Class 11 Number System Notes: Important Points
Class 11 Number System Notes: Important Points
Introduction
A number system is a method to represent (write) numbers. Every number system has a set of unique characters or literals. The count of these literals is called the radix or base of the number system. The four different number systems used in the context of computer are given below:
| Name of Number System | Base of Number System | Digits/Symbols included in Number System |
| Binary Number System | 2 | 0, 1 |
| Octal Number System | 8 | 0, 1, 2, 3, 4, 5, 6, 7 |
| Decimal Number System | 10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 |
| Hexa Decimal | 16 | 0 – 9 and A – F |
1. Decimal Number System
The decimal number system is used in our day-to-day life. It is known as base-10 system since 10 digits (0 to 9) are used. A number is presented by its two values — symbol value (any digit from 0 to 9) and positional value (in terms of base value).
Following table shows the integer and fractional part of decimal number 237.25 along with computation of the decimal number using positional values.
| Digit | 2 | 3 | 7 | . | 2 | 5 |
| Position Number | 2 | 1 | 0 | -1 | -2 | |
| Positional Value | 102 | 101 | 100 | 10-1 | 10-2 |
Add the product of positional value and corresponding digit to get decimal number.
2 x 102 + 3 x 101 + 7 x 100 + 2 x 10-1 + 5 x 10-2 = 237.25
2. Decimal Number System
The ICs (Integrated Circuits) in a computer are made up of a large number of transistors which are activated by the electronic signals (low/high) they receive.
The ON/ high and OFF/low state of a transistor is represented using the two digits 1 and 0, respectively. These two digits 1 and 0 form the binary number system.
This system is also referred as base-2 system as it has two digits only. Some examples of binary numbers are 101011, 110001, 1010.101
The following table shows Binary value for (0–9) digits of decimal number system.
| Decimal | Binary |
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
| 8 | 1000 |
| 9 | 1001 |
3. Octal Number System
Octal number system was devised for compact representation of the binary numbers. Octal number system is called base-8 system as it has total eight digits (0-7).
Three binary digits (8=23) are sufficient to represent any octal digit.
Following table shows the decimal and binary equivalent of 8 octal digits. Examples of octal numbers are (145)8, (245)8, and (235)8.
| Octal Digit | Decimal Value | 3 -bit Binary Number |
| 0 | 0 | 000 |
| 1 | 1 | 001 |
| 2 | 2 | 010 |
| 3 | 3 | 011 |
| 4 | 4 | 100 |
| 5 | 5 | 101 |
| 6 | 6 | 110 |
| 7 | 7 | 111 |
4. Hexadecimal Number System
Hexadecimal numbers are also used for compact representation of binary numbers. It consists of 16 unique symbols (0 – 9, A–F), and is called base-16 system.
Four binary digits (16=24) are sufficient to represent any hexadecimal number.
Note here that the decimal numbers 10 through 15 are represented by the letters A through F. Examples of Hexadecimal numbers are (43A)16, (1B)16
Following table shows Decimal and Binary equivalent of hexadecimal numbers 0–9, A–F
| Hexadecimal Symbol | Decimal Value | 4-bit Binary Number |
| 0 | 0 | 0000 |
| 1 | 1 | 0001 |
| 2 | 2 | 0010 |
| 3 | 3 | 0011 |
| 4 | 4 | 0100 |
| 5 | 5 | 0101 |
| 6 | 6 | 0110 |
| 7 | 7 | 0111 |
| 8 | 8 | 1000 |
| 9 | 9 | 1001 |
| A | 10 | 1010 |
| B | 11 | 1011 |
| C | 12 | 1100 |
| D | 13 | 1101 |
| E | 14 | 1110 |
| F | 15 | 1111 |
Applications of Hexadecimal Number System
1. To access 16-bit memory address, a programmer has to use 16 binary bits, which is difficult to deal with. To simplify the address representation, hexadecimal and octal numbers are used. Let us consider a 16- bit memory address 1100000011110001. Using the hexadecimal notation, this address is mapped to “C0F1” which is more easy to remember.
2. Hexadecimal numbers are also used for describing the colours on the webpage. Colour codes are written in hexadecimal form for compact representation. For example, 24-bit code for RED colour is 11111111,00000000,00000000. The equivalent hexadecimal notation is (FF,00,00), which can be easily remembered and used
Conversion between Number Systems
In this section we will learn how to convert a number from one number system to another number system for better understanding of the number representation in computers. Decimal number system is most commonly used by humans, but digital systems understand binary numbers; whereas Octal and hexadecimal number systems are used to simplify the binary representation.
