Grade-10 Ch-6 Number System RoseBud

 Chapter:6 Number System

1. Answer the following questions:

1. What is number system?
Answer :
The system concerned with the number and represented by sequence of digits is called number system. It plays vital role in computing and electronics. Number system also refers to the digits, its arrangement, positional value and base of number system.
2. What do you mean by decimal number system?
Answer:
The number system having base 10 and consists of digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 is called Decimal (Denary) number system.
3. What is Binary number system? Write its uses in computer system
Answer :
The number system having base two(2) and consists of digits: 0 and 1 is called Binary number system.
It has two bits 0 and 1. An electronic circuit has two states either ON & OFF state. The bit 1 represents high voltage (ON state) and the bit 0 represents the low voltage (OFF state) of an electronic circuit.
4. What do you meant by base or radix of number system?
Answer :
The total number of digits used by the particular number system is called base or radix of that number system. For Example: Base of Binary number system is 2 because it uses two digits 0 and 1 only.
5. What are the numbers and characters sets of hexadecimal number system?
Answer :
Numbers of hexadecimal are 0 to 9 (0,1,2,3,4,5,6,7,8,9) and characters of hexadecimal are A - F (A,B,C,D,E,F).
6. What do you mean by BCD and ASCII?
Answer :
BCD(Binary Coded Decimal)
The process that represents the digits and alphabets with the help of binary digits is called BCD. It is 4 bit code, which can represent a maximum of 64 different characters.
ASCII(American Standard Code for Information Interchange)
An 8 bit code and is exclusively used to represent the data internally in the micro computers is called ASCII.
7. Write the use of octal and hexadecimal number system in computer system?
Answer :
The octal and hexadecimal number systems are introduced to interpret large binary numbers in computer languages.

2. Convert the following binary number to decimal numbers:

a. (1101101)₂ = (?)₁₀

Answer:

Face Value	1	1	0	1	1	0	1
Place Value	26	25	24	23	22	21	20

=1×2⁶+1×2⁵+0×2⁴+1×2³+1×2²+0×2¹+1×2⁰
=64+32+0+8+4+0+1
=(109)₁₀
∴ (1101101)₂ = (109)₁₀

b. (101010)₂ = (?)₁₀

Answer:

Face Value	1	0	1	0	1	0
Place Value	25	24	23	22	21	20

=1×2⁵+0×2⁴+1×2³+0×2²+1×2¹+0×2⁰
=32+0+8+0+2+0
=(42)₁₀
∴ (101010)₂ = (42)₁₀

c. (1100110)₂ = (?)₁₀

Answer:

Face Value	1	1	0	0	1	1	0
Place Value	26	25	24	23	22	21	20

=1×2⁶+1×2⁵+0×2⁴+0×2³+1×2²+1×2¹+0×2⁰
=64+32+0+0+4+2+0
=(102)₁₀
∴ (1100110)₂ = (102)₁₀

d. (1100011)₂ = (?)₁₀

Answer:

Face Value	1	1	0	0	0	1	1
Place Value	26	25	24	23	22	21	20

=1×2⁶+1×2⁵+0×2⁴+0×2³+0×2²+1×2¹+1×2⁰
=64+32+0+0+0+2+1
=(99)₁₀
∴ (1100011)₂ = (99)₁₀

e. (111011)₂ = (?)₁₀

Answer:

Face Value	1	1	1	0	1	1
Place Value	25	24	23	22	21	20

=1×2⁵+1×2⁴+1×2³+0×2²+1×2¹+1×2⁰
=32+16+8+0+2+1
=(59)₁₀
∴ (1101101)₂ = (59)₁₀

f. (1010101)₂ = (?)₁₀

Answer:

Face Value	1	0	1	0	1	0	1
Place Value	26	25	24	23	22	21	20

=1×2⁶+0×2⁵+1×2⁴+0×2³+1×2²+0×2¹+1×2⁰
=64+0+16+0+4+0+1
=(85)₁₀
∴ (1010101)₂ = (85)₁₀

