The document explains number systems used in computer architecture, including binary, octal, decimal, and hexadecimal, along with their respective bases and digit representations. It provides conversion techniques among these systems, giving examples for converting binary to decimal, octal, and hexadecimal, as well as techniques for converting in the opposite direction. Additionally, the document details how to convert between decimal and other systems using various mathematical techniques.

1. What are Number Systems?
• Number systems are the technique to represent numbers in the computer system
architecture, every value that you are saving into or getting from computer memory
has a defined number system. Computer architecture supports following number
systems.
• Binary Number System (2 digits)
• Octal Number System (8 digits)
• Decimal Number System (10 digits )
• Hexa-decimal Number System (16 digits)

2. NUMBER SYSTEMS

3. Conversion Among Bases:
Hexadecimal
Decimal Octal
Binary
• The possibilities

4. Decimal Binary Octal
Hexa-
Decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
8 1000 10 8
9 1001 11 9
Decimal Binary Octal
Hexa-
Decimal
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F

5. Binary Number System
• Digital computers represents all kinds of data and information in the binary system.
• Binary Number System consists of two digits 0 and 1. Its base is 2. Each digit or bit
in binary number system can be 0 or 1.
• Binary to Decimal Conversion Techniques:
• Multiply each bit by 2n, where n is the “weight” of the bit.
• The weight is the position of the bit, starting from 0 on the right.
• Add the results.
• Example 1010112 = 4310

6. Binary Number System
• Binary to Octal Conversion Techniques:
• Group binary digits in a 3 bits, starting on right side.
• Convert to octal digits.
• Example 10110101112 = 13278
• Binary to Hexa-decimal Conversion Techniques:
• Group binary digits in a 4 bits, starting on right side.
• Convert to hexa-decimal digits.
• Example 10101110112 = 2BB16

7. Binary to Decimal Conversion
•

8. Binary to Octal Conversion
•

9. Binary to Hexa-decimal Conversion
•

10. Octal Number System
• Octal number system is the base 8 number system and uses the digits from 0 to 7.
This number system provides shortcut method to represent long binary numbers
• The number after 7 is 10. the number after 17 is 20 and so forth.
• Octal to Decimal Conversion Techniques:
• Multiply each bit by 8n, where n is the “weight” of the bit.
• The weight is the position of the bit, starting from 0 on the right.
• Add the results.
• Example 7248 = 46810

11. Octal Number System
• Octal to Binary Conversion Techniques:
• Convert octal digit in a 3 bits, starting on the right side.
• Example 7058 = 1110001012
• Octal to Hexa-decimal Conversion Techniques:
• Use binary as an intermediary
• Example 10768 = 23E16

12. Octal to Decimal Conversion
•

13. Octal to Binary Conversion
•

14. Octal to Hexa-decimal Conversion
•

15. Decimal Number System
• Decimal number system is the base 10 number system and uses the digits from 0 to
9. Using these digits you can express any quantity.
• It is what we most commonly use.
• Decimal to Binary Conversion Techniques:
• Divide each bit by 2, keep track of the remainder
• First remainder is bit 0 (LSB, least-significant bit)
• Example 12510 = 11111012

16. Decimal Number System
• Decimal to Octal Conversion Techniques:
• Divide each bit by 8, keep track of the remainder
• First remainder is bit 0 (LSB, least-significant bit)
• Example 123410 = 23228
• Decimal to Hexa-decimal Conversion Techniques:
• Divide each bit by 16, keep track of the remainder
• First remainder is bit 0 (LSB, least-significant bit)
• Example 123410 = 4D216

17. Decimal to Binary Conversion
• 125 = 2 125 --- 1
2 62 --- 0
2 31 --- 1
2 15 --- 1
2 7 --- 1
2 3 --- 1
1
= 11111012

18. Decimal to Octal Conversion
• 1234 =
8 1234 --- 2
8 154 --- 2
8 19 --- 3
8 2
= 23228

19. Decimal to Hexa-decimal Conversion
• 1234 =
16 1234 --- 2
16 77 --- 13
16 4
= 4 13 2
= 4D216

20. Hexa-decimal Number System
• Hexa-decimal number system is the base 16 and uses the digits from 0 to 9 and A to
F. This number system provides shortcut method to represent long binary numbers.
• Unlike binary and octal, hexa-decimal has six additional symbols that it uses beyond
the conventional ones found in decimal.
• Hexa-decimal to Decimal Conversion Techniques:
• Multiply each bit by 16n, where n is the “weight” of the bit.
• The weight is the position of the bit, starting from 0 on the right.
• Add the results.
• Example ABC16 = 274810

21. Hexa-decimal Number System
• Hexa-decimal to Binary Conversion Techniques:
• Convert hexa-decimal digit in a 4 bits, starting on the right side.
• Example 10AF16 = 00010000101011112
• Hexa-decimal to Octal Conversion Techniques:
• Use binary as an intermediary
• Example 1F0C16 = 00011111000011008

22. Hexa-decimal to Decimal Conversion
•

23. Hexa-decimal to Binary Conversion
•

24. Hexa-decimal to Octal Conversion
•

25. End