Introduction to Digital System Design
Z. Navabi
Topic 1
Number Systems
Zain Navabi
Slides prepared by Katayoon Basharkhah
Outline
2-1 Why Binary?
2-2 Decimal and Binary
2-3 Hex and Octal
2-4 Coding System
2-5 Binary arithmetic
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Why Binary?
• Computers don’t understand words or numbers the way humans do.
• In the lowest level, computers work with 0 and 1.
• Binary number is base 2.
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Decimal and Binary
Decimal Numbers
Decimal
Point
Binary Point
4𝟒
𝟏𝟎
2
3
𝟏𝟎
2
𝟑
2
𝟐
𝟏𝟎
2
25.13=
1
𝟏
𝟏𝟎
2
𝟎
𝟏𝟎
2
2
5
-1
0
𝟏𝟎−𝟏
.
-2
2
𝟏𝟎−𝟐
1
3
2
-3
𝟏𝟎−𝟑
2
-4
2𝟏𝟎
−𝟒
2*101 +5*100 + 1*10-1+3*10-2
Integer
Fraction
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Decimal and Binary
Binary Numbers
Binary Point
4
3
2
1
0
2
2
2
2
2
1
1
0
0
1
-1
.
-2
-3
-4
2
2
2
2
0
0
1
0
(11001.1010)2 = 1*24 + 1*23 +0*22 + 0*21 +1*20 + 0*2-1+0*2-2+ 1*2-3+0*2-4 = (25.125)10
Integer
Fraction
Binary to decimal conversion
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Decimal and Binary
Decimal to Binary Conversion (Integer Part)
(29.61)10=(?)2
Multiple divisions and collecting the remainders, the first remainder is the nearest
digit to the binary point.
2
14
29
28
1
2
07
14
14
2
0
03
07
06
1
2
1
03
02
1
(29)D = (11101.)B
Binary point
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Decimal and Binary
Decimal to Binary Conversion (Fractional Part)
Use multiplication to find the fractional part.
0.61*2 = 1 . 22
0.22*2 = 0 . 44
0.44*2 = 0 . 88
(0.61)10=(.100111)2
(29.61)10=(11101.100111)2
0.88*2 = 1 . 96
0.96*2 = 1 . 92
0.92*2 = 1 . 84
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Octal and Hex
To do conversions from base 2 to base 8(16), each 3(4) bits in
base 2 must be converted into it’s equivalent in base 8(16).
(35.37)10 = (00 100 011.010 111 00)2
(35.37)10 = (43.27)octal
(35.37)10 = (0010 0011.0101 1100)2
(35.37)10 = (23.5C)Hex
Introduction to Digital System Design - Copyright Zainalabedin Navabi
0000
0
0001
1
0010
2
0011
3
0100
4
0101
5
0110
6
0111
7
1000
8
1001
9
1010
A
1011
B
1100
C
1101
D
1110
E
1111
F
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Coding System
1- Signed Numbers
• We define the left-most digit as the number’s sign digit.
• The definition constructs the Sign and Magnitude representation of numbers.
• In this representation all digits except the left-most construct the magnitude of
the number, while the left-most digit being 1 shows that the number is negative
or positive otherwise.
Sign
Magnitude
0
1001001
+73
1
1001001
-73
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Coding System
2- Binary Coded Decimal
• A system of writing numerals.
• Assigns a four-digit binary code to each digit 0 through 9 in a decimal (base-10) numeral.
0000
0
0001
1
0010
2
0011
3
0100
4
0101
5
0110
6
0111
7
1000
8
1001
9
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Coding System
3- Seven Segment Display Code
• A system for simple display of digits 0 to 9.
• Assigns a seven-digit binary code to each digit 0 through 9 (base-10) numeral.
a
b
c
d
e
f
g
0
0000 1
1
1
1
1
1
0
1
0001 0
1
1
0
0
0
0
2
0010 1
1
0
1
1
0
1
3
0011 1
1
1
1
0
0
1
4
0100 0
1
1
0
0
1
1
5
0101 1
0
1
1
0
1
1
6
0110 1
0
1
1
1
1
1
7
0111 1
1
1
0
0
0
0
8
1000 1
1
1
1
1
1
1
9
1001 1
1
1
1
0
1
1
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Coding System
3- Seven Segment Display Code
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Coding System
Decimal Binary
4- Gray Code
• Physical switches are not ideal: it is very unlikely that physical switches
will change states exactly in synchrony.