3. Convert the following decimal numbers to binary numbers:

a. (44)₁₀=(?)₂

2	44	Rem
2	22	0
2	11	0
2	5	1
2	2	1
2	1	0
	0	1

∴ (44)₁₀ = (101100)₂
b. (404)₁₀=(?)₂

2	404	Rem
2	202	0
2	101	0
2	50	1
2	25	0
2	12	1
2	6	0
2	3	0
2	1	1
	0	1

∴ (404)₁₀ = (110010100)₂
c. (100)₁₀=(?)₂

2	100	Rem
2	50	0
2	25	0
2	12	1
2	6	0
2	3	0
2	1	1
	0	1

∴ (100)₁₀ = (1100100)₂
d. (145)₁₀=(?)₂

2	145	Rem
2	72	1
2	36	0
2	18	0
2	9	0
2	4	1
2	2	0
2	1	0
	0	1

∴ (145)₁₀ = (10010001)₂
e. (325)₁₀=(?)₂

2	325	Rem
2	162	1
2	81	0
2	40	1
2	20	0
2	10	0
2	5	0
2	2	1
2	1	0
	0	1

∴ (325)₁₀ = (101000101)₂
f. (505)₁₀=(?)₂

2	505	Rem
2	252	1
2	126	0
2	63	0
2	31	1
2	15	1
2	7	1
2	3	1
2	1	1
	0	1

∴ (505)₁₀ = (111111001)₂

4. Convert the following decimal numbers to octal numbers:

a. (44)₁₀=(?)₈

8	44	Rem
8	5	4
	0	5

∴ (44)₁₀ = (54)₈
b. (104)₁₀=(?)₈

8	104	Rem
8	13	0
8	1	5
	0	1

∴ (104)₁₀ = (150)₈
c. (540)₁₀=(?)₈

8	540	Rem
8	67	4
8	8	3
8	1	0
	0	1

∴ (540)₁₀ = (1034)₈
d. (230)₁₀=(?)₈

8	230	Rem
8	28	6
8	3	4
	0	3

∴ (230)₁₀ = (346)₈
e. (190)₁₀=(?)₈

8	190	Rem
8	23	6
8	2	7
	0	2

∴ (190)₁₀ = (276)₈
f. (980)₁₀=(?)₈

8	980	Rem
8	122	4
8	15	2
8	1	7
	0	1

∴ (980)₁₀ = (1724)₈

5. Convert the following octal numbers to decimal numbers.

a. (740)₈ = (?)₁₀

Answer:

Face Value	7	4	0
Place Value	82	81	80

=7×8²+4×8¹+0×8⁰
=7x64+4x8+0x1
=448+32+0
=(480)₁₀
∴ (740)₈ = (480)₁₀

b. (570)₈ = (?)₁₀

Answer:

Face Value	5	7	0
Place Value	82	81	80

=5×8²+7×8¹+0×8⁰
=5x64+7x8+0x1
=320+56+0
=(376)₁₀
∴ (570)₈ = (376)₁₀

c. (250)₈ = (?)₁₀

Answer:

Face Value	2	5	0
Place Value	82	81	80

=2×8²+5×8¹+0×8⁰
=2x64+5x8+0x1
=128+40+0
=(168)₁₀
∴ (250)₈ = (168)₁₀

d. (1230)₈ = (?)₁₀

Answer:

Face Value	1	2	3	0
Place Value	83	82	81	80

=1×8³+2×8²+3×8¹+0×8⁰
=1x512+2x64+3x8+0x1
=512+128+24+0
=(664)₁₀
∴ (1230)₈ = (664)₁₀

e. (330)₈ = (?)₁₀

Answer:

Face Value	3	3	0
Place Value	82	81	80

=3×8²+3×8¹+0×8⁰
=3x64+3x8+0x1
=192+24+0
=(216)₁₀
∴ (330)₈ = (216)₁₀

f. (2020)₈ = (?)₁₀

Answer:

