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EEL 3801

Learn about the binary nature of digital computers and how data is represented using 0s and 1s. Understand the concepts of bytes, words, and double words, as well as the conversion between binary and decimal numbers. Discover the use of hexadecimal numbers in representing computer memory and instructions.

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EEL 3801

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  1. EEL 3801 Part I Computing Basics EEL 3801C

  2. Data Representation Digital computers are binary in nature. They operate only on 0’s and 1’s. • Everything must be expressed in terms of 0’s & 1’s - Instructions, data, memory locs. • 1 = “on”  voltage at output of electronic device is high (saturated). • 0 = “off”  voltage at output of electronic device is zero. EEL 3801C

  3. Data Representation • The smallest element of information is a “bit” – 0 or 1. But by itself a bit does not convey much information. Therefore: • 8 bits in succession make a “byte”, the smallest addressable binary piece of information • 2 bytes (16 bits in succession) make a “word” EEL 3801C

  4. Data Representation • 2 words (32 bits in succession) make a “double word”. • We can easily understand base 10 numbers. So we need to learn how to convert between binary and decimal numbers. • Decimal numbers are not useful to the computer: a compromise – base 16 numbers (hexadecimal) and base 8 nos., or “octal”. EEL 3801C

  5. Binary Numbers • Each position in a binary number, starting from the right and going left, stands for the power of the number 2 (the base). • The rightmost position represents 20, which = 1. • The second position from the right represents 21= 2. • The third position from the right represents 22 = 4. • The fourth position from the right represents 23 = 8. EEL 3801C

  6. Binary Numbers • The value of an individual binary bit is multiplied by the corresponding base and power of 2, and all the resulting values for all the digits are added together. For ex. 0 0 1 0 1 0 0 1 = 0*27 + 0*26 + 1*25 + 0*24 + 1*23 + 0*22 + 0*21 + 1*20 = 0 + 0 + 32 + 0 + 8 + 0 + 0 +1 = 41 EEL 3801C

  7. Binary Numbers • Conversion from decimal to binary - Two methods: • Division by power of 2: • Find the value of the largest power of 2 that fits into decimal number. • Set 1 for position bit corresponding to that power. • Subtract that value from the decimal number. • Go to the first step, and re-do procedure until remainder is 0 EEL 3801C

  8. Binary Numbers • Example: • decimal number = 146 • Largest value of power of 2 that fits into 146 is 128, or 27 • Set 8th bit (power of 7) to 1 • Subtract 128 from 146 = 18 • Largest value that fits into 18 is 16 or 24 • Set 5th bit (power of 4) to 1 • 18 - 16 = 2 EEL 3801C

  9. Binary Numbers • Example (continued): • Largest power of 2 that fits into 2 is 2, or 21 • Set second bit (power of 1) to1 • Remainder is now 0 • Set all other bits to zero • The binary equivalent = 1 0 0 1 0 0 1 0 EEL 3801C

  10. Binary Numbers • Second Method • Division by 2 • Integer divide the decimal number by 2. • Note the remainder (if even, 0; if odd, 1) • The remainder represents the bit • Integer divide the quotient again by 2 and note remainder. • Continue until quotient = 0 EEL 3801C

  11. Binary Numbers • Example • Decimal number = 146. • 146/2 = 73, remainder = 0 • 73/2 = 36, remainder = 1 • 36/2 = 18, remainder =0 • 18/2 = 9, remainder = 0 • 9/2 = 4, remainder = 1 • 4/2 = 2, remainder = 0 • 2/2 = 1, remainder = 0 EEL 3801C

  12. Binary Numbers • 1/2 = 0, remainder = 1 • 0/2 = 0, remainder = 0 • Starting from the last one, the binary number is now the string of remainders: 0 1 0 0 1 0 0 10 Since the 0 to the left does not count, we can lop it off 1 0 0 1 0 0 1 0 EEL 3801C

  13. Hexadecimal Numbers • Large binary numbers are cumbersome to read. • ==> hexadecimal numbers are used to represent computer memory and instructions. • Hexadecimal numbers range from 0 to 15 (total of sixteen). • Octal numbers range from 0 to 7 (total of 8) EEL 3801C

  14. Hexadecimal Numbers • The letters of the alphabet are used to the numbers represent 10 through 15. • where A=10, B=11, C=12, D=13, E=14, and F=15 • But why use hexadecimal numbers? • 4 binary digits (half a byte) can have a maximum value of 15 (1111), and a minimum value of 0 (0000). EEL 3801C

  15. Hexadecimal Numbers • If we can break up a byte into halves, the upper and lowerhalves, each half would have 4 bits. • A single hexadecimal digit between 0 and F could more concisely represent the binary number represented by those half-bytes. • A byte could then be represented by two hexadecimal digits, rather than 8 bits EEL 3801C

  16. Hexadecimal Numbers • The advantage becomes more evident for larger binary numbers: • 00010110 00000111 10010100 11101010 • 1 6 0 7 9 4 D A •  160794DAh EEL 3801C

