The decimal number (127.25) 10, when converted to binary number, takes the form
(1111111.01) 2
Decimal to Binary Conversion Explained
Converting a decimal number with a fractional part to a binary number involves converting the integer part and the fractional part separately and then combining them using a radix point.
Converting the Integer Part: (127)10
To convert the integer part, (127)10, to binary, we use repeated division by 2. We record the remainders at each step. The binary representation is obtained by reading the remainders from bottom to top.
| Division | Quotient | Remainder |
|---|---|---|
| \( 127 \div 2 \) | \( 63 \) | \( 1 \) |
| \( 63 \div 2 \) | \( 31 \) | \( 1 \) |
| \( 31 \div 2 \) | \( 15 \) | \( 1 \) |
| \( 15 \div 2 \) | \( 7 \) | \( 1 \) |
| \( 7 \div 2 \) | \( 3 \) | \( 1 \) |
| \( 3 \div 2 \) | \( 1 \) | \( 1 \) |
| \( 1 \div 2 \) | \( 0 \) | \( 1 \) |
Reading the remainders from bottom to top (1111111), we find that \( (127)_{10} = (1111111)_{2} \).
Converting the Fractional Part: (0.25)10
To convert the fractional part, (0.25)10, to binary, we use repeated multiplication by 2. We record the integer part of the result at each step. The process continues until the fractional part becomes 0 or repeats. The binary representation is obtained by reading the integer parts from top to bottom.
| Multiplication | Result | Integer Part |
|---|---|---|
| \( 0.25 \times 2 \) | \( 0.50 \) | \( 0 \) |
| \( 0.50 \times 2 \) | \( 1.00 \) | \( 1 \) |
Since the fractional part became 0, we stop here. Reading the integer parts from top to bottom (01), we find that \( (0.25)_{10} = (0.01)_{2} \).
Combining Integer and Fractional Parts
Now, we combine the binary representations of the integer and fractional parts with a radix point:
\( (127.25)_{10} = (127)_{10} + (0.25)_{10} \)
\( (127.25)_{10} = (1111111)_{2} + (0.01)_{2} \)
\( (127.25)_{10} = (1111111.01)_{2} \)
Therefore, the decimal number (127.25)10 when converted to binary is (1111111.01)2.
Revision Table: Number System Conversions
| Concept | Description | Method for Decimal to Binary |
|---|---|---|
| Decimal Number System | Base-10 system using digits 0-9. Positional system where each digit's value depends on its position (powers of 10). | |
| Binary Number System | Base-2 system using digits 0 and 1. Positional system where each digit's value depends on its position (powers of 2). | |
| Integer Conversion | Converting the whole number part. | Repeatedly divide by 2 and record remainders. Read remainders from bottom up. |
| Fractional Conversion | Converting the part after the decimal point. | Repeatedly multiply by 2 and record the integer part of the result. Read integer parts from top down. |
| Radix Point | The point separating the integer part from the fractional part (decimal point in decimal, binary point in binary). | Used to combine the converted integer and fractional binary parts. |
Additional Information on Number Bases
Number systems are fundamental in computing. Different bases are used for various purposes.
- Binary (Base 2): Used internally by computers because electronic circuits can easily represent two states (on/off, high/low voltage, 0/1).
- Decimal (Base 10): The standard number system used by humans, based on ten fingers.
- Octal (Base 8): Uses digits 0-7. Sometimes used as a compact representation of binary numbers, as 3 binary digits can be represented by one octal digit (\( 2^3 = 8 \)).
- Hexadecimal (Base 16): Uses digits 0-9 and letters A-F (A represents 10, F represents 15). Commonly used in computing for memory addresses and data representation, as 4 binary digits can be represented by one hexadecimal digit (\( 2^4 = 16 \)).
Converting between these bases is a crucial skill in computer science and digital electronics.
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Important Questions from Binary Operations
Multiplication of 111 2by 101 2is
Let * be binary operation defined on R by \(\rm p * q=\frac{p+q}{2}, \forall \) p, q ∈ R. The operation is:
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A. 10101
B. 11110
C. 11101
D. 11001
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If G is the set of integer numbers, and a.b = a – b, ∀ a, b ∈ G, then G is
