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LOGODKT 122/3
DIGITAL SYSTEM 1
WEEK #2 NUMBER SYSTEMS, OPERATION & CODES
(PART 1)
Numbers & Codes
Numbering Systems Decimal numbering system (Base 10) Binary numbering system (Base 2) Hexadecimal numbering system (Base 16) Octal numbering system (Base 8)
Number Conversion Binary Arithmetic 1’s and 2’s Complements of Binary
Numbers
Numbers & Codes (cont..)
Signed Numbers Arithmetic Operations with Signed Numbers
Other Number Codes Binary-Coded-Decimal (BCD) ASCII codes Gray codes
Digital Codes & Parity
Numbering Systems
0 ~ 9
0 ~ 1
0 ~ 7
0 ~ F
Decimal(base 10)
Binary(base 2)
Octal(base 8)
Hexadecimal(base 16)
Num. Systems (Characteristics)
The digits are consecutive (berturutan).
The number of digits is equal to the size of the base.
Zero is always the first digit.
When 1 is added to the largest digit, a sum of zero and a carry of one results.
Numeric values determined by the implicit positional values of the digits.
00000000000000010000001000000011000001000000010100000110000001110000100000001001000010100000101100001100000011010000111000001111
000001002003004005006007010011012013014015016017
0123456789ABCDEF
0123456789
101112131415
BinaryOctalHexDec
Numbering Systems (Cont.)
Numbering System (Decimal)
Also called the Base 10 system Have 10 digits : 0 9 The position for each digit in the decimal
number indicates the magnitude of the quantity represented and can be assigned a weight
Numbering System (Decimal)
The weight for whole numbers are positive powers of ten that increase from right to left
105 104 103 102 101 100
For fractional numbers, the weights are negative powers of ten that decrease from left to right
102 101 100 . 10-1 10-2 10-3….
Decimal point
Numbering System (Binary)
Also called the Base 2 system The binary number system is used to
model the series of electrical signals computers use to represent information
0 represents the no voltage or an off state
1 represents the presence of voltage or an on state
Just think for a while..
“There are 10 kinds of mathematicians.
Those who can think binarily and those who can't...”
So, what is the meaning of this?
YOU FALL IN WHICH
CATEGORY?
Significant Digits
Binary: 11101101
Hexadecimal: 1D63A72A
Most significant digit (MSB) Least significant digit (LSB)
Question: How many bits does the numbers represent?
Number Conversion
Any Radix (base) to Decimal Conversion
Binary to Decimal Conversion
Decimal value of any binary number can be found by adding weights of all bits that are 1 and discarding the weights of all bits that are 0
Solve this..
(a) 10102
(b) 101112
Answer : ?
Answer : ?
(c) 10101102
Answer : ?
Convert the following binary numbers to decimal
Decimal to Binary Conversion
For whole number conversion, use the repeated division-by-2 process and record the remainder
For fractional number conversion, use repeated multiplication by 2 until the fractional product is 0 or until the desired number of decimal places is reached
2 5 = 12 + 1 2
1 2 = 6 + 0 2
6 = 3 + 0 2
3 = 1 + 1 2
1 = 0 + 1 2 MSB LSB 2510 = 1 1 0 0 1 2
Remainder
Decimal to Binary Conversion
Whole number
Decimal to Binary Conversion
Carry . 0 1 0 10.3125 x 2 = 0.625 0 0.625 x 2 = 1.25 1
0.25 x 2 = 0.50 0
0.5 x 2 = 1.00 1
The Answer: 1 1 0 0 1.0 1 0 1
MSB LSB
Fractional number
Solve this..
(a) 3910
(b) 5810
Answer : ?
Answer : ?
(c) 0.37510
Answer : ?