1. Conversion from Decimal to other Number Systems
Steps to convert a decimal number to any other number system (binary, octal or hexadecimal) are:
- Divide the given number by the base value (b) of the number system in which it is to be converted
- Note the remainder
- Keep on dividing the quotient by the base value and note the remainder till the quotient is zero
- Write the noted remainders in the reverse order (from bottom to top)
A. Decimal to Binary Conversion
The decimal number is repeatedly divided by 2 following the steps given in above till the quotient is 0. Record the remainder after each division and finally write the remainders in reverse order in which they are computed.
Example 1: Convert (122)10 to binary number.
Example 2: Convert (145)10 to binary number.
| Quotient | Remainder | |
| 145/2 | 72 | 1 |
| 72/2 | 36 | 0 |
| 36/2 | 18 | 0 |
| 18/2 | 9 | 0 |
| 9/2 | 4 | 1 |
| 4/2 | 2 | 0 |
| 2/2 | 1 | 0 |
| 1/2 | 0 | 1 |
(145)10 = 10010001 #Write the remainder in reverse order (from Bottom to Top)
Example 3: Convert (218)10 to binary number.
| Quotient | Remainder | |
| 218/2 | 109 | 0 |
| 109/2 | 54 | 1 |
| 54/2 | 27 | 0 |
| 27/2 | 13 | 1 |
| 13/2 | 6 | 1 |
| 6/2 | 3 | 0 |
| 3/2 | 1 | 1 |
| 1/2 | 0 | 1 |
(218)10 = 11011010
Q1. Express the following decimal numbers into Binary numbers.
(i) (715)10
(ii) (625)10
(iii) (660)10
(iv) (548)10
(v) (268)10
B. Decimal to Octal Conversion
Base value of octal is 8, the decimal number is repeatedly divided by 8 to obtain its equivalent octal number.
Example 4: Convert (122)10 to Octal number.
| Quotient | Remainder | |
| 122/8 | 15 | 2 |
| 15/8 | 1 | 7 |
| 1/8 | 0 | 1 |
(122)10 = (172)8 #Write the remainder in reverse order(Bottom to Top)
Example 5: Convert (278)10 to Octal number.
| Quotient | Remainder | |
| 278/8 | 34 | 6 |
| 34/8 | 4 | 2 |
| 4/8 | 0 | 4 |
(278)10 = (426)8
Example 6: Convert (328)10 to Octal number.
| Quotient | Remainder | |
| 328/8 | 41 | 0 |
| 41/8 | 5 | 1 |
| 5/8 | 0 | 5 |
(328)10 = (510)8
Q2. Express the following Decimal numbers into Octal numbers.
(i) (269)10
(ii) (780)10
(iii) (590)10
(iv) (400)10
(v) (983)10
C. Decimal to Hexadecimal Conversion
The base value of hexadecimal is 16, the decimal number is repeatedly divided by 16 to obtain its equivalent hexadecimal number.
Example 7: Convert (785)10 to Hexadecimal number.
| Quotient | Remainder | |
| 785/16 | 49 | 1 |
| 49/16 | 3 | 1 |
| 3/16 | 0 | 3 |
(785)10 = (311)16
Example 8: Convert (485)10 to Hexadecimal number.
| Quotient | Remainder | |
| 485/16 | 30 | 5 |
| 30/16 | 1 | 14 |
| 1/16 | 0 | 1 |
(485)10 = (1E5)16 #10 – 15 are represented by Alphabet A – F respectively
Q3. Express the following Decimal numbers into Hexadecimal numbers.
(i) (970)10
(ii) (520)10
(iii) (750)10
(iv) (350)10
(v) (820)10
Conversion from other Number Systems to Decimal Number System
We can use the following steps to convert the given number with base value b to its decimal equivalent. Base value b can be 2, 8 and 16 for binary, octal and hexadecimal number system, respectively.
- Write the position number for each alphanumeric symbol in the given number.
- Get positional value for each symbol by raising its position number to the base value b symbol
in the given number. - Multiply each digit with the respective positional value to get a decimal value.
- Add all these decimal values to get the equivalent decimal number.
(A) Binary Number to Decimal Number
Since binary number system has base 2, the positional values are computed in terms of powers of 2.
