Electric Power Transmission System Engineering (Turan Gonen)
Assoc. Prof. Dr. Ahmet Turan ÖZCERİT. The base of numbers Conversion between number bases ...
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Transcript of Assoc. Prof. Dr. Ahmet Turan ÖZCERİT. The base of numbers Conversion between number bases ...
Introduction to Computer Engineering
WEEK-4
Number Systems
Assoc. Prof. Dr. Ahmet Turan ÖZCERİT
2
The base of numbers
Conversion between number bases
Arithmetic operations on different bases
Number Systems
You will learn:
The Base of Numbers
He/She can define the term of number bases
3
Computers use a number base other than base-10, namely binary
Each data used and stored in computer represented in binary numbers
Binary numbers are not easy to do arithmetic operations, so we use hex
and octal numbers for the sake of simplicity
All characters, images, audio and video samples are also presented in
binary numbers
The General Term of Number Systems
He/She can define the term of number bases
4
N: Digit value
d: Number digit
R: Number radix(base)
N=dnRn+ dn-1Rn-1 + …+ d2R2 + d1R1+d0R0 (for integers)
N=dnRn+dn-1Rn-1+…+d1R1+d0R0 , d1R-1+ d2R-2 +…+ dnR-n (for real numbers)
The number of digits in the R-based number system is R, the largest digit
is R-1, and the least digit is 0.
The largest number for n-digit is Rn-1 and the number of different value
for n-digit is Rn
Binary NumbersHe/She can make operation on binary numbers
5
The largest digit in binary system R-1 => 2-1 => 1
The least digit in binary system is 0
Each radix in binary number systems is called BIT (BInary DigiT).
The most significant bit (MSB)
The binary number general form:
B= dn2n + dn-12n-1 +…….+ d222 + d121 + d020 , d12-1 +d22-2 +dn2-n
Binary System Decimal System
00 0
01 1
10 2
11 3
Binary Numbers
He/She can make operation on binary numbers
6
What is binary value of following binary number: 1 0110 1101
B= 1*28 +0*27+1*26 + 1*25 + 0*24 + 1*23 + 1*22 + 0*21 + 1*20
B= 256 + 0 + 64 + 32 + 0 + 8 + 4 + 0 + 1
B= 36510
What is binary value of following binary number: 1 0110 1101, 1101
B= 1*28 +0*27+1*26+1*25+0*24+1*23+1*22+0*21+1*20, 1*2-1+1*2-2+0*2-
3+1*2-4
B= 256 + 0 + 64 + 32 + 0 + 8 + 4 + 0 + 1 + 0.5 +0.25 + 0+ 0.0625
B= 365,812510
Octal Numbers
He/She can make operation on octal numbers
7
Octal numbers are used to present binary numbers with 3-digit format
The digits used in Octal Systems: 0, 1, 2, 3, 4, 5, 6, 7
O= dn8n + dn-18n-1 +…….+ d282 + d181 + d080 , d18-1 +d28-2 +dn8-n
Binary Octal
000 0
001 1
010 2
011 3
100 4
101 5
110 6
111 7
Hexadecimal Numbers
He/She can make operation on hex numbers
8
Hex numbers are used to present binary numbers with 4-digit format
The digits used in Octal Systems: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
O= dn16n + dn-116n-1 +…….+ d2162 + d1161 + d0160 , d116-1 +d216-2 +dn16-n
Binary Hex
0000 0
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
Binary Hex
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F
Decimal to Binary Conversion
He/She can convert number bases each other
9
The steps of converting of 117,8610 decimal number into a binary number
First, the integer part of the number is resolved then real partDivision Remainde
rResult
117/2 =58 1 B0=1
58/2 = 29
0 B1=0
29/2 =14 1 B2=1
14/2 =7 0 B3=0
7/2 =3 1 B4=1
3/2 =1
1 B5=1
1/2 =0 1 B6=1
Multiply Integer Result
0.86*2=1.72 1 b1=1
0.72*2=1.44 1 b2=1
0.44*2=0.88 0 b3=0
0.88*2=1.76 1 b4=1
0.76*2=1.52 1 b5=1
0.52*2=1.04 1 b6=1
0.04*2=0.08 0 b7=0
(117,86)10 = (1110101,1101110….)2
Decimal to Octal Conversion
He/She can convert number bases each other
10
Convert a real decimal number (0,513)10 into an octal number
Operation Multiply Integer Result
0.513*8 4.104 4 o0=1
0.104*8 0.832 0 o1=0
0.832*8 6.656 6 o2=6
0.656*8 5.248 5 o3=5
0.248*8 1.984 1 o4=1
(0,513)10 ≅ (0,40651)8
Decimal to Hex Conversion
He/She can convert number bases each other
11
Convert a decimal number (214)10 into a hex numberOperation Division Remainder Result
214/16 13 6 O0=6
13/16 0 13 O1=D (214)10 = (D6)16
Convert a decimal number (423)10 into a hex numberOperation Division Remainder Result
423/16 26 7 O0=7
26/16 1 10 O1=A
1 1 O2=1 (423)10 = (1A7)16
Binary to Decimal Conversion
He/She can convert number bases each other
12
Convert a binary number (100.01)2 into a decimal number
100.01= 1*22 + 0*21 + 0*20 , 0*2-1 + 1*2-2
= 1*4 + 0 + 0 , 0 + 1* ¼
= 4 + 0 +0 , 0+ 0.25
= (4,25)10
Binary to Octal Conversion
He/She can convert number bases each other
13
Converting a binary number (110111011.1100111)2 into an octal number
For integer part; start from just after the dot towards to the leftmost digit and combine 3-bits as a group. Fill zeros if the group is less than 3-bit
1 5 6 6.