0111
1111
1101
1001
0100
• Solution: Two successive values differ in only one bit.
Introduction to Digital System Design - Copyright Zainalabedin Navabi
Gray
0
0000
0000
1
0001
0001
2
0010
0011
3
0011
0010
4
0100
0110
5
0101
0111
6
0110
0101
7
0111
0100
8
1000
1100
9
1001
1101
10
1010
1111
11
1011
1110
12
1100
1010
13
1101
1011
14
1110
1001
15
1111
1000
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Coding System
5- ASCII
• In some application storing text is the
most important of all. The ASCII
code is used for this mean.
• Digits 0 through 9, lowercase and
uppercase letters all have specific codes
in this system.
• Today, an extended ASCII table is also
used alongside the basic one which
contains the codes of foreigner letters.
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Coding System
5- ASCII
• 8-bit codes are used to represent the ASCII codes, where the leftmost digit
specifies which table a specific code belongs to. Codes starting with 1 are
from the extended ASCII table and those starting with 0 belongs to the
basic one.
• Special characters are also included in the ASCII code range. It’s also true
about the control characters.
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Binary arithmetic
1- Addition
• Adding numbers in base 2 can be considered the same as decimal
Addition.
• The main problems can occur when doing subtractions. But even
in this case the borrowing is just as it is in decimal numbers.
+2 +2 +2 +2
+1 +1
+1
101101
+ 011001
1000110
Addition
1 01101
+1 +1 +1 +1
- 0 01110
0 11111
Subtraction
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Binary arithmetic
2- Subtraction
• With another look at the arithmetic in our decimal number
system:
The Largest number with the same number
of digits plus 1 as our operand is added
and subtracted
52 - 37 = 52 - 37 + 100 - 100 = 52 - (37 + 99) + 1 - 100 =
52 + (62 + 1) - 100 = 52 + 63 - 100 = 115 - 100 = 15
Negative of 37
• We have actually found a new way to negate 37 and that is the
result of subtracting it from 99 and adding 1 to it.
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Binary arithmetic
2- Subtraction
• With another look at the arithmetic in our decimal number
system:
00110100 - 00100101 = 00110100 - 00100101 + 100000000 - 100000000 =
00110100 - 00100101 + 11111111 + 1 - 100000000 =
00110100 + 11011011 - 100000000 = 100001111 - 100000000 = 00001111
The negative of 00100101
• This representation for negative numbers is used in the 2’s
Complement System.
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Binary arithmetic
3- Two’s complement
• 2’s complementing a number is consisted of complementing it and then
adding 1.
• The same rule as the sign and magnitude representation for distinguishing
positive numbers from negative ones applies in the 2’s complement
system.
• In the 2’s complement system, the positive numbers have the same form
as they have in the sign and magnitude system.
• 2’s complementing a negative number is performed as starting from the
right-most digit until the first ‘1’ is reached and then complementing the
rest of the digits.
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Binary arithmetic
4- Two’s complement arithmetic
+1+1+1 +1 +1
00011011
+ 01001101
01101000
Addition
+1 +1
+1 +1
0 0 0 1 1 0 1 1 negating 0 0 0 1 1 0 1 1
- 01001101
+ 10110011
-(00110010)
11001110
Binary
Subtraction
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Binary arithmetic
5- Overflow
0 0 1 1 0 1 0 0
+
0 1 1 0 1 0 0 0
1 0 0 1 1 1 0 0
• A negative result has occurred from the addition of 2 positive numbers.
• The result could have been shown correctly in 9 bits, but our system has at most 8.
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Binary arithmetic
5- Overflow
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Binary arithmetic
5- Overflow
• A way for pointing out an overflow occurrence is when the sign bit of the result
differs from that of the operands (if the operand’s sign bits are the same).
• Solution: Sign-Extension Process
+
0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0
0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0
0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0
Sign-Extension
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Conclusion
We have covered:
•
•
•
•
•
Binary number representation.
Binary to decimal conversion and vs.
Coding systems like BCD, Gray code and ASCII.
Binary arithmetic.
Two’s complement system.
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