EEL 3801 Part I Computing Basics EEL 3801

EEL 3801 Part I Computing Basics EEL 3801 C

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 3801 C

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 3801 C

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 3801 C

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 3801 C

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. 00101001 = 0*27 + 0*26 + 1*25 + 0*24 + 1*23 + 0*22 + 0*21 + 1*20 = 0 + 32 + 0 + 8 + 0 +1 = 41 EEL 3801 C

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 3801 C

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

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

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 3801 C

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 3801 C

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: 010010010 Since the 0 to the left does not count, we can lop it off 10010010 EEL 3801 C

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 3801 C

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 3801 C

Hexadecimal Numbers • If we can break up a byte into halves, the upper and lower halves, 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 3801 C

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

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 3801 C

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 3801 C

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 3801 C

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: • 26 AF 0010 0110 1010 1111 EEL 3801 C

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 3801 C

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 (0000 to 1111). • If a word, the total range is 0 to 65, 535 (00000000 to 11111111). EEL 3801 C

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 3801 C

Signed numbers (cont. ) • The range of a signed integer in a byte is therefore, -128 to 127, remembering that the high bit does not count. • The range of a signed integer in a word is – 32, 768 to 32, 767. EEL 3801 C

The One’s Complement • The one’s complement of a binary number is when all digits are reversed in value. • For example, 00011011 has a one’s complement of 11100100 EEL 3801 C

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 3801 C

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 (number)) = number. EEL 3801 C

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 3801 C

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

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

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

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 3801 C

Character Representation – ASCII (cont. ) • Bits 00 h to 7 Fh 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 80 h to FFh. EEL 3801 C

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 3801 C