Number system
- 1. DIGITAL ELECTRONICS DR. R. M. THOMBRE HEAD OF DEPT OF PHYSICS M.G. COLLEGE ARMORI
- 2. NUMBER SYSTEMS • Decimal • Binary • Octal • Hexadecimal
- 3. • A digital system can understand positional number system only where there are a few symbols called digits and these symbols represent different values depending on the position they occupy in the number. • A value of each digit in a number can be determined using • The digit • The position of the digit in the number • The base of the number system (where base is defined as the total number of digits available in the number system).
- 5. DECIMAL SYSTEM • The number system that we use in our day-to-day life is the decimal number system. Decimal number system has base 10 as it uses 10 digits from 0 to 9. In decimal number system, the successive positions to the left of the decimal point represents units, tens, hundreds, thousands and so on. • Each position represents a specific power of the base (10). For example, the decimal number 1234 consists of the digit 4 in the units position, 3 in the tens position, 2 in the hundreds position, and 1 in the thousands position, and its value can be written as
- 6. (1×1000) + (2×100) + (3×10) + (4×l) (1×103) + (2×102) + (3×101) + (4×l00) 1000 + 200 + 30 + 1 1234
- 8. BINARY NUMBER SYSTEM • Characteristics • Uses two digits, 0 and 1. • Also called base 2 number system • Each position in a binary number represents a 0 power of the base (2). Example: 20 • Last position in a binary number represents an x power of the base (2). Example: 2x where x represents the last position - 1.
- 9. BINARY SYSTEM • In mathematics and digital electronics, a binary number is a number expressed in the base-2numeral system or binary numeral system, which uses only two symbols: typically 0 (zero) and 1 (one). • The base-2 numeral system is a positional notation with a radix of 2. • Each digit is referred to as a bit. I will teach about it briefly.
- 10. Step Binary Number Decimal Number Step 1 101012 ((1 × 2 4 ) + (0 × 2 3 ) + (1 × 2 2 ) + (0 × 2 1 ) + (1 × 2 0 ))10 Step 2 101012 (16 + 0 + 4 + 0 + 1)10 Step 3 101012 2110 Example Binary Number: 101012 Calculating Decimal Equivalent − . Note: 101012 is normally written as 10101
- 11. OCTAL NUMBER SYSTEM • Characteristics • Uses eight digits, 0,1,2,3,4,5,6,7. • Also called base 8 number system • Each position in an octal number represents a 0 power of the base (8). Example: 80 • Last position in an octal number represents an x power of the base (8). Example: 8x where x represents the last position - 1.
- 12. Step Octal Number Decimal Number Step 1 125708 ((1 × 8 4 ) + (2 × 8 3 ) + (5 × 8 2 ) + (7 × 8 1 ) + (0 × 8 0 ))10 Step 2 125708 (4096 + 1024 + 320 + 56 + 0)10 Step 3 125708 549610 Example Octal Number − 125708 Calculating Decimal Equivalent − Note: 125708 is normally written as 12570.
- 13. HEXADECIMAL NUMBER SYSTEM • Characteristics • Uses 10 digits and 6 letters, 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. • Letters represents numbers starting from 10. A = 10, B = 11, C = 12, D = 13, E = 14, F = 15. • Also called base 16 number system. • Each position in a hexadecimal number represents a 0 power of the base (16). Example 160. • Last position in a hexadecimal number represents an x power of the base (16). Example 16x where x represents the last position - 1.
- 15. Step Hexadecimal Number Decimal Number Step 1 19FDE16 ((1 × 16 4 ) + (9 × 16 3 ) + (F × 16 2 ) + (D × 16 1 ) + (E × 16 0 ))10 Step 2 19FDE16 ((1 × 16 4 ) + (9 × 16 3 ) + (15 × 16 2 ) + (13 × 16 1 ) + (14 × 16 0 ))10 Step 3 19FDE16 (65536 + 36864 + 3840 + 208 + 14)10 Step 4 19FDE16 10646210 Example − Hexadecimal Number: 19FDE16 Calculating Decimal Equivalent − . Note − 19FDE16 is normally written as 19FDE
- 16. DECIMAL FRACTION • To the right of the decimal point, the digit 1 has a weight 0.1 (1/10), the digit 2 has a weight of 0.01 (1/100) and the digit 5 has a weight of 0.001(1/1000)
- 18. BINARY FRACTION • The binary numbering system is a base-2 numbering system which contains only two digits, a “0” or a “1”. Thus each digit of a binary number can take the “0” or the “1” value with the position of the 0 or 1 indicating its value or weighting. • Similar to decimal fractions, binary numbers can also be represented as unsigned fractional numbers by placing the binary digits to the right of the decimal point or in this case, binary point. Thus all the fractional digits to the right of the binary point have respective weightings which are negative powers of two, creating a binary fraction. In other words, the powers of 2 are negative. • So for the fractional binary numbers to the right of the binary point, the weight of each digit becomes more negative giving: 2-1, 2-2, 2-3, 2-4, and so on as shown.