  17. Numbers • A radix is placed after the number to indicate the base of the number. • These are always in lower case. • If binary, the radix is a “b”; • if hexadecimal, “h”, • if octal, “o” or “q”. • Decimal is the default, so if it has no indication, it is assumed to be decimal. A “d” can also be used. EEL 3801C

  18. Hexadecimal to Decimal Conversion • Similarly to binary-to-decimal conversion, each digit (position from right to left) of the hex number represents a power of the base (16), starting with power of 0. • 2 F 5 B  2*163 + 15*162 + 5*161 + 11*160 = 2*4096 + 15*256 + 5*16 + 11*1 • = 8192 + 3840 + 80 + 11 = 12,123 EEL 3801C

  19. Binary Conversion into Hexadecimal • Binary to hex is somewhat different, because we in reality, take each 4 bits starting from the right, and convert it to a decimal number. • We then take the hexadecimal equivalent of the decimal number (i.e., 10 = A, 11 = B, etc.) and assign it to each 4 bit sequence. • Each digit in a hex number = “hexadized” decimal equivalent of 4 binary bits. EEL 3801C

  20. Hexadecimal Conversion into Binary • This conversion is also rather simple. • Each hex digit represents 4 bits. The corresponding 4-bit binary sequence replaces the hex digit. For example: • 26AF  0010 0110 1010 1111 EEL 3801C

  21. Signed and Unsigned Integers • Integers are typically represented by one byte (8 bits) or by one word (16 bits). • There exist two types of binary integers: signed and unsigned EEL 3801C

  22. Unsigned Integers • Unsigned integers are easy – they use all 8 or 16 bits in the byte or word to represent the number. • If a byte, the total range is 0 to 255 (00000000 to 11111111). • If a word, the total range is 0 to 65,535 (0000000000000000 to 1111111111111111). EEL 3801C

  23. Signed Integers • Are slightly more complicated, as they can only use 7 or 15 of the bits to represent the number. The highest bit is used to indicate the sign. • A high bit of 0  positive number • A high bit of 1  negative number - counter intuitive, but more efficient. EEL 3801C

  24. Signed numbers (cont.) • The range of a signed integer in a byte is therefore, -128 to127, remembering that the high bit does not count. • The range of a signed integer in a word is –32,768 to 32,767. EEL 3801C

  25. The One’s Complement • The one’s complement of a binary number is when all digits are reversed in value. • For example, 0 0 0 1 1 0 1 1 has a one’s complement of 1 1 1 0 0 1 0 0 EEL 3801C

  26. Storage of Numbers • Unsigned numbers and positive signed numbers are stored as described above. • Negatively signed numbers, however, are stored in a format called the “Two’s complement” which allows it to be added to another number as if it was positive. EEL 3801C

  27. Storage of Numbers (cont.) • The Two’s Complement of a number is obtained by adding 1 to the lowest bit of the one’s complement of the number. • The Two’s Complement is perfectly reversible • TC (TC (number)) = number. EEL 3801C

  28. Storage of Numbers (cont.) • Therefore, if the high bit is set (to 1), the number is a negatively signed integer. • But, its decimal value can only be obtained by taking the two’s complement, and then converting to decimal. • If the high bit is not set (= 0), then the number can be directly converted into decimal. EEL 3801C

  29. Example • 0 0 0 0 1 0 1 0 is a positive number, as the high bit is 0. • 0 0 0 0 1 0 1 0 can be easily converted to 10 decimal in a straightforward fashion. • 0 0 0 0 1 0 1 0 = 10 decimal EEL 3801C

  30. Example (cont.) • 1 0 0 0 1 0 1 0 is a negative number because of the high bit being set. • 1 0 0 0 1 0 1 0, however, is not –10, as we first have to determine its two’s complement. EEL 3801C

  31. Example (cont.) • One’s Complement of (1 0 0 0 1 0 1 0)  (0 1 1 1 0 1 0 1), • add 1  (0 1 1 1 0 1 1 0) •  64 + 32 + 16 + 4 + 2 = -118 EEL 3801C

  32. Character Representation – ASCII • How do we represent non-numeric characters as well as the symbols for the decimal digits themselves if we want to get an alphanumeric combination? • Typically, characters are represented using only one byte minus the high bit (7-bit code). EEL 3801C

  33. Character Representation – ASCII (cont.) • Bits 00h to 7Fh represent the possible values. The ASCII table maps the binary number designated to be a character with a specific character. The back inside cover of the textbook contains that mapping. • If the eighth bit is used (as is done in the IBM PC to extend the mapping to Greek and graphics symbols), then the hex numbers used are 80h to FFh. EEL 3801C

  34. Character Representation – ASCII (cont.) • The programmer has to keep track of what a binary number in a program stands for. • It is not inherent in the hardware or the operating system. • High level languages do this by forcing you to declare a variable as being of a certain type. • Different data types have different lengths EEL 3801C

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