1. Numbering Systems

2. Introduction to Numbering Systems
Decimal System
We are all familiar with the decimal number system (Base 10). Some other number systems that we will work with are:
Binary  Base 2النظام الثنائي
Octal  Base 8النظام الثماني
Hexadecimal  Base 16النظام السداسي عشر

3. Characteristics of Numbering Systems
The digits are consecutive.الارقام متسلسلة
The number of digits is equal to the size of the base عدد الأرقام = الأساس
Zero is always the first digit.الصفر هو دائما اول رقم
The base number is never a digit.الأساس ليس ضمن العناصر
When 1 is added to the largest digit, a sum of zero and a carry of one results عند اضافة واحد لأكبر رقم نحصل علي صفر ونحمل واحد
Numeric values are determined by the implicit positional values of the digits.تعتمد قيمة العدد على موضع الرقم

4. Most significant digit Least significant digit
Significant Digits
Binary:
Most significant digit Least significant digit
Hexadecimal: 1D63A7A

5. Binary System

6. Binary Number System 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

7. Binary Decimal

8. Decimal to Binary Conversion
The easiest way to convert a decimal number to its binary equivalent is to use the Division Algorithm
This method repeatedly divides a decimal number by 2 and records the quotient and remainder 
The remainder digits (a sequence of zeros and ones) form the binary equivalent in least significant to most significant digit sequence

9. An algorithm for finding the binary representation of a positive integer

10. Division Algorithm
Convert 67 to its binary equivalent:
6710 = x2
Step 1: 67 / 2 = 33 R Divide 67 by 2. Record quotient in next row
Step 2: 33 / 2 = 16 R Again divide by 2; record quotient in next row
Step 3: 16 / 2 = 8 R Repeat again
Step 4: 8 / 2 = 4 R 0 Repeat again
Step 5: 4 / 2 = 2 R Repeat again
Step 6: 2 / 2 = 1 R Repeat again
Step 7: 1 / 2 = 0 R 1 STOP when quotient equals 0

11. Binary to Decimal Conversion
The easiest method for converting a binary number to its decimal equivalent is to use the Multiplication Algorithm
Multiply the binary digits by increasing powers of two, starting from the right
Then, to find the decimal number equivalent, sum those products

12. Multiplication Algorithm
Convert ( )2 to its decimal equivalent:
Binary
Positional Values x x x x x x x x 27 26 25 24 23 22 21 20
Products
17310

13. BINARY TO DECIMAL CONVERTION
Any binary number can be converted to its decimal equivalent simply by summing together the weights of the various positions in the binary number which contain 1.
Example 1:
convert to decimal value
Solve: 1 1 1 1 = =

14. Convert 101101012 to decimal value
Example 2 :
Convert to decimal value
Solve: 1 1 1 1 1 = =
You should noticed the method is find the weights (i.e., powers of 2) for each bit position that contains 1, and then to add them up.

15. DECIMAL TO BINARY CONVERTION
Example : convert 2510 to binary
Solve =
2510 = ?2 25 2 =
12 balance 1
LSB 12 2 =
6 balance 0 6 2 =
3 balance 0 3 2 =
1 balance 1 1 2 =
0 balance 1
MSB . . .
Answer =

16. Octal System

17. Octal Number System Also known as the Base 8 System Uses digits 0 - 7
Readily converts to binary
Groups of three (binary) digits can be used to represent each octal digit
Also uses multiplication and division algorithms for conversion to and from base 10

18. Octal Decimal

19. OCTAL TO DECIMAL CONVERTION
Convert from octal to decimal by multiplying each octal digit by its positional weight.
Example 1: Convert 1638 to decimal value
Solve = =
1 x x x 1 =
11510
Example 2: Convert 3338 to decimal value
Solve = =
3 x x x 1 =
21910

20. DECIMAL TO OCTAL CONVERTION
Convert from decimal to octal by using the repeated division method used for decimal to binary conversion.
Divide the decimal number by 8
The first remainder is the LSB and the last is the MSB.
Example : convert to Decimal Value
Solve =
35910 = ?8
359 8 =
44 balance 7
LSB 44 8 =
5 balance 4 5 8 =
0 balance 5
MSB . . .
Answer = 5478

21. Octal Binary

22. OCTAL TO BINARY CONVERTION
Convert from octal to binary by converting each octal digit to a three bit binary equivalent
Octal digit 1 2 3 4 5 6 7
Binary Equivalent
000
001
010
011
100
101
110
111
Convert from binary to octal by grouping bits in threes starting with the LSB.
Each group is then converted to the octal equivalent
Leading zeros can be added to the left of the MSB to fill out the last group.

24. BINARY TO OCTAL CONVERSION
Can be converted by grouping the binary bit in group of three starting from LSB
Octal is a base-8 system and equal to two the power of three, so a digit in Octal is equal to three digit in binary system.

26. Hexadecimal System

27. Hexadecimal Number System
Base 16 system
Uses digits 0-9 &
letters A,B,C,D,E,F
Groups of four bits represent each base 16 digit

28. Hexadecimal Decimal

29. Decimal to Hexadecimal Conversion
Convert to its hexadecimal equivalent:
830 / 16 = 51 R14
51 / 16 = 3 R3
3 / 16 = 0 R3
= E in Hex
33E16

30. Hexadecimal to Decimal Conversion
Convert 3B4F16 to its decimal equivalent:
Hex Digits
3 B F x x x x
Positional Values
Products
15,18310

31. Hexadecimal Binary

32. Binary to Hexadecimal Conversion
The easiest method for converting binary to hexadecimal is to use a substitution code
Each hex number converts to 4 binary digits

33. Binary Arithmetic

34. The binary addition facts

35. Binary Arithmetic
The individual digits of a binary number are referred to as bits
Each bit represents a power of two
= 0 • • • • • 20 = 11
= 0 • • • • • 20 = 2
00010
01101 2
+ 11 13
Equivalent
decimal addition
Binary
addition