Convert the following binary numbers to equivalent decimal numbers. • (a) (1001)2 • (b) (10011)2 • (c) (01101101)2 • (d) (10001101)2 • (e) (11111111)2 • (f) (00111011)2 • (g) (101110010001)2 • (h) (11011.101)2 2. Convert the following decimal numbers to equivalent binary numbers. • (a) (45)10 • (b) (62)10 • (c) (255)10 • (d) (256)10 • (e) (2356)10 • (f) (4195)103. Convert the following octal numbers to equivalent decimal numbers. • (a) (45)8 • (b) (2173)8 4. Convert the following decimal numbers to equivalent octal numbers. • (a) (19)10 • (b) (332)10 • (c) (513)10 5. Convert the following hexadecimal numbers to equivalent decimal numbers. • (a) (B6)16 • (b) (1FF)16 • (c) (2D3D)16 6. Convert the following decimal numbers to equivalent hexadecimal numbers. • (a) (19)10 • (b) (332)10 • (c) (513)107. Convert the following binary numbers to equivalent octal numbers. • (a) (10110)2 • (b) (101101110)2 • (c) (10110111)2 8. Convert the following binary numbers to equivalent hexadecimal numbers. • (a) (10110)2 • (b) (101101110)2 • (c) (10110111)2
The Correct Answer and Explanation is :
Here are the answers and explanations for each conversion:
1. Convert the following binary numbers to equivalent decimal numbers:
- (a) (1001)₂
Decimal: ( 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 0 + 0 + 1 = 9 ) - (b) (10011)₂
Decimal: ( 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 16 + 0 + 0 + 2 + 1 = 19 ) - (c) (01101101)₂
Decimal: ( 0 \times 2^7 + 1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 64 + 32 + 0 + 8 + 4 + 0 + 1 = 109 ) - (d) (10001101)₂
Decimal: ( 1 \times 2^7 + 0 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 128 + 8 + 4 + 1 = 141 ) - (e) (11111111)₂
Decimal: ( 1 \times 2^7 + 1 \times 2^6 + 1 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255 ) - (f) (00111011)₂
Decimal: ( 0 \times 2^7 + 0 \times 2^6 + 1 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 32 + 16 + 8 + 2 + 1 = 59 ) - (g) (101110010001)₂
Decimal: ( 1 \times 2^{11} + 0 \times 2^{10} + 1 \times 2^9 + 1 \times 2^8 + 1 \times 2^7 + 0 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 2048 + 512 + 256 + 128 + 16 + 1 = 2961 ) - (h) (11011.101)₂
Decimal: ( 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 16 + 8 + 0 + 2 + 1 = 27 )
For the fractional part: ( 1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} = 0.5 + 0 + 0.125 = 0.625 )
Total: ( 27 + 0.625 = 27.625 )
2. Convert the following decimal numbers to equivalent binary numbers:
- (a) (45)₁₀
Binary: ( 45 \div 2 = 22 ) remainder 1
( 22 \div 2 = 11 ) remainder 0
( 11 \div 2 = 5 ) remainder 1
( 5 \div 2 = 2 ) remainder 1
( 2 \div 2 = 1 ) remainder 0
( 1 \div 2 = 0 ) remainder 1
Binary: (101101)₂ - (b) (62)₁₀
Binary: (111110)₂ - (c) (255)₁₀
Binary: (11111111)₂ - (d) (256)₁₀
Binary: (100000000)₂ - (e) (2356)₁₀
Binary: (100100101100)₂ - (f) (4195)₁₀
Binary: (1000001101011)₂
3. Convert the following octal numbers to equivalent decimal numbers:
- (a) (45)₈
Decimal: ( 4 \times 8^1 + 5 \times 8^0 = 32 + 5 = 37 ) - (b) (2173)₈
Decimal: ( 2 \times 8^3 + 1 \times 8^2 + 7 \times 8^1 + 3 \times 8^0 = 1024 + 64 + 56 + 3 = 1147 )
4. Convert the following decimal numbers to equivalent octal numbers:
- (a) (19)₁₀
Octal: (23)₈ - (b) (332)₁₀
Octal: (514)₈ - (c) (513)₁₀
Octal: (777)₈
5. Convert the following hexadecimal numbers to equivalent decimal numbers:
- (a) (B6)₁₆
Decimal: ( 11 \times 16^1 + 6 \times 16^0 = 176 + 6 = 182 ) - (b) (1FF)₁₆
Decimal: ( 1 \times 16^2 + 15 \times 16^1 + 15 \times 16^0 = 256 + 240 + 15 = 511 ) - (c) (2D3D)₁₆
Decimal: ( 2 \times 16^3 + 13 \times 16^2 + 3 \times 16^1 + 13 \times 16^0 = 8192 + 3328 + 48 + 13 = 10581 )
6. Convert the following decimal numbers to equivalent hexadecimal numbers:
- (a) (19)₁₀
Hexadecimal: (13)₁₆ - (b) (332)₁₀
Hexadecimal: (14C)₁₆ - (c) (513)₁₀
Hexadecimal: (201)₁₆
7. Convert the following binary numbers to equivalent octal numbers:
- (a) (10110)₂
Octal: (26)₈ - (b) (101101110)₂
Octal: (266)₈ - (c) (10110111)₂
Octal: (267)₈
8. Convert the following binary numbers to equivalent hexadecimal numbers:
- (a) (10110)₂
Hexadecimal: (16)₁₆ - (b) (101101110)₂
Hexadecimal: (BE)₁₆ - (c) (10110111)₂
Hexadecimal: (B7)₁₆
Explanation:
In binary-to-decimal and other conversions, we use the positional number system, where the value of each digit is determined by the base raised to the power of the position. For example, in binary, each digit is a power of 2, and in octal, it is a power of 8, and in hexadecimal, it is a power of 16. To convert between bases, you can either break the number down into parts (like in binary to decimal) or use division (like in decimal to binary).