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# Binary numbers

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### Binary numbers

1. 1. An Introduction to Computer Hardware - BKHS Binary Numbers 1 01/08/14
2. 2. Introduction  Importance of the Binary Number  Use of binary numbers in computers 2 01/08/14
3. 3. Agenda  Binary Theory  Binary to Decimal Conversion  Decimal to Binary Conversion  Data Flow using Binary Numbers 3 01/08/14
4. 4. Overview  Binary numbers are used extensively in digital electronics  Binary numbers are the foundation of other numbering systems such as Hexadecimal and Octal when used in digital electronics. 4 01/08/14
5. 5. Binary Numbers Defined  A single “bit” is the foundation  Only 2 states possible  Hi – Lo, On – Off, True – False, Open – Closed  8 “bits” makeup a single “byte”  Data typically stored in “bytes” 5 01/08/14
6. 6. Converting Decimal to Binary  LSB – least significant bit  MSB – Most significant bit  Divide the number by 2  If no remainder, record a zero (0) for LSB  If there is a remainder, record a one (1) for LSB  Divide the previous answer by 2  If no remainder, record a zero in the next bit position (to the left of the LSB)  If there is a remainder, record a one.  Repeat previous 3 steps until the answer is no longer divisible by 2. 6 01/08/14
7. 7. Decimal to Binary Example Convert 5267 to binary 5267/2 = 2633 2633/2 = 1316 1316/2 = 658 658/2 = 329 329/2 = 164 164/2 = 82 82/2 = 41 41/2 = 20 20/2 = 10 10/2 = 5 5/2 =2 2/2 =1 1/2 =0 r-1 r-1 r–0 r–0 r–1 r–0 r–0 r–1 r–0 r–0 r–1 r–0 r–1 LSB = 1 next = 1 next = 0 next = 0 next = 1 next = 0 next = 0 next = 1 next = 0 next = 0 next = 1 next = 0 MSB = 1 Binary Number – 1010010010011 7 01/08/14
8. 8. Converting Binary to Decimal Convert 1101001 to decimal Each bit position is calculated using the formula: (value in position) x 2^(position #) so, if bit position 2 = 1 then, applying the formula 1 x 2^2 = 4 Any bit position containing a zero is skipped Bit position 0 is the LSB. LSB = 1, so 2^0 = 1, add it. Bit position 1 is 0, so skip it Bit position 2 is 0, so skip it also Bit position 3 is 1, so 2^3 = 8, add it. Bit position 4 is 0, so skip it Bit position 5 is 1, so 2^5 = 32, add it. Bit position 6 is the MSB, MSB = 1 so 2^6 = 64, add it. 1 + 8 + 32 + 64 = 105. Decimal value = 105 8 01/08/14
9. 9. Adding Binary Numbers 1010 +1111 ______  Step one: Column 2^0: 0+1=1. Record the 1. Temporary Result: 1; Carry: 0  Step two: Column 2^1: 1+1=10. Record the 0, carry the 1. Temporary Result: 01; Carry: 1  Step three: Column 2^2: 1+0=1 Add 1 from carry: 1+1=10. Record the 0, carry the 1. Temporary Result: 001; Carry: 1  Step four: Column 2^3: 1+1=10. Add 1 from carry: 10+1=11. Record the 11. Final result: 11001 9 01/08/14
10. 10. Binary Multiplication Multiplication in the binary system works the same way as in the decimal system:  1*1=1  1*0=0  0*1=0 101 * 11 -----101 1010 -----1111 10 01/08/14
11. 11. Data Streams using Binary numbers In the diagram, a start bit is sent, followed by eight data bits, no parity bit and one stop bit, for a 10-bit character frame. The number of data and formatting bits, and the transmission speed, must be pre-agreed by the communicating parties. After the stop bit, the line may remain idle indefinitely, or another character may immediately be started: 11 01/08/14