Face Value	2	0	2	0
Place Value	83	82	81	80

=2×8³+0×8²+2×8¹+0×8⁰
=2x512+0x64+2x8+0x1
=1024+0+16+0
=(1040)₁₀
∴ (2020)₈ = (1040)₁₀

6. Convert the following decimal numbers to hexadecimal numbers.

a. (25)₁₀=(?)₁₆

16	25	Rem
16	1	9
	0	1

∴ (25)₁₀ = (19)₁₆
b. (40)₁₀=(?)₁₆

16	40	Rem
16	2	8
	0	2

∴ (40)₁₀ = (28)₁₆
c. (45)₁₀=(?)₁₆

16	45	Rem
16	2	13 (D)
	0	2

∴ (45)₁₀ = (2D)₁₆
d. (46)₁₀=(?)₁₆

16	46	Rem
16	2	14 (E)
	0	2

∴ (25)₁₀ = (2E)₁₆
e. (145)₁₀=(?)₁₆

16	145	Rem
16	9	1
	0	9

∴ (145)₁₀ = (91)₁₆
f. (115)₁₀=(?)₁₆

16	115	Rem
16	7	3
	0	7

∴ (115)₁₀ = (73)₁₆

7. Convert the following hexa-decimal numbers to decimal numbers.

a. (130)₁₆ = (?)₁₀

Answer:

Face Value	1	3	0
Place Value	162	161	160

=1×16²+3×16¹+0×16⁰
=1x256+3x16+0x1
=256+48+0
=(304)₁₀
∴ (130)₁₆ = (304)₁₀

b. (1A7)₁₆ = (?)₁₀

Answer:

Face Value	1	A	7
Place Value	162	161	160

=1×16²+A×16¹+7×16⁰
=1x256+10x16+7x1
=256+160+7
=(423)₁₀
∴ (1A7)₁₆ = (423)₁₀

c. (240)₁₆ = (?)₁₀

Answer:

Face Value	2	4	0
Place Value	162	161	160

=2×16²+4×16¹+0×16⁰
=2x256+4x16+0x1
=512+64+0
=(576)₁₀
∴ (240)₁₆ = (576)₁₀

d. (2AB)₁₆ = (?)₁₀

Answer:

Face Value	2	A	B
Place Value	162	161	160

=2×16²+A×16¹+B×16⁰
=2x256+10x16+11x1
=512+160+11
=(683)₁₀
∴ (2AB)₁₆ = (683)₁₀

e. (4FD)₁₆ = (?)₁₀

Answer:

Face Value	4	F	D
Place Value	162	161	160

=4×16²+F×16¹+D×16⁰
=4x256+15x16+13x1
=1024+240+13
=(1277)₁₀
∴ (4FD)₁₆ = (1277)₁₀

f. (530F)₁₆ = (?)₁₀

Answer:

Face Value	5	3	0	F
Place Value	163	162	161	160

=5x16³+3×16²+0×16¹+F×16⁰
=5x4096+3x256+0x16+15x1
=20480+768+0+15
=(21263)₁₀
∴ (530F)₁₆ = (21263)₁₀

8. Convert the following octal numbers to binary numbers.

a. (77)₈ = (?)₂

Answer:

Octal Value	7	7
Binary value in 3 digits	111	111

∴ (77)₈ = (111111)₂

b. (76)₈ = (?)₂

Answer:

Octal Value	7	6
Binary value in 3 digits	111	110

∴ (76)₈ = (111110)₂

c. (47)₈ = (?)₂

Answer:

Octal Value	4	7
Binary value in 3 digits	100	111

∴ (47)₈ = (100111)₂

d. (17)₈ = (?)₂

Answer:

Octal Value	1	7
Binary value in 3 digits	001	111

∴ (17)₈ = (1111)₂

e. (65)₈ = (?)₂

Answer:

Octal Value	6	5
Binary value in 3 digits	110	101

∴ (65)₈ = (1110101)₂

f. (127)₈ = (?)₂

Answer:

Octal Value	1	2	7
Binary value in 3 digits	001	010	1111

∴ (127)₈ = (1010111)₂

9. Convert the following binary numbers to octal numbers.

a. (1010)₂ = (?)₈

Answer:

Binary numbers in group of 3 bits (From Right to left)	001	010
Octal Value	1	2

∴ (1010)₂ = (12)₈

b. (1110)₂ = (?)₈

Answer:

Binary numbers in group of 3 bits (From Right to left)	001	110
Octal Value	1	6

∴ (1010)₂ = (16)₈

c. (111)₂ = (?)₈

Answer:

Binary numbers in group of 3 bits (From Right to left)	111
Octal Value	7

∴ (111)₂ = (7)₈

d. (10111)₂ = (?)₈

Answer:

Binary numbers in group of 3 bits (From Right to left)	010	111
Octal Value	2	7

∴ (10111)₂ = (27)₈

e. (11101)₂ = (?)₈

Answer:

Binary numbers in group of 3 bits (From Right to left)	011	101
Octal Value	3	5

∴ (1101)₂ = (35)₈

f. (10111)₂ = (?)₈

Answer:

Binary numbers in group of 3 bits (From Right to left)	010	101
Octal Value	2	7

∴ (10111)₂ = (27)₈

10. Convert the following binary numbers to hexadecimal numbers.

a. (10101)₂ = (?)₁₆

Answer:

Binary numbers in group of 4 bits (From Right to left)	0001	0101
Hexadecimal Value	1	5

∴ (10101)₂ = (15)₁₆

b. (1111)₂ = (?)₁₆

Answer:

Binary numbers in group of 4 bits (From Right to left)	1111
Hexadecimal Value	15 (F)

∴ (1111)₂ = (F)₁₆

c. (11110)₂ = (?)₁₆

Answer:

Binary numbers in group of 4 bits (From Right to left)	0001	1110
Hexadecimal Value	1	14 (E)

∴ (10101)₂ = (1E)₁₆

d. (110)₂ = (?)₁₆

Answer:

Binary numbers in group of 4 bits (From Right to left)	0110
Hexadecimal Value	6

∴ (110)₂ = (6)₁₆

e. (1101)₂ = (?)₁₆

Answer:

Binary numbers in group of 4 bits (From Right to left)	1101
Hexadecimal Value	13 (D)

∴ (1101)₂ = (D)₁₆

f. (1111)₂ = (?)₁₆

Answer:

Binary numbers in group of 4 bits (From Right to left)	1111
Hexadecimal Value	15 (F)

∴ (1111)₂ = (F)₁₆

11. Convert the following hexa-decimal numbers to binary numbers.

a. (56)₁₆ = (?)₂

Answer:

Hexadecimal Value	5	6
Decimal Value	5	6
Binary value	0101	0110

∴ (56)₁₆ = (1010110)₂

b. (A27)₁₆ = (?)₂

Answer:

Hexadecimal Value	A	2	7
Decimal Value	10	2	7
Binary value	1010	0010	0111

∴ (A27)₁₆ = (101000100111)₂

c. (76)₁₆ = (?)₂

Answer:

Hexadecimal Value	7	6
Decimal Value	7	6
Binary value	0111	0110

∴ (76)₁₆ = (1110110)₂

d. (3B)₁₆ = (?)₂

Answer:

Hexadecimal Value	3	B
Decimal Value	3	11
Binary value	0011	1011

∴ (3B)₁₆ = (111011)₂

e. (A16)₁₆ = (?)₂

Answer:

Hexadecimal Value	A	1	6
Decimal Value	10	1	6
Binary value	1010	0001	0110

∴ (A16)₁₆ = (101000010110)₂

f. (F45)₁₆ = (?)₂

Answer:

Hexadecimal Value	F	4	5
Decimal Value	15	4	5
Binary value	1111	0100	0101

∴ (F45)₁₆ = (111101000101)₂

12. Convert the following octal numbers to hexadecimal numbers.

a. (45)₈ = (?)₁₆

Answer:

Octal number	4	5
3 bits binary number	100	101

∵ (45)₈ = (0010 0101)₂
Again,

Binary numbers in 4 bits	0010	0101
Hexadecimal number	2	5

∴(45)₈=(25)₁₆

b. (27)₈ = (?)₁₆

Answer:

Octal number	2	7
3 bits binary number	010	111

∵ (27)₈ = (0001 0111)₂
Again,

Binary numbers in 4 bits	0001	0111
Hexadecimal number	1	7

∴(27)₈=(17)₁₆

c. (136)₈ = (?)₁₆

Answer:

Octal number	1	3	6
3 bits binary number	001	011	110

∵ (136)₈ = (0101 1110)₂
Again,

Binary numbers in 4 bits	0101	1110
Hexadecimal number	5	14 (E)

∴(136)₈=(5E)₁₆

d. (100)₈ = (?)₁₆

Answer:

Octal number	1	0	0
3 bits binary number	001	000	000

∵ (100)₈ = (0100 0000)₂
Again,

Binary numbers in 4 bits	0100	0000
Hexadecimal number	4	0

∴(100)₈=(40)₁₆

e. (237)₈ = (?)₁₆

Answer:

Octal number	2	3	7
3 bits binary number	010	011	111

∵ (237)₈ = (1001 1111)₂
Again,

Binary numbers in 4 bits	1001	1111
Hexadecimal number	9	15 (F)

∴(237)₈=(9F)₁₆

f. (321)₈ = (?)₁₆

Answer:

Octal number	3	2	1
3 bits binary number	011	010	001

∵ (321)₈ = (1101 0001)₂
Again,

Binary numbers in 4 bits	1101	0001
Hexadecimal number	13 (D)	1

∴(321)₈=(D1)₁₆

13. Convert the following hexa-decimal numbers to octal numbers.

a. (45)₁₆ = (?)₈

Answer:

Hexadecimal number	4	5
Binary number in 4 bits	0100	0101

∵ (45)₁₆ = (001 000 101)₂
Again,

Binary numbers in 3 bits	001	000	101
Octal number	1	0	5

∴(45)₁₆=(105)₈

b. (245)₁₆ = (?)₈

Answer:

Hexadecimal number	2	4	5
Binary number in 4 bits	0010	0100	0101

∵ (245)₁₆ = (001 001 000 101)₂
Again,

Binary numbers in 3 bits	001	001	000	101
Octal number	1	1	0	5

∴(245)₁₆=(1105)₈

c. (1259)₁₆ = (?)₈

Answer:

Hexadecimal number	1	2	5	9
Binary number in 4 bits	0001	0010	0101	1001

∵ (1259)₁₆ = (001 001 001 011 001)₂
Again,

Binary numbers in 3 bits	001	001	001	011	001
Octal number	1	1	1	3	1

∴(1259)₁₆=(11131)₈

d. (A16)₁₆ = (?)₈

Answer:

Hexadecimal number	A	1	6
Binary number in 4 bits	1010	0001	0110

∵ (A16)₁₆ = (101 000 010 110)₂
Again,

Binary numbers in 3 bits	101	000	010	110
Octal number	5	0	2	6

∴(A16)₁₆=(5026)₈

e. (5B)₁₆ = (?)₈

Answer:

Hexadecimal number	5	B
Binary number in 4 bits	0101	1011

∵ (5B)₁₆ = (001 011 011)₂
Again,

Binary numbers in 3 bits	001	011	011
Octal number	1	3	3

∴(5B)₁₆=(133)₈

f. (F20)₁₆ = (?)₈

Answer:

Hexadecimal number	F	2	0
Binary number in 4 bits	1111	0010	0000

∵ (F20)₁₆ = (111 100 100 000)₂
Again,

Binary numbers in 3 bits	111	100	100	000
Octal number	7	4	4	0

∴(F20)₁₆=(7440)₈