Example 9. Convert (1101)2 into decimal number.
| Digit | 1 | 1 | 0 | 1 |
| Position Number | 3 | 2 | 1 | 0 |
| Position Value | 23 | 22 | 21 | 20 |
| Decimal Number | 1 × 23 | 1 x 22 | 0 x 21 | 1 x 20 |
| 8 | 4 | 0 | 1 |
(1101)2 = 1 × 23 + 1 x 22 + 0 x 21 + 1 x 20
(1101)2 = 8 + 4 + 0 + 1
(1101)2 = (13)10
Example 10. Convert (1011010)2 into decimal number.
(1011010)₂ = (1 × 2⁶) + (0 × 2⁵) + (1 × 2⁴) + (1 × 2³) + (0 × 2²) + (1 × 2¹) + (0 × 2⁰)
(1011010)₂ = 64 + 0 + 16 + 8 + 0 + 2 + 0
(1011010)₂ = (90)10
Example 11. Convert (1000111)2 into decimal number.
(1000111)₂ = (1 × 2⁶) + (0 × 2⁵) + (0 × 2⁴) + (0 × 2³) + (1 × 2²) + (1 × 2¹) + (1 × 2⁰)
(1000111)₂ = 64 + 0 + 0 + 0 + 4 + 2 + 1
(1000111)₂ = (71)10
Example 12. Convert (11110101)2 into decimal number.
(11110101)₂ = (1 × 2⁷) + (1 × 2⁶) + (1 × 2⁵) + (1 × 2⁴) + (0 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰)
(11110101)₂ = 128 + 64 + 32 + 16 + 0 + 4 + 0 + 1
(11110101)₂ = (245)10
(B) Octal Number to Decimal Number
Since Octal number system has base 8, the positional values are computed in terms of powers of 8.
Example 13. Convert (173)8 into decimal number.
| Digit | 1 | 7 | 3 |
| Position Number | 2 | 1 | 0 |
| Position Value | 82 | 81 | 80 |
| Decimal Number | 1 × 82 | 7 x 81 | 3 x 80 |
| 64 | 56 | 3 |
(173)8 = (1 × 8²) + (7 × 8¹) + (3 × 8⁰)
(173)8 = 64 + 56 + 3
(173)8 = (123)10
Example 14. Convert (350)8 into decimal number.
(350)8 = (3 × 8²) + (5 × 8¹) + (0 × 8⁰)
(350)8 = 192 + 40 + 0
(350)8 = (232)10
Example 15. Convert (725)8 into decimal number.
(725)8 = (7 × 8²) + (2 × 8¹) + (5 × 8⁰)
(725)8 = 448 + 16 +5
(725)8 = (469)10
(B) Hexadecimal Number to Decimal Number
Since Hexadecimal number system has base 16, the positional values are computed in terms of powers of 16.
Example 16. Convert (3A5)16 into decimal number
| Digit | 3 | A | 5 |
| Position Number | 2 | 1 | 0 |
| Position Value | 162 | 161 | 160 |
| Decimal Number | 3 × 162 | 10 x 161 | 5 x 160 |
| 768 | 160 | 5 |
(3A5)16 = (3 × 16²) + (10 × 16¹) + (5 × 16⁰)
(3A5)16 = 768 + 160 + 5
(3A5)16 = (933)10
Example 17. Convert (2B)16 into decimal number
(2B)₁₆ = (2 × 16¹) + (11 × 16⁰)
(2B)₁₆ = 32 + 11
(2B)₁₆ = (43)10
Example 18. Convert (97)16 into decimal number
(97)₁₆ = (9 × 16¹) + (7 × 16⁰)
(97)₁₆ = 144 + 7
(97)₁₆ = (151)10
Q1. Why 3 bits in a binary number are grouped together to get octal number?
Ans. The base value of octal number system is 8. Convert value 8 in terms of exponent of 2, i.e., 8=23. Hence,
three binary digits are sufficient to represent all 8 octal digits.
Conversion from Binary Number to Octal/Hexadecimal Number and Vice-Versa
A binary number is converted to octal or hexadecimal number by making groups of 3 and 4 bits, respectively,
and replacing each group by its equivalent octal/hexadecimal digit.
