Number system on various number tyoes decimal
- 1. Number Systems DECIMAL, BINARY, AND HEXADECIMAL 1
- 2. Base-N Number System Base N N Digits: 0, 1, 2, 3, 4, 5, …, N-1 Example: 1045N Positional Number System 2 • Digit do is the least significant digit (LSD). • Digit dn-1 is the most significant digit (MSD). 1 4 3 2 1 0 1 4 3 2 1 0 n n N N N N N N d d d d d d
- 3. Decimal Number System Base 10 Ten Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 104510 Positional Number System Digit d0 is the least significant digit (LSD). Digit dn-1 is the most significant digit (MSD). 3 1 4 3 2 1 0 1 4 3 2 1 0 10 10 10 10 1010 n n d d d d d d
- 4. Binary Number System Base 2 Two Digits: 0, 1 Example: 10101102 Positional Number System Binary Digits are called Bits Bit bo is the least significant bit (LSB). Bit bn-1 is the most significant bit (MSB). 4 1 4 3 2 1 0 1 4 3 2 1 0 2 2 2 2 2 2 n n b b b b b b
- 5. nybble = 4 bits byte = 8 bits (short) word = 2 bytes = 16 bits (double) word = 4 bytes = 32 bits (long) word = 8 bytes = 64 bits 1K (kilo or “kibi”) = 1,024 1M (mega or “mebi”) = (1K)*(1K) = 1,048,576 1G (giga or “gibi”) = (1K)*(1M) = 1,073,741,824 1 Petabyte = 1024 Terabytes. 1 terabyte (TB) equals 1,000 gigabytes (GB) or 1,000,000 megabytes (MB). 5 NVIDIA A100? Upto 80 gigabytes (GB) 16 and 32GB configurations, and offers the performance of up to 32 CPUs in a single GPU.
- 6. Hexadecimal Number System Base 16 Sixteen Digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F Example: EF5616 Positional Number System 6 0000 0 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 8 1001 9 1010 A 1011 B 1100 C 1101 D 1110 E 1111 F 1 4 3 2 1 0 16 16 16 16 1616 n
- 7. Binary Addition 7 •Single Bit Addition Table 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10 Note “carry”
- 8. Hex Addition 8 • 4-bit Addition 4 + 4 = 8 4 + 8 = C 8 + 7 = F F + E = 1D Note “carry”, Refer HEX Addition Table
- 9. Hex Digit Addition Table + 0 1 2 3 4 5 6 7 8 9 A B C D E F 0 0 1 2 3 4 5 6 7 8 9 A B C D E F 1 1 2 3 4 5 6 7 8 9 A B C D E F 10 2 2 3 4 5 6 7 8 9 A B C D E F 10 11 3 3 4 5 6 7 8 9 A B C D E F 10 11 12 4 4 5 6 7 8 9 A B C D E F 10 11 12 13 5 5 6 7 8 9 A B C D E F 10 11 12 13 14 6 6 7 8 9 A B C D E F 10 11 12 13 14 15 7 7 8 9 A B C D E F 10 11 12 13 14 15 16 8 8 9 A B C D E F 10 11 12 13 14 15 16 17 9 9 A B C D E F 10 11 12 13 14 15 16 17 18 A A B C D E F 10 11 12 13 14 15 16 17 18 19 B B C D E F 10 11 12 13 14 15 16 17 18 19 1A C C D E F 10 11 12 13 14 15 16 17 18 19 1A 1B D D E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C E E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D F F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 9
- 10. 1’s Complements 1’s complement (or Ones’ Complement) To calculate the 1’s complement of a binary number just “flip” each bit of the original binary number. E.g. 0 1 , 1 0 01010100100 10101011011 10
- 12. 2’s Complements 2’s complement To calculate the 2’s complement just calculate the 1’s complement, then add 1. 01010100100 10101011011 + 1= 10101011100 Handy Trick: Leave all of the least significant 0’s and first 1 unchanged, and then “flip” the bits for all other digits. Eg: 01010100100 -> 10101011100 12
- 13. Complements Note the 2’s complement of the 2’s complement is just the original number N EX: let N = 01010100100 (2’s comp of N) = M = 10101011100 (2’s comp of M) = 01010100100 = N 13
- 14. Signed Binary Numbers Two methods: First method: sign-magnitude Use one bit to represent the sign 0 = positive, 1 = negative Remaining bits are used to represent the magnitude Range - (2n-1 – 1) to 2n-1 - 1 where n=number of digits Example: Let n=4: Range is –7 to 7 or 1111 to 0111 14
- 15. Signed Binary Numbers Second method: Two’s-complement Use the 2’s complement of N to represent -N Note: MSB is 0 if positive and 1 if negative Range - 2n-1 to 2n-1 -1 where n=number of digits Example: Let n=4: Range is –8 to 7 Or 1000 to 0111 15
- 16. Signed Numbers – 4-bit example Decimal 2’s comp Sign-Mag 7 0111 0111 6 0110 0110 5 0101 0101 4 0100 0100 3 0011 0011 2 0010 0010 1 0001 0001 0 0000 0000 16 Pos 0
- 17. Signed Numbers-4 bit example Decimal 2’s comp Sign-Mag -8 1000 N/A -7 1001 1111 -6 1010 1110 -5 1011 1101 -4 1100 1100 -3 1101 1011 -2 1110 1010 -1 1111 1001 -0 0000 (= +0) 1000 17
- 18. Notes: “Humans” normally use sign-magnitude representation for signed numbers Eg: Positive numbers: +N or N Negative numbers: -N Computers generally use two’s- complement representation for signed numbers First bit still indicates positive or negative. If the number is negative, take 2’s complement to determine its magnitude Or, just add up the values of bits at their positions, remembering that the first bit is implicitly negative. 18
