Data Storage and manipulation

Data Storage

• Computers store and manipulate data in the form of electronic pulses (high and Low voltages).

• This digitised data is represented numerically using the binary number system (base 2) which uses only 1 and 0.

Converting decimal numbers to binary numbers

• Divide the number by 2 noting the answer and the remainder.

• Repeat the division of the answer by 2, again noting the answer and the remainder.

• When the final answer is zero, write the remainders from the final remainder to the first. This is the binary value.

Convert 19 from base 10 to binary

Converting binary numbers todecimal numbers

• Starting from the right, each number is twice the one before it.

• Write the digits of the binary number in the columns of the table, starting from the right.

• Add up the decimal values of the columns where there is a 1 to get the decimal equivalent of your number.

Example

• You can add binary numbers using four simple rules:

• 0+0=0• 0+1=1• 1+0=1• 1 + 1 = 10 (10 is the binary value of 2: you say ‘one

zero’, not ‘ten’.)• (and 1 + I + 1 = ii: you say ‘one one’, not ‘eleven’.)• Example 3• Add 1101 + 1001• 1101 +1001 10110

Binary Addition

• You can add binary numbers using four simple rules:

Representing positive and negative numbers

• There are three systems used to represent positive and negative numbers:

• Binary Coded Decimal

• Sign and magnitude

• Two’s complement.

Binary Coded Decimal (BCD)

A four-bit code is also assigned for the + and — signs:+ = 1110— = 1111

Binary Coded Decimal (BCD)

• A four-bit code is also assigned for the + and — signs:

• + = 1110

• — = 1111

Example 4

• Convert —910 to BCD.• Write down the four-bit binary code for the minus

sign and each decimal digit. The bits are written all together.

Sign and Magnitude System • The sign and magnitude system works as

follows.• 1 Write a binary 1 for a minus sign or a 0

for a plus sign. This will be the leftmost bit in the answer

• 2 Ignore the sign of the given number and convert to binary as shown in Example 1.

• 3 Write the binary value, adding any zeros on the left if a specific number of bits is required. Remember to include the sign bit on the left as one of the required bits in the answer.

Example 5

• Convert -14 to 8-bit sign and magnitude. The number is negative, so the sign bit is 1.

• 14 converted into binary = 1110

• Therefore the answer for -14 in 8 bits is 10001110 in 8-bit sign magnitude

• 1 0001110

• Sign Value=14

Two’s complement system

• Complements are commonly used to represent negative numbers.

• You can think of a number as the ‘mirror image’ of its complement written to the same base.

• In the binary system, the complement of 1 is 0, and vice versa.

Two’s complement

• Two’s complement is very important when you want to subtract one binary number from another.

• The logic circuits of the ALU can readily add two numbers, but not subtract them.

• Adding one binary number to the two’s complement of another binary number is equivalent to subtracting the second one from the first one.

Two’s complement

• . Therefore two’s complement not only allows the computer to store both positive and negative integers but also allows it to do both addition and subtraction operations.

• In this system it is most important to note the number of bits required for the answer. The two’s complement system works like this.

1 Ignore the sign of the given value and convert the number to binary.

2 Add any required zeros on the left of the binary value so that it has the required number of bits.

3 If the number is positive then this is the final answer. If it is negative than continue to the next step.

4 Invert the bits of the binary number by changing every ito 0 and every 0 to 1.

5 Using binary addition, add 1 to this number. This gives the final answer.

Example 6Represent -14 in 8-bit two’s complement.First convert 14 to binary:

14 = 1110 in binarywhich is = 00001110 in 8 bitsInverting each bit gives= 11110001 Finally add 1:= 11110001

+ 1 11110010Therefore -14 is 11110010 in 8-bit two’s

complement.

Parity

• This is an error checking system which makes sure that when data is sent from one computer to another no errors have crept into the code.

• An extra ‘parity’ bit is added to the end of each binary code. If the system used is odd parity, the extra bit is chosen to ensure there is an odd number of 1-bits in the pattern.

Parity

• Even parity adds a bit to each binary code to ensure there is an even number of 1-bits in the pattern.

• If one of the digits in the code is wrong when the information is received, then the parity bit will be wrong.

Exainple

• 1101111001 for even parity the parity bit shown is 1 (six 1s)

• 1101111000 for odd parity; the parity bit shown is 0 (five 1s)

Parity

Representing alphanumeric characters

• When data in the form of numbers, letters and symbols is input to a computer, for example using a keyboard, it must be represented inside the computer by binary codes.

• A different code is used for each symbol.

Code System• There are two coding systems which

provide each character with its own unique binary character code for use on the computer:

• ASCII — American Standard Code for Information Interchange

• EBCDIC — Extended Binary Code Decimal Interchange Code

ASCII Code• ASCII is used on most microcomputers

and represents each symbol as a 7- or 8-bit binary code. These codes are sequential for the alphabet and for numbers.

• That is, if’a’ is represented as 1000001 then ‘b’ is represented as 1000010, ‘c’ is represented as 1000011 and so on.

• Adding 1 to each pattern gives the code for the next letter.

• EBCDIC is a similar coding system used mainly on IBM mainframe systems.

Example 8

If the ASCII code for ‘d’ is 1000100, find the ASCII code for ‘i’.

1 Convert the binary code to decimal.(1000100 = 64 + 4 = 68)

2 Determine how far the required letter is from the letter given. (‘i’ is 5 letters after ‘d’)

3 Add this number to the decimal value. (68 + 5 = 73)

4 Convert this final decimal value to binary.73 = 1001001 in binaryTherefore the ASCII code for ‘i’ is 1001001.

Computer Data• Computers represent their data as digitised

electronic pulses called discrete data. • That is, it can only take one of a finite set of

values such as 0 or 1.• This is different from continuous data, also

called analog data, which can take all in-between values: examples are measurements such as height or weight.

• So the waist sizes of students in a class form continuous data, whereas the number of students in the class represents discrete data.

1 Convert the following to binary:

a 63 b 25 c 54 d 81 e 12

2 Convert the following to decimal:

a 10110111 b 10101010 c 10001111

3 Add the following number:

a 0101 and 001 b 0111 and 0010

c 0100 and 0011

4 Name the three methods by which positive and negative numbers may be represented.

5 Convert the following numbers to BCD:

a-456 b 512 c 987 d 600

6 Convert the following to 8-bit sign and magnitude:a-21 b-55 a81 a32

• 7 Represent in 8-bit two’s complement the following:

• A-21 b-55 a81 a32

• 8 If the ASCII code for ‘d’ is 1000100, find the ASCII code for ‘a’