[#3] – Number systems – Questions

As we already know that a computer stores data internally in a format that is not easily readable by human beings. This is the reason why input and output (I/O) interfaces are required. Every computer stores numbers, letters, and other special characters in coded form. Before going into the details of these codes, it is essential to have a basic understanding of number system.

Number systems are of two types – Non-positional and Positional. In early days, human beings counted on fingers when couting beyond ten fingers, they used stones, pebbles, or sticks to indicate values. This method of counting uses an additional approach or non-positional number system. In this system, we have symbols such as I for 1, II for 2, III for 3, IIII for 4, IIIII for 5 etc. Each symbol represents the same value regardless of its position in a number, and to find the value of a number, one has to count the number of symbols presents in the number. Since it is very difficult to perform arithmetic with such a number system, positional number systems were developed.

Also Read:

• [#1] – Introduction to Computer Fundamentals
• [#2] – Basic computer organization
• [#3] – Number systems
• [#4] – Computer Codes
• [#5] – Computer Arithmetic
• [#6] – Processor and Memory
• [#7] – Secondary Storage Devices
• [#8] – Input-Output Devices
• [#9] – Computer software
• [#10] – Planning the Computer Program
• [#11] – Computer Languages
• [#12] – System Implementation and Operation
• [#13] – Operating Systems
• [#14] – Application Software packages
• [#15] – Business Data Processing
• [#16] – Data Communications and Computer Networks
• [#17] – The Internet
• [#18] – Multimedia
• [#19] – Classification of Computers
• [#20] – Introduction to C Programming Language

Points To Remember:

• Number systems are of two types – non-positional and positional
• In a non-positional number system, each symbol represents the same value regardless of its position in the number and to find the value of a number, one has to count the number of symbols present in the number. It is very difficult to perform arithmetic with such a number system.
• In a positional number system, there are only a few symbols called digits. These symbols represent different values depending on the position they occupy in a number. The value of each digit in such a number is determined by three considerations:
  – The digit itself.
  – The position of the digit in the number, and
  – The base of the number system (where base is defined as the total number of digits available in the number system).
• In our day-to-day life, we use the decimal number system. In this system, base is equal to 10 because there are altogether ten symbols or digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9).
• Some positional number systems that are used in computer design and by computer professionals are binary (having base 2), octal (having base 8), and hexadecimal (having base 16).
• Readers must know the techniques for following types of conversions to convert numbers from one base to another:
  – Converting from another base to decimal
  – Converting from decimal to another base
  – Converting from a base other than 10 to another base other than 10
  – Shortcut method for binary to octal conversion
  – Shortcut method for octal to binary to hexadecimal conversion
  – Shortcut method for octal to binary conversion
  – Shortcut method for hexadecimal to binary conversion

List of Questions

1. Differentiate between positional and non-positional number systems? Write examples of both types of number systems.
2. What is base of a number system? Write examples to illustrate the role of base in positional number systems.
3. What is the value of base for decimal, hexadecimal, binary, and octal number systems?
4. Write an example for octal number system to show that the same digit may signify different values depending on the position it occupies in the number.
5. What will be the total number of different symbols or digits and the maximum value of a single digit for the following number systems:
   – Number system with base 5
   – Number system with base 20
   – Number system with base 9
   – Number system with base 12
6. What is a ‘bit’ in computer terminology? How many different patterns of bits are possible with
   (a) 6 bits
   (b) 7 bits
   (c) 8 bits
7. Explain the meaning of the term “memory dump”.
8. Why do we use octal and/or hexadecimal number systems as shortcut notations?
9. Find decimal equivalent of following binary numbers:
   – 1101011 1000
   – 11010 10110001100
   – 10110011 110001
   – 11011101 1010101100
   – 1110101 111
10. Find octal equivalent of the binary numbers of question 9.
11. Find hexadecimal equivalent of the binary numbers of question 9.
12. Convert the following numbers to decimal numbers:
    – 110110₂
    – 2A3B₁₆
    – 2573₆
    – 1234₉
13. Convert the following decimal numbers to binary numbers:
    – 435₁₀
    – 32₁₀
    – 1694₁₀
    – 135₁₀
14. Convert the decimal numbers of question 13 to octal numbers.
15. Convert the decimal numbers of question 13 to hexadecimal numbers.
16. Carry out the following conversions:
    – 125₆ =?₄
    – 24₉ =?₃
    – ABC₁₆ =?₈
17. Convert the following numbers to their binary equivalent:
    – 2AC₁₆
    – 2614₈
    – FAB₁₆
    – 562₈
18. Find decimal equivalent of following numbers:
    – 111.01
    – 247.65
    – 1001.011
    – A2B.D4

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