Numircal Analysis

1. 1
Numerical Analysis
Lecture 1

2. Assessment
2
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)

3. Assignments
• Google Classroom
• Code:
3
x3n5el6

4. Numbers
4
• A number is a mathematical object used to count, measure, and
label.
• Numbers can be represented in languages with symbols.
e.g.:

5. 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.
5

6. Number Systems
6
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

7. Conversion Table
7
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

8. Conversions
• 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
8
We'll demonstrate here the following:

9. Decimal to Other Base System
• 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.
9
Steps:

10. Conversion from decimal to any base (Base-x)
10
❑ For integer number → divide by x
Example:
Convert from decimal to binary:
(45)10
45
2
22
1
2
11
2
5
2
2
2
1
2
0
0
1
1
0
1
101101
‫أعلى‬ ‫إلى‬ ‫أسفل‬ ‫من‬ ‫الرقم‬ ‫نكتب‬
(69)10
69
2
34
1
2
17
2
8
2
4
2
2
2
1
0
1
0
0
0
1000101
2
0
1
‫الباقي‬

11. Conversion from decimal to any base (Base-x)
11
Convert from decimal to octal:
901
901
8
112
5
8
14
0
8
1
6
8
0
1
1605
(901)10 → (1605)8

12. Conversion from decimal to any base (Base-x)
12
Convert from decimal to hexadecimal:
1066
1066
16
66
10
16
4
2
16
0
4
(1066)10 → (42A)16

13. 13
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
 pos
5 4 3 2 1 0
= 1*20+1*21+0*22+1*23+0*24+1*25
=1+2+0+8+0+32
=43
 value
(101011)2 → (43)10

14. Conversion from any base (Base-x) to decimal
14
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

15. Conversion from octal to binary
15
Replace each octal digit into its equivalent 3-bit binary number
Convert from octal to binary:
(537)8
(537)8= (101 011 111)2

16. Conversion from binary to octal
16
‫كل‬ ‫ل‬ّ‫نحو‬
3bit
‫بال‬ ‫المكافئ‬ ‫الرقم‬ ‫الى‬
octal
Convert from binary to octal:
a) 111110110
(111 110 110)2=(766)8
b) 1000001
(001 000 001)2=(101)8
‫الصحيح‬ ‫العدد‬ ‫في‬
‫بنبدأ‬
‫الشمال‬ ‫على‬ ‫أصفار‬ ‫نزود‬ ‫وممكن‬ ‫اليمين‬ ‫من‬
‫علشان‬
‫نكمل‬
3
bit

17. Conversion from hexadecimal to binary
17
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

18. 18
Conversion from binary to hexadecimal
‫كل‬ ‫ل‬ّ‫نحو‬
4bit
‫بال‬ ‫المكافئ‬ ‫الرقم‬ ‫الى‬
hexadecimal
‫الصحيح‬ ‫العدد‬ ‫في‬
‫بنبدأ‬
‫الشمال‬ ‫على‬ ‫أصفار‬ ‫نزود‬ ‫وممكن‬ ‫اليمين‬ ‫من‬
‫علشان‬
‫نكمل‬
4
bit
Convert from binary to hexadecimal:
a) 10101001
(1010 1001)2= (A9)16
d) 01101111
(0110 1111)2= (6F)16

19. Conversion from octal to hexadecimal
19
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

20. 20
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

21. 21