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COMPUTER
PLAY WORKORGANIZE THEIR LIVES
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DIGITAL CIRCUITS are the
brains of the technologicalworld
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PC have subdivided the circuit
board into four areas
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FOUR AREAS
1. BASIC LOGIC AREA. This consists of the
basic gates (glue chips).
2. MEMORY AREA. This consists of bothRAMs and ROMs,
3. MICROPROCESSOR. This is the heart of
the PC and contains several millions oftransistors and be able to execute over 100
millions instruction/second.
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4. PERIPHERAL ICs. These ICs
primarily involved withInput/Output (I/O)
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How this digital
circuits works in a
computer?
To understand the operation
and application digital
electronics mastered theMATHEMATICS AND
LOGIC DEVICES, then
BASIC GATES
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COMPUTERS are designed to handle numbers, not
letters
Use a more efficient code called theBINARY NUMBERS
To understand the basic operation ofcomputer it is a prerequisite to knowthe conversion of each number system
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Binary system- uses only two
distinctive numbers: LOGIC 1 ------------ON
LOGIC 0 -----------OFF
An action of a switch0 and 1 called BIT
Bit Quantities:
4 bits = One nibble or hexadecimal digit
8 bits = Two nibble or one byte
16 bits = Two bytes or one word
32 bits = Two words or one longword
(Motorolas Term) one Double word (Intels term)
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BINARY TO DECIMAL
CONVERSION
To find the equivalent decimal
numbers, we will use thepolynomials expansion:
N = +(B2R2) + (B1R
1) + (B0R0)
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CONVERSION OF BINARY
FRACTIONS TO DECIMALSN = (B-1R
-1) + (B-2R-2) +
MIXED NUMBERS
can be converted from binary to
decimal by working on the integersand fraction portion separately.
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EXAMPLES:1. 101112
2. 1101012
3. 1100010112
4. 11011.101112
5. 0.000100112
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DECIMAL TO BINARY
CONVERSION1. Obtain the N ( the decimal number to be
converted_
2. Determine if N is odd or even.3. (a) If N is odd, write 1 and subtract 1 from N.
Go to step 4. (b) If N is even, write 0. Go to step4.
4. Obtain a new value of N by dividing the N ofstep 3 by 2.
5. (a) If N >1, go back to step 2 and repeat theprocedure. (b) If N = 1 , write 1.
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NOTE: The number written is thebinary equivalent of the original
decimal number. The number
written first is the least
significant bit (LSB), and the
number written last is the mostsignificant bit (MSB)
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EXAMPLES:1. 1010
2. 2510
3. 25010
4. 7710
5. 8910
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CONVERTING DECIMAL
FRACTION TO BINARY
FRACTIONS1. Obtain N.
2. Double N.3. (a) If the new value of N is greater than 1,
write 1 as the next most significant bit,subtract 1 from N, and go back to step 2.(b) If the new value of N is less than 1,write 0 as the next most significant bit andgo back to step 2.
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MIXED NUMBERSMixed numbers can be converted from
decimal and fraction portions separately.
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EXAMPLES:1. 0.510
2. 0.2510
3. 3.72510
4. 0.69310
5. 0.35610
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TWO ALTERNATIVES
SYSTEMS USEDF
ORREPRESENTING BINARY:
1. OCTAL NUMBER SYSTEM-uses 0 to 7 and base 8
2. HEXADECIMAL NUMBERSYSTEM- uses 0 to 9, and Athrough F corresponding tonumbers 10 through 15 and base16
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BINARY TO OCTAL
CONVERSION1. Converting binary to octal is grouped into
three from the least significant bit (LSB)
to the most significant (MSB)Examples:
1. 111111112
2. 1000111023. 0000111124. 101010102
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OCTAL TO BINARY
CONVERSION Converting octal to binary just write the binary
equivalent of the numbers in octal form
EXAMPLES:
1. 778
2. 578
3. 6584. 468
5. 528
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DECIMAL TO OCTAL
CONVERSION1. Convert the decimal number to its binary
equivalent
2. Then grouped the binary equivalent into
three and convert each group separately to
octal equivalent
Examples:
1. 4410, 5410, 6610
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OCTAL TO DECIMAL
CONVERSION1. Convert the octal number to its binary equivalent
by assigning a 3-bit binary equivalent to each
octal digit2. Then convert the binary back to decimal using
the usual method.
