Introduction to Computer Science Lecture 4.1: Nibble, Byte, Word, and Memory Units

1. Computer Math CPS120 Introduction to Computer Science Lecture 4

2. 1 nibble 1 byte 1 word 1 long word 1 quad word 1 octa-word 4 consecutive bits 8 consecutive bits 2 consecutive bytes 4 consecutive bytes 8 consecutive bytes 16 consecutive bytes Memory Units

3. 1 Kilobyte 1 Megabyte 1 Gigabyte 1 Terabyte 1 Petabyte 1 Exabyte 1024 bytes ~106 bytes ~109 bytes ~1012 bytes ~1015 bytes ~1018 bytes Larger Units of Memory 32 Mb = 32*103 Kb = 32 * 103 *1024 bytes = 32,768,000 bytes

4. Representing Data • The computer knows the type of data stored in a particular location from the context in which the data are being used; • i.e. individual bytes, a word, a longword, etc • 01100011 01100101 01000100 01000000 • Bytes: 99(10, 101 (10, 68 (10, 64(10 • Two byte words: 25,445 (10 and17,472 (10 • Longword: 1,667,580,992 (10

5. Alphanumeric Codes • American Standard Code for Information Interchange (ASCII) • 7-bit code • Since the unit of storage is a bit, all ASCII codes are represented by 8 bits, with a zero in the most significant digit • H e l l o W o r l d • 48 65 6C 6C 6F 20 57 6F 72 6C 64 • Extended Binary Coded Decimal Interchange Code (EBCDIC)

6. Number Systems • We use the DECIMAL (10 system • Computers use BINARY (2 or some shorthand for it like OCTAL (8 or HEXADECIMAL (16

7. Codes • Given any positive integer base (RADIX) N, there are N different individual symbols that can be used to write numbers in the system. The value of these symbols range from 0 to N-1 • All systems we use in computing are positional systems • 495 = 400 + 90 +5

8. Conversions

9. Decimal Equivalents • Assuming the bits are unsigned, the decimal value represented by the bits of a byte can be calculated as follows: • Number the bits beginning on the right using superscripts beginning with0 and increasing as you move left • Note: 20, by definition is 1 • Use each superscript as an exponent of a power of2 • Multiply the value of each bit by its corresponding power of 2 • Add the products obtained

10. Horner’s Method • Another procedure to calculate the decimal equivalent of a binary number • Note: This method works with any base • Horner’s Method: • Step 1: Start with the first digit on the left • Step 2: Multiply it by the base • Step 3: Add the next digit • Step 4: Multiply the sumby the base • Step 5: Continue the process until you add the last digit

11. Binary to Hex • Step 1: Form four-bit groups beginning from the rightmost bit of the binary number • If the last group (at the leftmost position) has less than four bits, add extra zeros to the left of the group to make it a four-bit group • 0110011110101010100111 becomes • 0001 1001 1110 1010 1010 0111 • Step 2: Replace each four-bit group by its hexadecimal equivalent • 19EAA7(16

12. Converting Decimal to Other Bases • Step 1: Divide the number by the base you are converting to (r) • Step 2: Successively divide the quotients by (r) until a zero quotient is obtained • Step 3: The decimal equivalent is obtained by writing the remainders of the successive division in the opposite order in which they were obtained • Know as modulus arithmetic • Step 4: Verify the result by multiplying it out

13. Addition & Subtraction Terms • A + B • A is the augend • B is the addend • C – D • C is the minuend • D is the subtrahend

14. Addition Rules – All Bases Addition • Step 1: Add a column of numbers • Step 2: Determine if there is asingle symbol for the result • Step 3: If so, write it and go to the next column. If not, write the accompanying number and carry the appropriate value to the next column

15. Subtraction Rules – All Bases • Step1: Start with the rightmost column, if the column of the minuend is greater than that of the subtrahend, do the subtraction, if not… • Step 2: Borrow one unit from the digit to the left of the once being processed • The borrowed unit is equal to “borrowing” the radix • Step 4: Decrease the column form which you borrowed by one • Step 3: Subtract the subtrahend from the minuend and go to the next column

