1. Phys 4330 Digital ElectronicsBinary System Digital circuits process signals that contain just two voltage levels or states, labeled logic "0" and logic "1". These discrete voltage levels are commonly known as Binary digits and are normally referred to as BITS. Because there are only two valid Boolean values for representing either logic "1" or logic "0", the Binary Numbering system is ideal for use in digital or logic circuits and systems. The Binary Numbers system is a Base-2 system which follows the same rules in mathematics as the common decimal system meaning instead of powers of ten it uses a power of two, 1, 2, 4, 8, 16, 32 etc.

2. Phys 4330 Digital ElectronicsBinary System Basic Concepts In the decimal system, things are organized into columns: H | T | O 9 | 2 | 4 10 2 | 10 1 | 10 0 9 | 2 | 4 The number 924 is really {(9*10 2 )+(2*10 1 )+(4*10 0 )}. Each column has a value equals to its weight multiply by its frequency The frequency of each column can be 0 or 1 or 2 or ………. or 9 (ten possible values) The column weight is a power of ten; 1 (or Ones), 10 1 (or Tens), 10 2 (or Hundreds), and so on

3. Phys 4330 Digital ElectronicsBinary System 2 2 | 2 1 | 2 0 1 | 0 | 1 The binary number 101 is really equivalent to the decimal number {(1*1)+(2*0)+(4*1)} = 1+0+4 =5. Each column has a value equals to its weight multiply by its frequency The frequency of each column can be 0 or 1 (two possible values) The column weight is a power of 2; 1, 2, 4, 8, 16, 32, 64 and so on But because we have only two states or two frequency; 0 and 1, the weight of consecutive columns is a power of two instead of tens The binary system works under the exact same principles as the decimal system, only it operates in base 2 rather than base 10.

4. Phys 4330 Digital ElectronicsBinary System Example: What would the binary number 10, 111, 10101 and 11110 be in Decimal notation? Least Significant Number LSN and Highest Significant Number HSN As in decimal the LSN is the one bit to the most right of the digital number while the HSN is the bit to the most left of the digital number Binary2424 23232 2121 2020 Decimal 10101*2+0*1=2 1111111*1+1*2+1*4=7 10101101011*16+0*8+1*4+0*2+1*1=21 11110111101*16+1*8+1*4+1*2+1*0=30

5. Phys 4330 Digital ElectronicsBinary System The same approach applies to non-integral numbers. 110.101 = 1*2 2 +1*2 1 +0*2 0 +1*2 -1 +0*2 -2 +1*2 -3 = 4 + 2 + 0 + 0.5 + 0 + 0.125 = 6.625 Convert the following number from binary form to decimal form: 100011 100000 111111 101010 101.101 110.011 23232 2121 2020.2 -1 2 -2 2 -3 Example:

6. Phys 4330 Digital ElectronicsBinary System Binary Addition In decimal system we add the ones together and tens together and hundred together and so on. If the sum exceeds the bit value (9 fold), we carry the exceeded to the left bit. Example: 273 465 ----- 738 In this example when we add the tens (7 tens + 6 tens) we get 13 tens. We put the SUM, 3 tens in tens bit and add the CARRY 10 tens as 1 to the hundred bit. In adding binary number we do the same, we put the SUM and take the CARRY to left bit. Because 1 is the maximum value in any binary bits, the CARRY is 1 only if both bits is 1.

7. Phys 4330 Digital ElectronicsBinary System Adding rules: 0011 0101 -------- SUM 0111 CARRY 0001 ---------------------------------------------------------- Result00010 ++++++++ The process is the same for multiple-bit binary numbers: Number 11101 Number 21110 SUM0011 CARRY1100 Total11011

8. Phys 4330 Digital ElectronicsBinary System Try the following examples of binary addition: 11101 10111 10011 +11010 +11100 +11100 _____ ______ ______

9. Phys 4330 Digital ElectronicsBinary System Decimal to Binary conversion Example: Convert the decimal number 44 into its corresponding binary number. To convert a decimal number to binary, first subtract the largest possible power of two, and keep subtracting the next largest possible power form the remainder, marking 1s in each column where this is possible and 0s where it is not. 44 - 32 ____ 12 - 8 ____ 4 - 4 ____ 0 32168421 101100 Remember, the power of two is: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, ……….. Method one:

10. Phys 4330 Digital ElectronicsBinary System Method two: Keep dividing the decimal number by two; put 0 if there is no remaining and put 1 if there is until you done Example: convert 44 into binary 44/2220 22/2110 11/251 5/221 2/210 1/201 LSN HSN 101100