(A) Binary Number to Octal Number
| Octal Digit | Decimal Value | 3 -bit Binary Number |
| 0 | 0 | 000 |
| 1 | 1 | 001 |
| 2 | 2 | 010 |
| 3 | 3 | 011 |
| 4 | 4 | 100 |
| 5 | 5 | 101 |
| 6 | 6 | 110 |
| 7 | 7 | 111 |
Example 19. Convert (10101100)2 to octal number
Make group of 3-bits of the given binary number (right to left) and write octal number for each 3-bit group.
010 —- 101 —- 100
2 —- 5 —– 4
(10101100)2 = (254)8
NOTE: In case number of bits in a binary number is not multiple of 3, then add required number of 0s on most significant position of the binary number.
Example 20. Convert (111101010)2 to octal number
111—–101—–010
7 ——-5 ——–2
(111101010)2 = (752)8
(B) Octal Number to Binary Number
Each octal digit is an encoding for a 3-digit binary number. Octal number is converted to binary by replacing each octal digit by a group of three binary digits.
Example 21. Convert (705)8 to binary number.
7 ——- 0 ——-5
111 —-000 —101
(705)8 = (111000101)2
Example 22. Convert (134)8 to binary number.
1———-3———–4
001——-011———100
(134)8 = (001011100)2
(C) Binary Number to Hexadecimal Number
| Hexadecimal Symbol | Decimal Value | 4-bit Binary Number |
| 0 | 0 | 0000 |
| 1 | 1 | 0001 |
| 2 | 2 | 0010 |
| 3 | 3 | 0011 |
| 4 | 4 | 0100 |
| 5 | 5 | 0101 |
| 6 | 6 | 0110 |
| 7 | 7 | 0111 |
| 8 | 8 | 1000 |
| 9 | 9 | 1001 |
| A | 10 | 1010 |
| B | 11 | 1011 |
| C | 12 | 1100 |
| D | 13 | 1101 |
| E | 14 | 1110 |
| F | 15 | 1111 |
Make group of 4-bits of the given binary number (right to left) and write hexadecimal number for each 4-bit group.
Example 23. Convert (0110101100)2 to Hexadecimal number.
#Make group of 4-bits and write hexadecimal symbol for each group
Binary Digits: 0001---- 1010 ---- 1100
Hexadecimal Digits: 1 ---------A-----------C
(0110101100)2 = (1AC)16
Example 24. Convert (110100110101)2 to Hexadecimal number.
Binary Digits: 1101 ------ 0011 -------0101
Hexadecimal Digits: D ----------3 -----------5
(110100110101)2 = (D35)16
(D) Hexadecimal Number to Binary Number
Each hexadecimal symbol is an encoding for a 4-digit binary number. Hence, the binary equivalent of a hexadecimal number is obtained by substituting 4-bit binary equivalent of each hexadecimal digit and combining them together.
Example 25. Convert (23D)16 to binary number.
Hexadecimal digits: 2 ---------- 3 ------------D
Binary Digits : 0010 -------0011 -------- 1101
(23D)16 = (001000111101)2
Example 26. Convert (F018)16 to binary number.
Hexadecimal digits: F ---------- 0 ------------1 ------------ 8
Binary Digits : 1111 -------0000 -------- 0001 --------1000
(F018)16 = (1111000000011000)2
Example 27. Convert (172)16 to binary number.
Hexadecimal digits: 1 ----------- 7 ----------- 2
Binary Digits: 0001 ------0111 ----------0010
(172)16 = (000101110010)2
Conversion of a Number with Fractional Part
We will learn about conversion of numbers with a fractional part.
(A) Decimal Number with Fractional Part to another Number System
To convert the fractional part of a decimal number, repeatedly multiply the fractional part by the base value b till the fractional part becomes 0. Write integer part from top to bottom to get equivalent number in that number system.
Example 28. Convert (0.25)10 to binary
Integer Part
0.25 × 2 = 0.50 0
0.50 × 2 = 1.00 1
(0.25)10 = (0.01)2
NOTE: If the fractional part does not become 0 in successive multiplication, then stop after, say 10 multiplications.
NOTE: In some cases, fractional part may start repeating, then stop further calculation.
Example 29. Convert (0.675)10 to binary.
Integer part
0.675 × 2 = 1.350 1
0.350 × 2 = 0.700 0
0.700 × 2 = 1.400 1
0.400 × 2 = 0.800 0
0.800 × 2 = 1.600 1
0.600 × 2 = 1.200 1
0.200 × 2 = 0.400 0
Since the fractional part (.400) is the repeating value in the calculation, the multiplication is stopped. Write the integer part from top to bottom.