- 19. BINARY FRACTIONS
- 21. Now lets suppose we have the following binary number of: 1101.01112, what will be its decimal number equivalent. 1101.0111 = (1×23) + (1×22) + (0×21) + (1×20) + (0×2-1) + (1×2-2) + (1×2-3) + (1×2-4) = 8 + 4 + 0 + 1 + 0 + 1/4 + 1/8 + 1/16 = 8 + 4 + 0 + 1 + 0 + 0.25 + 0.125 + 0.0625 = 13.437510
- 22. DECIMAL TO BINARY CONVERSION • A decimal like 19 can be converted into binary by repeatedly dividing the number by 2 and collecting the remainders (double dabble method) • 2 19 2 9 – 1 LSB 2 4 – 1 2 2 – 0 2 1 -0 0 -1 MSB
- 23. • For decimal fractions, the fractional part has to be multiplied by 2 successively and collecting the carries from top to bottom. The multiplication can be repeated till the fractional part becomes zero. If the fractional part is not zero after four or five steps the process can be stopped and we have to be satisfied with the nearest value. • For example the decimal fraction .625 is converted into binary as
- 24. • 0.625 × 2 = 1.250; carry is 1 (MSB) • 0.250 × 2 = 0.500; carry is 0 • 0.500 × 2 = 1.000; carry is 1 (LSB) • Therefore (0.625)10 = (0.101)2
- 25. PROBLEMS- DECIMAL TO BINARY • Convert (107.6875) • Convert (52.4) 2 107 2 53 -1 2 26-1 2 13 -0 2 6 -1 2 3 -0 2 1 -1 0 -1
- 26. • 0.6875 = 0.10112 • 52.4 = 110100.01102
- 27. HEXADECIMAL TO DECIMAL • Convert D5H to decimal • (D5)H = (13×161 + 5×160) = (13 ×16 + 5× 1) = (208+5) = 21310
- 28. • (E9)H = (233)H • (3FC.8)H = (1020.5)H • (FFFF)H = (65535)H
- 29. DECIMAL TO HEXADECIMAL • Convert 213 to Hex • 16 213 • 13 -5 • 0 -13 (D) • (213)10 = D5H
- 30. HEXADECIMAL TO BINARY • Convert (25)H to Binary • 25H = (0010 0101)2 • Convert (3A.7) to Binary • (3A.7)H = (0011 1010. 0111)2 • Convert (CD.E8) to Binary • (CD.E8) = (1100 1101. 1110 1000)2
- 31. BINARY TO DECIMAL • (1010.1101)2 = (A.D)H • (110.101) = (0110.1010) = (6.A)H • (1110.11)2 = (1110.1100) = (E.C)H
- 32. OCTAL TO DECIMAL Decimal Octal Binary 0 0 000 1 1 001 2 2 010 3 3 011 4 4 100 5 5 101 6 6 110 7 7 111
- 33. OCTAL TO DECIMAL • Convert (75)8 to decimal • (75)8 = (7×81 + 5×80) = (7×8 + 5× 1) = (56+5) = 6110
- 34. DECIMAL TO OCTAL • Convert 6810 to octal • 8 68 • 8 8 - 4 • 8 1 – 0 • 0 – 1 • (68)10 = (104)8
- 35. OCTAL TO BINARY Octal Binary Decimal 27 010 111 2 3 135 001 011 101 93 45.5 100 101.101 37.625
- 36. BINARY TO OCTAL • (10 101)2 = (010 101)2 = (25)8 • (10111.1) = (010 111 . 100) = (2 7 . 4)8
- 37. • Binary to decimal • (11101) • (111.11) • Hex to decimal • (50)H • (3EF)H • Decimal to Hex • (48)10 • (1024)10 • Hex to binary (99)H , (FF.E6)H