- 19. Examples Let N=4: two’s-complement What is the decimal equivalent of 01012 Since MSB is 0, number is positive 01012 = 4+1 = +510 What is the decimal equivalent of 11012 = Since MSB is one, number is negative Must calculate its 2’s complement 11012 = −(0010+1)= − 00112 or −310 19
- 20. Very Important!!! – Unless otherwise stated, assume two’s-complement numbers for all problems, quizzes, HW’s, etc. The first digit will not necessarily be explicitly underlined. 20
- 21. Arithmetic Subtraction Borrow Method This is the technique you learned in grade school For binary numbers, we have 21 0 - 0 = 0 1 - 0 = 1 1 - 1 = 0 0 - 1 = 1 with a “borrow” 1
- 22. Binary Subtraction Note: A – (+B) = A + (-B) A – (-B) = A + (-(-B))= A + (+B) In other words, we can “subtract” B from A by “adding” –B to A. However, -B is just the 2’s complement of B, so to perform subtraction, we 1. Calculate the 2’s complement of B 2. Add A + (-B) 22
- 23. Binary Subtraction - Example Let n=4, A=01002 (410), and B=00102 (210) Let’s find A+B, A-B and B-A 23 0 1 0 0 + 0 0 1 0 (4)10 (2)10 0 11 0 6 A+B
- 24. Binary Subtraction - Example 24 0 1 0 0 - 0 0 1 0 (4)10 (2)10 10 0 1 0 2 A-B 0 1 0 0 + 1 1 1 0 (4)10 (-2)10 A+ (-B) “Throw this bit” away since n=4
- 25. Binary Subtraction - Example 25 0 0 1 0 - 0 1 0 0 (2)10 (4)10 1 1 1 0 -2 B-A 0 0 1 0 + 1 1 0 0 (2)10 (-4)10 B + (-A) 1 1 1 02 = - 0 0 1 02 = -210
- 27. Binary-to-Decimal Conversion The decimal value of any binary number can be found by adding the weights of all bits that are 1 and discarding the weights of all bits that are 0. Example Let’s convert the binary whole number 101101 to decimal. Weight: 2 2 2 2 2 2º Binary no: 1 0 1 1 0 1 101101= 2 + 2 + 2 + 2º = 32+8+4+1=45 27
- 28. Decimal-to-Binary Conversion One way to find the binary number that is equivalent to a given decimal number is to determine the set of binary weights whose sum is equal to the decimal number. For example decimal number 9, can be expressed as the sum of binary weights as follows: 9 = 8 + 1 or 9 = 2³ + 2º Placing 1s in the appropriate weight positions, 2³ and 2º, and 0s in the 2² and 2¹ positions determines the binary number for decimal 9. 2³ 2² 2¹ 2º 1 0 0 1 Binary number for nine 28
- 29. Binary-to-Hexadecimal Conversion Simply break the binary number into 4-bit groups, starting at the right-most bit and replace each 4- bit group with the equivalent hexadecimal symbol as in the following example. Convert the binary number to hexadecimal: 1100101001010111 Solution: 1100 1010 0101 0111 C A 5 7 = CA57 29
- 30. Hexadecimal-to- Decimal Conversion One way to find the decimal equivalent of a hexadecimal number is to first convert the hexadecimal number to binary and then convert from binary to decimal. Convert the hexadecimal number 1C to decimal: 1 C 0001 1100 = 2 + 2 ³ + 2² = 16 +8+4 = 28 30
- 31. Convert the decimal number 650 to hexadecimal by repeated division by 16. 650 = 40.625 0.625 x 16 = 10 = A (LSD) 16 40 = 2.5 0.5 x 16 = 8 = 8 16 2 = 0.125 0.125 x 16 = 2 = 2 (MSD) 16 The hexadecimal number is 28A 31
- 32. Octal System Like the hexadecimal system, the octal system provides a convenient way to express binary numbers and codes. However, it is used less frequently than hexadecimal in conjunction with computers and microprocessors to express binary quantities for input and output purposes. The octal system is composed of eight digits, which are: 0, 1, 2, 3, 4, 5, 6, 7 To count above 7, begin another column and start over: 10, 11, 12, 13, 14, 15, 16, 17, 20, 21 and so on. Counting in octal is similar to counting in decimal, except that the digits 8 and 9 are not used. 32
- 33. Octal-to-Decimal Conversion Since the octal number system has a base of eight, each successive digit position is an increasing power of eight, beginning in the right- most column with 8º. The evaluation Of an octal number in terms of its decimal equivalent is accomplished by multiplying each digit by its weight and summing the products. Let’s convert octal number 2374 in decimal number. Weight 8³ 8² 8 8º Octal number 2 3 7 4 2374 = (2 x 8³) + (3 x 8²) + (7 x 8) + (4 x 8º)=1276 33
- 34. Decimal-to-Octal Conversion A method of converting a decimal number to an octal number is the repeated division-by-8 method, which is similar to the method used in the conversion of decimal numbers to binary or to hexadecimal. Let’s convert the decimal number 359 to octal. Each successive division by 8 yields a remainder that becomes a digit in the equivalent octal number. The first remainder generated is the least significant digit (LSD). 359 = 44.875 0.875 x 8 = 7 (LSD) 8 44 = 5.5 0.5 x 8 = 4 8 5 = 0.625 0.625 x 8 = 5 (MSD) 8 The number is 547. 34
- 35. Octal-to-Binary Conversion Because each octal digit can be represented by a 3-bit binary number, it is very easy to convert from octal to binary.. Octal/Binary Conversion Octal Digit 0 1 2 3 4 5 6 7 Binary 000 001 010 011 100 101 110 111 Let’s convert the octal numbers 25 and 140. 2 5 1 4 0 010 101 001 100 000 35