(0.675)10 = (0.1010110)2
Example 30. Convert (0.675)10 to Octal
Integer part
0.675 × 8 = 5.400 5
0.400 × 8 = 3.200 3
0.200 × 8 = 1.600 1
0.600 × 8 = 4.800 4
0.800 × 8 = 6.400 6
Since the fractional part (.400) is the repeating value in the calculation, the multiplication is stopped.
(0.675)10= (0.53146)8
Example 31. Convert (0.70)10 to Octal
Integer Part
0.70 x 8 = 5.60 5
0.60 x 8 = 4.80 4
0.80 x 8 = 6.40 6
0.40 x 8 = 3.20 3
0.20 x 8 = 1.60 1
(0.70)10 = (0.54631)8
Example 32. Convert (0.675)10 to hexadecimal form.
Integer Part
0.675 × 16 = 10.800 A (Hexadecimal symbol for 10)
0.800 × 16 = 12.800 C (Hexadecimal symbol for 12)
Since the fractional part (.800) is repeating, the multiplication is stopped. Write the integer part from
top to bottom to get hexadecimal equivalent for the fractional part.
(0.675)10 = (0.AC)16
(B) Non-decimal Number with Fractional Part to Decimal Number System
Example 33. Convert (100101.101)2 into decimal
| 1 | 0 | 0 | 1 | 0 | 1 | . | 1 | 0 | 1 |
| 25 | 24 | 23 | 22 | 21 | 20 | 2-1 | 2-2 | 2-3 | |
| 1 x 25 | 0 x 24 | 0 x 23 | 1 x 22 | 0 x 21 | 1 x 20 | 1 x 2-1 | 0 x 2-2 | 1 x 2-3 | |
| 32 | 0 | 0 | 4 | 0 | 1 | . | 0.5 | 0 | 0.125 |
(100101.101)2 = (37.625)10
Example 34. Convert (605.12)8 into decimal number.
| 6 | 0 | 5 | . | 1 | 2 |
| 82 | 81 | 80 | . | 8-1 | 8-2 |
| 6 x 82 | 0 x 81 | 5 x 80 | . | 1 x 8-1 | 2 x 8-2 |
| 384 | 0 | 5 | .125 | .03125 |
(605.12)8 = (389.15625)10
(C) Fractional Binary Number to Octal or Hexadecimal Number
To convert the fractional binary number into octal or hexadecimal value, substitute groups of 3-bit or 4-bit in integer part( start from right to left) by the corresponding digit. Similarly, make groups of 3-bit or 4-bit for fractional part starting from left to right, and substitute each group by its equivalent digit or symbol in Octal or Hexadecimal number system.
Example 35. Convert (10101100.01011)2 to octal number.
Make perfect group of 3 bits
010------101-----100 . 010-----110
2 ---------5--------4 . 2 --------6
Convert (10101100.01011)2 = (254.23)8
NOTE: Make 3-bit groups from right to left for the integer part and left to right for the fractional part.
Example 36. Convert (10101100.010111)2 to Hexadecimal number.
Make perfect group of 4 bits
1010 ---- 1100 . 0101 ----- 1100
A --------C . 5 ---------C
(10101100.010111)2 = (AC.5C)16
Click for Class 11 Number System MCQ
Click for Class 11 Number System NCERT Solutions
Class XI & XII IP Syllabus 2024-25
Class XI & XII CS Syllabus 2024-25
Chapter Wise MCQ
2. Flow of Control (Loop and Conditional statement)
3. 140+ MCQ on Introduction to Python
4. 120 MCQ on String in Python
7. 100+ MCQ on Flow of Control in Python
8. 60+ MCQ on Dictionary in Python
Important Links
100 Practice Questions on Python Fundamentals
120+ MySQL Practice Questions
90+ Practice Questions on List
50+ Output based Practice Questions
100 Practice Questions on String
70 Practice Questions on Loops
70 Practice Questions on if-else
Disclaimer : I tried to give you the correct notes of ” Class 11 Number System Notes: Important Points” , but if you feel that there is/are mistakes in the handouts or explanation of “Class 11 Number System Notes: Important Points“ given above, you can directly contact me at csiplearninghub@gmail.com. The above Notes of “Class 11 Number System Notes: Important Points ” are created by referring NCERT Book of Class 11. Screenshots used in this article are taken from NCERT book.