Number Systems PDF

NUMBER SYSTEMS Lesson 2 Objectives Understand Decimal Numbers System Binary Number System Conversion Arithmetic operations (Addition, Subtraction, Division and Multiplication) 2 Decimal Number ▪ It uses 10 different sy...

NUMBER SYSTEMS Lesson 2 Objectives Understand Decimal Numbers System Binary Number System Conversion Arithmetic operations (Addition, Subtraction, Division and Multiplication) 2 Decimal Number ▪ It uses 10 different symbols to express any number. Those 10 symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and called as digits of a number. ▪ Since there are 10 different numbers it referred to as Base 10 number system or Radix of 10 number system. ▪ In decimal, the position of each digit will give different value. Express decimal 593 using radix of 10. 5 9 3 Solution 5  100 + 9  10 + 31 5 10 2 + 9 10 1 + 3 10 0 3 Example Express Decimal 37.46 using radix of 10. Solution 3 7. 4 6 3  10 + 7  1+ 4  0.1 + 6  0.01 3 101 +7 10 + 0 4 10−1 +6 10−2 4 Binary Number System ▪ Unlike in decimal, binary use only 2 symbols 0 and 1. ▪ Binary system can be called as base of 2 system or radix of 2. ▪ Same as in decimal, in binary the LSB gives 1s place from 0 2. Second column 1 gives 2s-place (2 ), that gives the decimal value of 2. Third column gives 4s- 2 place (2 ), that gives the decimal value of 4. ▪ Likewise every added column is an added power of 2 5  6 Binary Number System ▪ Fractional binary numbers can be place right to the decimal point and it decreases by negative power of 2 for each digit 2n − 1 n − 2 …2 2 3 22 21 0 2. 2 -1 2 −2 2−3 −4 2 …2 -(n+1) Decrease with negative power of 2  7 Binary Number System Binary−to−Decimal Conversion Any binary number can be converted to decimal by: 1. Multiplying the weight of each position with the binary digit 2. Adding them together Convert the binary number 101102 to its Decimal equivalent. Solution Binary number 1 0 1 1 0 Power of 2 position 2 4 + 23 + 2 2 + 21 + 20 (2 1)+(2 0)+(2 1)+(2 1)+(2  4 3 2 1 0 0) Decimal value 16 + 0 + 4 + 2 + 0 = 2210 8  9 Example Convert the fractional binary number 101.102 to its decimal equivalent. Solution Binary number 1 0 1. 1 0 Power of 2 position 22 + 2 1 + 20. 2 -1 + 2-2 (2 1)+ 2 (2 0)+(2 1) 1 0. (2 1)+ -1 (2 0) -2 Decimal value 4 + 0 + 1. 0.5 + 0 = 5. 510  10 Decimal–to–Binary Conversion The most convenient method is called division by 2 method. In which first decimal number will be divided by 2. The quotient will be dividend for the next step. In each step the remainder part will be recorded separately. The 1st remainder of the 1st division will be the LSB in the binary number.  The quotient should repeatedly divide by 2 until the quotient becomes 0. The final remainder will be the MSB in binary number. Decimal–to–Binary Conversion Convert decimal 2010 to its binary equivalent Solution 2 20 + remainder of 0 2 10 + remainder of 0 2 5 + remainder of 1 2 2 + remainder of 0 21 + remainder of 1 0 1 0 1 0 02 11  12 Decimal–to–Binary Conversion When converting a decimal fractional number to its binary, the decimal fractional part will be multiply by 2 till the fractional part gets 0 or till the number of decimal places reached. Example Convert Decimal 0.62510 to its binary equivalent. Solution Step−1: 0.625 will be multiply by 2 (0.625  2 = 1.25) Step−2: The integer part will be the MSB in the binary result Step−3: The fractional part of the earlier result will be multiply again. (0.25  2 = 0.5)  13 Decimal–to–Binary Conversion Cont.. Step−4: Each time after the multiplication the integer part of the result will be written as the binary number. Step−5: The procedure should continue till the fractional part gets 0. Binary Addition ▪ Adding of two binary numbers follows same as addition of two decimal numbers. ▪ There are mainly 4 rules should be followed in the process of addition in binary numbers. Sum Carryout Rule 1 0 + 0 = 0 0 Rule 2 0 + 1 = 1 0 Rule 3 1 + 0 = 1 0 Rule 4 1 + 1 = 0 1 14 Binary Subtraction ▪ When subtracting one binary number A (subtrahend) from another binary number B (Minuend) where B > A, the answer is called the difference. ▪ There are mainly 4 rules should be followed Minuend (B) Subtrahend (A) Difference Borrow out Rule 1 0 − 0 = 0 Rule 2 0 − 1 = 1 1 Rule 3 1 − 0 = 1 Rule 4 1 − 1 = 0 15 Binary Division  Binary division process is the same as that which followed for decimal number.  When we divide one binary number (dividend) by another binary number (division), the process is shown in the following standard format. dividend / divisor = Quotient 16  Multiplication in Binary Multiplying two binary numbers (Multiplicand and multiplier) is same as in decimal multiplication. Same as in binary addition and subtraction, binary multiplication should perform according to four main rules. 17