EXAMPLES:
1. 7782. 5783. 6584. 468
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BINARY TO HEX
CONVERSION To convert binary to hex, grouped the
binary digits into four bits (from LSB to the
MSB) and get the equivalent in hex.
EXAMPLES:
1. 1010101111002
2. 1100000101111111012
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HEX TO BINARY
CONVERSION Get the binary equivalent of each digit in
the proper order.
EXAMPLES:
1. 1CB0916
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DECIMAL TO HEX
CONVERSION1. Convert the decimal number to binary
using the usual approach.
2. Then, grouped the binary digits into 4 bits
and take the hex equivalent of each group
separately.
EXAMPLES:
1. 76410
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HEX TO DECIMAL
CONVERSION1. Using the direct approach, the hex digit
are multiplied with the weighed bit
equivalent of the number (Polynomialexpansion)
2. Convert the hex number to binary by
assigning 4-bit equivalent for each hexdigit and then converting the binary to
decimal number.
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EXAMPLE 1. 2FC16
Since C = 12, F = 15
H2 + H1 + H0
= 162 x 2 + 161 x 15 + 160 x 12
= 512 + 240 + 12
= 764
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BCD Code The BCD is one popular form of coding used to
simplify the conversion of binary into its decimalequivalent.
This is generally used for coding binary in 7-segment displays.
Note: this code is weighed code, not a numbersystem.
Similar to the hexadecimal number system, thebinary bits are grouped into four to facilitateconversion; but unlike hex, the acceptable binarycodes are obviously for digits 0 to 9 only.
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BCD 1. 321 = 0011 0010 0001
2. 10 = 0001 0000
3. 11 = 0001 0001
4. 22 = 0010 0010
5. 100 = 0001 0000 0000
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BASIC BINARY ARITHMETIC ADDTION
Rules of Binary Addition:
0+0 = 00+1 = 1
1+0 = 1
1+1 = 0, and carry 1 to the next most significantbit
Examples: 1. 00011010 + 00001100
2] 11+11 , 3] 11 + 101 , 4] 1001 + 1101
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SUBTRACTION Rules of binary subtraction:
0 0 = 0
1 0 = 1
1 1 = 0
0 1 = 1, and borrow 1 from the next most significant
bit.
EXAMPLES:
A] 00100101 00010001
B] 00110011 00010110
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MULTIPLICATIONRules of binary multiplication:
0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1, and no carry or borrow bits
EXAMPLES:A] 00101001 * 00000110
B] 00010111 * 00000011
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DIVISIONBinary division is the repeated process of
subtraction, just as in decimal division.
EXAMPLES:
A] 00101010
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2s Complementing Numbers The 2s complement number system is used in all
modern computers to express numbers.
It is similar to the binary number system, but bothpositive and negative numbers can be represented.
MSB of a 2s complement number denotes:
0 means the number is positive
1 means the number is negativeNOTE: 2s complement notation, positive numbers are
represented as simple binary numbers with therestriction that the MSB is 0.
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To obtain the representation of a
negative number, use the following
algorithm:1. Represent the number as a positive binary
number.
2. Complement it. (Write 0s where there are
1s and 1s where there are 0s in the
positive number)
3. Add 1.
4. Ignore any carries out of the MSB.
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Example:1. Given 8-bit word, find the 2s complement
representation of:
a. 25 = 000110012
b. 25
c. 1
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Adding 2s Complement Numbers
C = A + B
If A and B are both positive, addition is
required.
But if one of the operands is negative and
the other is positive, a subtraction operation
must be performed.
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Example:1. Express the numbers 19 and 11 as 8-bit, 2s
complement numbers, and add them.
+ 19 = 00010011
Get the 2s complement of - 11
= 11110101Then add:
00001000
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Example:2. Add 11 and 19.
- 11 = 11110101
- 19 = 11101101
Then add:
11100010
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Subtraction of Binary Numbers The 2s complement of the subtrahend is taken
and added to the minuend.
Example:
Subtract 30 from 53. Use 8-bit numbers.
53 = 00110101 (Minuend)
2s complement of 30 = 11100010 (Subtrahend)
Then add:
00010111
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Subtract 30 from 19.- 19 = 11101101
- 30 = 11100010 = 2s complement =
00011110
Then add:
00001011