic is simple to learn as binary number system has only two digits – 0 and 1 Following slides show rules and example for the four basic arithmetic operations using binary numbers
- 9. Binary Addition Rule for binary addition is as follows: ① 0 + 0 = 0 ② 0 + 1 = 1 ③ 1 + 0 = 1 ④ 1 + 1 = 0 plus a carry of 1 to next higher column
- 10. Example 1: 000110102 + 000011002 = 001001102 Binary Addition
- 11. Example 2: 000100112 + 001111102 = 010100012 Binary Addition
- 12. Binary Addition (Example 3)
- 13. Binary Subtraction Rule for binary subtraction is as follows: ① 0 - 0 = 0 ② 0 - 1 = 1 with a borrow from the next column ③ 1 - 0 = 1 ④ 1 - 1 = 0
- 14. Binary Subtraction Example 1: 001001012 - 000100012 = 000101002
- 15. Example 2: 001100112 - 000101102 = 000111012 Binary Subtraction
- 16. Binary Multiplication Table for binary multiplication is as follows: ① 0 x 0 = 0 ② 0 x 1 = 0 ③ 1 x 0 = 0 ④ 1 x 1 = 1
- 17. Example 1: 001010012 × 000001102 = 111101102 Binary Multiplication
- 18. Example 2: 000101112 × 000000112 = 010001012 Binary Multiplication
- 21. Binary Division Table for binary division is as follows: ① 0 ÷ 0 = Divide by zero error ② 0 ÷ 1 = 0 ③ 1 ÷ 0 = Divide by zero error ④ 1 ÷ 1 = 1 As in the decimal number system (or in any other number system), division by zero is meaningless The computer deals with this problem by raising an error condition called ‘Divide by zero’ error
- 22. Rules for Binary Division ① Start from the left of the dividend ② Perform a series of subtractions in which the divisor is subtracted from the dividend ③ If subtraction is possible, put a 1 in the quotient and subtract the divisor from the corresponding digits of dividend ④ If subtraction is not possible (divisor greater than remainder), record a 0 in the quotient ⑤ Bring down the next digit to add to the remainder digits. Proceed as before in a manner similar to long division
- 23. Binary Division (Example 1)
- 24. Binary Division (Example 2) Example: 001010102 ÷ 000001102 = 000001112
- 25. Example: 100001112 ÷ 000001012 = 000110112 Binary Division (Example 3)
- 26. Complement of a Number
- 27. Complement of a Decimal Number
- 28. Complement of a Octal Number
- 29. Complement of a Binary Number
- 30. Complementary Method of Subtraction Involves following 3 steps: Step 1: Find the complement of the number you are subtracting (subtrahend) Step 2: Add this to the number from which you are taking away (minuend) Step 3: If there is a carry of 1, add it to obtain the result; if there is no carry, recomplement the sum and attach a negative sign Complementary subtraction is an additive approach of subtraction
- 33. Binary Subtraction Using Complementary Method (Example 1)
- 34. Binary Subtraction Using Complementary Method (Example 2)
- 35. Addition/Subtraction of Numbers in 2’s Complement Notation 35 Represent all negative numbers in 2’s complement form. Now we have the same procedure for addition and subtraction. Subtraction of a number is achieved by adding the 2’s complement of the number. This is illustrated in the following example where the carry, if any, from the most significant bit, during addition, should be ignored. The result has to be interpreted appropriately using the same convention.
- 36. Addition/Subtraction of Numbers in 2’s Complement Notation…
- 37. Key Words/Phrases Additive method of division Additive method of multiplication Additive method of subtraction Binary addition Binary arithmetic Binary division Binary multiplication Binary subtraction Complementary subtraction Complement Computer arithmetic