Understanding Binary Numbers. Different Number Systems. Conversion: Bin Hex. Conversion MAP. Binary (0, 1) Hexadecimal 0 9, A(10), B(11),, F(15) :

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1 Understanding Binary Numbers Computers operate on binary values (0 and 1) Easy to represent binary values electrically Voltages and currents. Can be implemented using circuits Create the building blocks of modern computers Binary numbers are made of binary digits (bits): 0 and 1 How many items does an binary number represent? (1011) 2 1x x x x2 0 (11) 10 What about fractions? (110.10) 2 1x x x x x2-2 Groups of eight bits are called a byte ( ) 2 Decimal: Different Number Systems (0 9) 1 10, , , , Binary (0, 1) 0 2, 1 2, , , Hexadecimal 0 9, A(10), B(11),, F(15) : 10F 16, ABCDEF 16, Octal 0 7: 137 8, 100 8, 777 8,.. N-base (0, 1., N-1): , , DT-009/1. Digital Electronics Number system & Conversion 1 DT-009/1. Digital Electronics Number system & Conversion 2 Conversion MAP Conversion: Bin Hex Octal(base 8) 1. group into 4's starting at least significant symbol (if the number of bits is not evenly divisible by 4, then add 0's at the most significant end) 2. write 1 hex digit for each group Decimal(base 10) Binary Coded Decimal (BCD) Binary(base 2) Hexadecimal (base16) Example: E F A 3 Gray Code DT-009/1. Digital Electronics Number system & Conversion 3 DT-009/1. Digital Electronics Number system & Conversion 4

2 Conversion: Bin BCD BCD format needs 4 bits for each decimal digit Is a way to represent binary numbers in decimal format BCD is not the same as binary representation! BCD BCD D2 16 Conversion: Bin Dec Multiply each 1 bit by the appropriate power of 2 and add them together ? ? DT-009/1. Digital Electronics Number system & Conversion 5 DT-009/1. Digital Electronics Number system & Conversion 6 Conversion: Dec Bin For each digit position: 1. Divide decimal number by the base (e.g. 2) 2. The remainder is the lowest-order digit 3. Repeat first two steps until no divisor remains. Conversion: Dec N Base (16) For each digit position: 1. Divide decimal number by the base N (e.g. 16) 2. The remainder is the lowest-order digit 3. Repeat first two steps until no divisor remains. Example: Convert Decimal 13 (13 10 ) to Binary : Integer Quotient Remainder Coefficient 13/ a 0 1 6/ a 1 0 3/ a 2 1 1/ a 3 1 Example for (1615) 10: Integer Quotient Remainder Coefficient 1615/ /16 a 0 15(F) 100/ /16 a 1 4 6/ /16 a 2 6 Answer: (13) 10 (a 3 a 2 a 1 a 0 ) 2 (1101) 2 DT-009/1. Digital Electronics Number system & Conversion 7 Answer (1615) 10 (64F) 16 DT-009/1. Digital Electronics Number system & Conversion 8

3 Conversion: Dec Bin Number Representation - Binary to decimal A decimal number can be converted to binary by repeated division by 2 number /2 remainder Least Significant Bit Most Significant bit DT-009/1. Digital Electronics Number system & Conversion 9 Gray Code Grey-code is one where only one bit changes at a time. The following tables show the difference between a three-bit Binary numbers and Gray-coded numbers Note that in the Gray-coded sequence, only one bit changes at a time. Yet, we have still represented all the possible bit combinations. Gray codes for 4 or more bits are not unique!!! Binary Grey-coded DT-009/1. Digital Electronics Number system & Conversion 10 Conversion: Bin Gray Conversion of Binary to Gray Code Start with the most significant bit of the binary number. Copy this bit as the MSB of the Gray code number. Add the MSB of the binary to the next bit of the binary number. The sum (ignoring carry) is the next bit of the gray code number. Continue adding each bit of the binary to the next bit to its right to get the gray code for that position as shown below: Binary: Gray: DT-009/1. Digital Electronics Number system & Conversion 11 Conversion: Gray Bin Start with the most significant bit of the Gray code number. Copy this bit as the MSB of the binary number. Add the MSB of the binary number result to the next bit of the Gray code number. The sum (ignoring carry) is the next bit of the binary. Continue adding each bit of the binary to the next Gray code bit to its right to get the binary bit for that position as shown below: Gray Binary DT-009/1. Digital Electronics Number system & Conversion 12

4 Exam (May 2002) (a) Convert the hexadecimal numbers 4A, FF and 6C to binary, octal, decimal and BCD forms. [6 marks] (b) Convert the decimal numbers 27, 99 and 01 to BCD, binary, hexadecimal and octal forms. [6 marks] (c) Find the twos complement of the decimal numbers 100 and 28 and hence show how the following arithmetic operations can be performed by an 8-bit binary adder. (i) (ii) (iii) Solution 1a (May 2002) (a) Convert the hexadecimal numbers 4A, FF and 6C to binary, octal, decimal and BCD forms. 1. 4A Bin Oct: 4 A or or Bin Dec: x x x Dec BCD: or BCD 1. FF Bin Oct: FF or or Bin Dec: Dec BCD: or BCD 1. 6C Bin Oct: 6C or or Bin Dec: x64 +1x16+1x8+1x Dec BCD: or BCD DT-009/1. Digital Electronics Number system & Conversion 13 DT-009/1. Digital Electronics Number system & Conversion 14 Solution 1b (May 2002) (b) Convert the decimal numbers 27, 99 and 01 to BCD, binary, hexadecimal and octal forms. 1. Dec BCD: BCD or BCD 2. Dec Bin Hex Oct : / 2 13 & 1 27 / 16 1 & 11(B) 27 / 8 3 & / 2 6 & 1 1 / 16 0 & 1 3 / 8 0 & / 2 3 & / 2 1 & / 2 0 & 1 Therefore B O B Solution 1c (May 2002) (c) Find the twos complement of the decimal numbers 100 and 28 and hence show how the following arithmetic operations can be performed by an 8-bit binary adder. (i) (ii) (iii) Comp Comp (64 +8) -72 DT-009/1. Digital Electronics Number system & Conversion 15 DT-009/1. Digital Electronics Number system & Conversion 16

5 (iii) c (128) Exam (May 2002) DT-009/1. Digital Electronics Number system & Conversion 17

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