16. Addition of Binary Numbers • Rules for adding or subtracting very similar to the ones in decimal system • Limited to only two digits • 0 + 0 = 0 • 0 + 1 = 1 • 1 + 0 = 1 • 1 + 1 = 0 carry 1

17. Addition & Subtraction of Hex • Due to the propensity for errors in binary, it is preferable to carry out arithmetic in hexadecimal and convert back to binary • If we need to borrow in hex, we borrow 16 • It is convenient to think “in decimal” and then translate the results back to hex

18. Representing Signed Numbers • Remember, all numeric data is represented inside the computer as 1s and 0s • Arithmetic operations, particularly subtraction raise the possibility that the result might be negative • Any numerical convention needs to differentiate two basic elements of any given number, its sign and its magnitude • Conventions • Sign-magnitude • Two’s complement • One’s complement

19. Representing Negatives • It is necessary to choose one of the bits of the “basic unit” as a sign bit • Usually the leftmost bit • By convention, 0 is positive and 1 is negative • Positive values have the same representation in all conventions • However, in order to interpret the content of any memory location correctly, it necessary to know the convention being used used for negative numbers

20. Comparing the Conventions

21. Sign-Magnitude • For a basic unit of N bits, the leftmost bit is used exclusively to represent the sign • The remaining (N-1) bits are used for the magnitude • The range of number represented in this convention is –2 N+1 to +2 N-1 -1

22. Sign-magnitude Operations • Addition of two numbers in sign-magnitude is carried out using the usual conventions of binary arithmetic • If both numbers are the same sign, we add their magnitude and copy the same sign • If different signs, determine which number has the larger magnitude and subtract the other from it. The sign of the result is the sign of the operand with the larger magnitude • If the result is outside the bounds of –2 n+1 to +2 n-1 –1, an overflow results

23. One’s Complement Convention • Devised to make the addition of two numbers with different signs the same as two numbers with the same sign • Positive numbers are represented in the usual way • For negatives • STEP 1: Start with the binary representation of the absolute value • STEP 2: Complement all of its bits

24. One's Complement Operations • Treat the sign bit as any other bit • For addition, carry out of the leftmost bit is added to the rightmost bit – end-around carry

25. Two’s Complement Convention • A positive number is represented using a procedure similar to sign-magnitude • To express a negative number • Express the absolute value of the number in binary • Change all the zeros to ones and all the ones to zeros (called “complementing the bits”) • Add one to the number obtained in Step 2 • The range of negative numbers is one larger than the range of positive numbers • Given a negative number, to find its positive counterpart, use steps 2 & 3 above

26. Two’s Complement Operations • Addition: • Treat the numbers as unsigned integers • The sign bit is treated as any other number • Ignore any carry on the leftmost position • Subtraction • Treat the numbers as unsigned integers • If a "borrow" is necessary in the leftmost place, borrow as if there were another “invisible” one-bit to the left of the minuend

27. Overflows in Two’s Complement • The range of values in two’s-complement is –2 n+1 to +2 n-1 –1 • Results outside this band are overflows • In all overflow conditions, the sign of the result of the operation is different than that of the operands • If the operands are positive, the result is negative • If the operands are negative, the result is positive

28. Transmission Errors • When binary data is transmitted, there is a possibility of an error in transmission due to equipment failure or "noise" • Bits change from0 to 1 or vice-versa

29. Categorizing Coding Schemes • The number of bits that have to change within a byte before it becomes invalid characterizes the code • Single-error-detecting code • To detect single errors have occurred we use an added parity check bit – makes each byte either even or odd • Two-error-detecting code

30. Bytes Transmitted 01100011 11100001 01110100 11110011 00000101Parity Block B I T Bytes Received 01100011 11100001 11111100 11110011 00000101Parity Block B I T Error Detection: Even Parity