Convert the following decimal numbers to binary
Binary Arithmetics
Binary Addition Binary Subtraction Binary Multiplication Binary Division
Binary Addition
Four basic rules for adding binary digits (bits) are:
0 + 0 = 0 (Sum of 0 with a carry of 0)
0 + 1 = 1 (Sum of 1 with a carry of 0)
1 + 0 = 1 (Sum of 1 with a carry of 0)
1 + 1 = 1 0 (Sum of 0 with a carry of 1)
Examples
Perform the following binary additions:
(a) 100 + 10
1 0 0
1 0+
1 1 0 (Answer)
(b) 111 + 11
1 1 1
1 1+
1 0 1 0 (Answer)
Solve this..
(a) 11 + 01
(b) 111 + 110
Answer : ?
Answer : ?
(c) 1001 + 101:Answer : ?
Perform the following binary additions:
Binary Arithmetics
Binary Addition Binary Subtraction Binary Multiplication Binary Division
Binary Subtraction
Four basic rules for subtracting binary digits (bits) are:
0 - 0 = 0
1 - 1 = 0
1 - 0 = 1
1 0 - 1 = 1 (0 – 1 with a borrow of 1)
Examples
Perform the following binary subtractions:
(a) 101 – 011
1 0 1
0 1 1-
0 1 0 (Answer)
(b) 110 – 101
1 1 0
1 0 1-
0 0 1 (Answer)
Solve this..
(a) 101 – 100
(b) 1110 - 11
Answer : ?
Answer : ?
(c) 1100 - 1001:Answer : ?
Perform the following binary subtractions
Binary Arithmetics
Binary Addition Binary Subtraction Binary Multiplication Binary Division
Binary Multiplication
Four basic rules for muliplying binary digits (bits) are:
0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1
Examples
Multiply 111 and 101:
1 1 1
1 0 1x
1 1 10 0 0
1 1 1
1 0 0 0 1 1 (Answer)
Solve this..
(b) 110 x 111:
(c) 1101 x 1010:
Answer : ?
Answer : ?
(a) 11 x 11:Answer : ?
Binary Arithmetics
Binary Addition Binary Subtraction Binary Multiplication Binary Division
Binary Division
Division in binary follows the same procedure as division in decimal
Example: Perform the binary divisions of 110 11
1 1 01 1
1 0
1 1
0 0 0
(Answer)
Solve this..
(a) 100 10Answer : ?
(b) 1100 100:
Answer : ?
Divide the binary numbers as indicated:
Changing all the 1s to 0s and all the 0s to 1s
Example:
1 1 0 1 0 0 1 0 1 Binary number
0 0 1 0 1 1 0 1 0 1’s complement
1’s Complement
2’s Complements
Find the 1’s complements of the numbers
1 1 1 0 1 0 1 0 1 Binary number 0 0 0 1 0 1 0 1 0 1’s complement
Step 1: Step 1:
Step 2: Step 2: Add ‘1’ to the 1’s complements
0 0 0 1 0 1 0 1 0 1’s complement + 1 Add 1 0 0 0 1 0 1 0 1 1 2’s complement
Solve this..
(a) 00010110
Answer : ?
(b) 10010001
Answer : ?
Determine the 2’s complement of each binary number:
Left most is the sign bit 0 is for positive, 1 is for negative
Sign & magnitude
00011001 = +25
sign bit magnitude bits
Signed Numbers
Sign-Magnitude Numbers
Sign bit
0 = positive
1 = negative
31 bits for magnitude
0110010.. …00101110010101
The left-most is the sign bit and the remaining bits are the magnitude bits
1’s complement
The negative number is the 1’s complement of the corresponding positive number
Example
+25 is 00011001 So, -25 is 11100110
2’s complement
The positive number – same as sign magnitude and 1’s complement
The negative number is the 2’s complement of the corresponding positive number.
Example
+25 is 00011001 So, -25 is 11100111
Signed Numbers (Cont.)
Solve this..
Express +19 and -19 (as an 8-bit number) ini. sign magnitude
ii. 1’s complement
iii. 2’s complement
Answer : ?
Answer : ?
Answer : ?
END