Numerical Analysis
Lecture 1
1
Assessment
Total: 100 marks
• Final Exam: 60 marks
• Mid-term: 15 marks
• Oral : 10 marks
• Assignments: 10 marks
• Participation: 5 marks
You will pass if you get 50 marks (PASS)
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Assignments
• Google Classroom
• Code:
x3n5el6
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Numbers
• A number is a mathematical object used to count, measure, and
label.
• Numbers can be represented in languages with symbols.
e.g.:
4
Digits and Numbers
• Digit
A digit is a single numerical symbol.
For example: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• Number
A number is a string of one or more digits.
For example, the number 23 is written with two digits, 2 and 3.
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Number Systems
System
Digits
Base
Binary
0,1
Base-2
Octal
0,1,2,3,4,5,6,7
Base-8
Decimal
0,1,2,3,4,5,6,7,8,9
Base-10
Hexadecimal
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Base-16
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Conversion Table
Hexadecimal
Octal
Decimal
Binary
0
0
0
0000
1
1
1
0001
2
2
2
0010
3
3
3
0011
4
4
4
0100
5
5
5
0101
6
6
6
0110
7
7
7
0111
8
10
8
1000
9
11
9
1001
A
12
10
1010
B
13
11
1011
C
14
12
1100
D
15
13
1101
E
16
14
1110
F
17
15
1111
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Conversions
We'll demonstrate here the following:
•
Decimal to Other Base System
•
Other Base System to Decimal
•
Other Base System to Non-Decimal
•
Shortcut method − Binary to Octal
•
Shortcut method − Octal to Binary
•
Shortcut method − Binary to Hexadecimal
•
Shortcut method − Hexadecimal to Binary
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Decimal to Other Base System
Steps:
•
•
•
•
Step 1 − Divide the decimal number to be converted by the value of the
new base.
Step 2 − Get the remainder from Step 1 as the rightmost digit (least
significant digit) of new base number.
Step 3 − Divide the quotient of the previous divide by the new base.
Step 4 − Record the remainder from Step 3 as the next digit (to the left) of
the new base number.
Repeat Steps 3 and 4, getting remainders from right to left, until the quotient
becomes zero in Step 3.
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Conversion from decimal to any base (Base-x)
❑ For integer number → divide by x
Example:
Convert from decimal to binary:
(45)10
(69)10
‫الباقي‬
1000101
2 45
1
2 69
1
2 34
0
2 22
0
2 17
1
2 11
1
2
8
0
2
5
1
2
4
0
2
2
0
2
2
0
2
1
1
2
1
1
0
‫نكتب الرقم من أسفل إلى أعلى‬
101101
0
10
Conversion from decimal to any base (Base-x)
Convert from decimal to octal:
901
1605
8
901
5
8
112
0
8
14
6
8
1
1
0
(901)10 → (1605)8
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Conversion from decimal to any base (Base-x)
Convert from decimal to hexadecimal:
1066
16 1066
10
16
66
2
16
4
4
0
(1066)10 → (42A)16
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Conversion from any base (Base-x) to decimal
Result = SUM (value * xpos)
Example:
Convert from binary to decimal:
101011
1 0 1 0 1 1  value
5 4 3 2 1 0  pos
= 1*20+1*21+0*22+1*23+0*24+1*25
=1+2+0+8+0+32
=43
(101011)2 → (43)10
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Conversion from any base (Base-x) to decimal
Convert from octal to decimal:
672
(672)8=2*80+7*81+6*82
=2+56+384
=(442)10
5061
(5561)8=1*80+6*81+0*82+5*83
=1+48+0+2560
=(2609)10
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Conversion from octal to binary
Replace each octal digit into its equivalent 3-bit binary number
Convert from octal to binary:
(537)8
(537)8= (101 011 111)2
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‫‪Conversion from binary to octal‬‬
‫نحول كل ‪ 3bit‬الى الرقم المكافئ بال ‪octal‬‬
‫ّ‬
‫في العدد الصحيح بنبدأ من اليمين وممكن نزود أصفار على الشمال علشان نكمل ‪bit 3‬‬
‫‪Convert from binary to octal:‬‬
‫‪a) 111110110‬‬
‫‪(111 110 110)2=(766)8‬‬
‫‪b) 1000001‬‬
‫‪(001 000 001)2=(101)8‬‬
‫‪16‬‬
Conversion from hexadecimal to binary
Replace each hexadecimal digit into its equivalent 4-bit binary number
Convert from hexadecimal to binary:
(A8C3)16
(A8C3)16= (1010 1000 1100 0011)2
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Conversion from binary to hexadecimal
hexadecimal ‫ الى الرقم المكافئ بال‬4bit ‫نحول كل‬
ّ
bit 4 ‫في العدد الصحيح بنبدأ من اليمين وممكن نزود أصفار على الشمال علشان نكمل‬
Convert from binary to hexadecimal:
a) 10101001
(1010 1001)2= (A9)16
d) 01101111
(0110 1111)2= (6F)16
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Conversion from octal to hexadecimal
Octal → Binary → Hexadecimal
Convert from octal to hexadecimal:
A) 777
(777)8 → (111111111)2 → (0001 1111 1111)2 → (1FF)16
B) 605
(605)8 → (110000101)2 → (0001 1000 0101)2 → (185)16
C) 443
(443)8 → (100100011)2 → (0001 0010 0011)2 → (123)16
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Conversion from hexadecimal to octal
Hexadecimal → Binary → Octal
Convert from hexadecimal to octal:
A) A9
(A9)16=(10101001)2=(010 101 001)2=(251)8
B) E7
(E7)16=(11100111)2=(011 100 111)2=(347)8
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