For real part; start from just after the dot towards to the rightmost digit and
combine 3-bits as a group. Fill zeros if the group is less than 3-bit
.6 3 4
1 1 0 1 1 1 0 1 1 0 .
. 1 1 0 0 1 1 1
Binary to Hex Conversion
He/She can convert number bases each other
14
Converting a binary number (110111011.1100111)2 into a hex number
For integer part; start from just after the dot towards to the leftmost digit and combine 4-bit as a group. Fill zeros if the group is less than 4-bit
3 7 6.
For real part; start from just after the dot towards to the rightmost digit and
combine 4-bit as a group. Fill zeros if the group is less than 4-bit
.C E
1 1 0 1 1 1 0 1 1 0 .
. 1 1 0 0 1 1 1
Octal to Binary and Decimal
He/She can convert number bases each other
15
Converting an Octal number into a binary numberConvert each digit into its 3-bit binary counterpart.
Example(673.124)8 =(110 111 011. 001 010 100)2
Converting an octal number into a decimal numberMultiply each digit’s value by radix and sum all terms
Example(372.2)8 = 3x82 + 7x81 + 2x80 + 2x8-1
= 3x64 + 7x8 + 2x1 +2x0.125=(250.25)10
Octal to Hex Conversion
He/She can convert number bases each other
16
Converting an Octal number into a hex numberStep1. Convert each digit into its 3-bit binary counterpart.Step2. Compose 4-bit groupsStep3. Convert 4-bit into hex counterpart
Integer part group direction . Real part group direction
Example(5431)8 = ( ? )16
Step1. (101 100 011 001)2
Step2. 1011 0001 1001
Step3. B 1 9 (5431)8 = (B19)16
Hex to Binary, Decimal Conversion
He/She can convert number bases each other
17
Converting an hex number into a binary numberConvert each digit into its 4-bit binary counterpart.
Example(673.124)16 =(0110 0111 0011. 0001 0010 0100)2
Converting an hex number into a decimal numberMultiply each digit’s value by radix and sum all terms
Example(372)16 = 3x162 + 7x161 + 2x160
= 3x256+ 7x16 + 2x1= 768 + 112 + 1= (881)10
Hex to Octal Conversion
He/She can convert number bases each other
18
Converting an Hex number into a Octal numberStep1. Convert each digit into its 4-bit binary counterpart.Step2. Compose 3-bit groupsStep3. Convert 3-bit into octal counterpart
Integer part group direction . Real part group direction
Example(E0CA)16 = ( ? )8
Step1. (1110 0000 1100 1010)2
Step2. (001 110 000 011 001 010)2
Step3. ( 1 6 0 3 1 2)8
(E0CA)16 = (160312)8
Binary Addition
He/She can do arithmetic operations on various radix
19
0+0 =0
1+0 =1
1+1 =0 Carry=1
1 1 1 1
1 0 1 0
0 1 1 0
0 0 1 1
1 0 0 1 0
Carry
Carry
Carry
Carry
Carry
Binary Subtraction
He/She can do arithmetic operations on various radix
20
0-0 =0
1-0 =1
1-1 =0
0-1 =1 borrow=1
1 0 0 1
1 1 1
0 0 1 0
Binary Subtraction by r complement
He/She can do arithmetic operations on various radix
21
In digital electronics, it is easier to create adder than subtractor unit.
M-N can be redefined as M+( r complement of N)
N can be negated by 1’s complement but 1’s complement contain
both +0 and -0.
STEP-1:
Convert subtraction operation into addition by using r complement
STEP-2:
a. If an extra carry is obtained at the end, discard it and
the number is assumed as positive.
b. If no carry is obtained, apply r complement to result and add 1
Binary Subtraction by 2’s complement
He/She can do arithmetic operations on various radix
22
Example with extra carry:
N can also be negated by 2’s complement (1’s complement+1)
M=1010100, N=1000100 M-N=?
M-N = M+(-N)
= M+ (2’s complement of N)
= M+ (1’s complement+1)
1’s Complement of N= (0111011)2
2s complement of N (1’s complement of N + 1) = (0111100)2
1010100
0111100
1 0010000 MSB is discarded, the result= (0010000)2
Binary Subtraction by 2’s complement
He/She can do arithmetic operations on various radix
23
Example without extra carry:
M=10001002 (68)10
N =10101002 (84)10
M-N =? for 2’s complement
1’s complement of N= (0101011)
2’s complement of N (1’s complement of N + 1) = (0101100)
1000100
0101100
1110000 No carry (-16)10
1s comp 0001111
0000001
-(0010000)2 (-16)10
Binary Multiplication
He/She can do arithmetic operations on various radix
24
1101 13
0101 5
1101 65 0000 1101
100001
Binary Division110010 101
101 1010
00101 101
0000
QUESTIONS
He/She can do arithmetic operations on various radix
25
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