If the number 235 in decimal system is converted into binary system, then what is the resulting number?
(11101011) 2
Converting Decimal Number 235 to Binary System
Converting a number from the decimal system (base-10) to the binary system (base-2) is a fundamental concept in number systems. The most common method for converting a whole decimal number to binary is the repeated division-by-2 method.
Here’s how to convert the decimal number 235 into its binary equivalent:
We repeatedly divide the decimal number by 2 and record the remainder at each step. The remainders, read from bottom to top, form the binary number.
| Division | Result | Remainder |
|---|---|---|
| \(235 \div 2\) | \(117\) | \(1\) |
| \(117 \div 2\) | \(58\) | \(1\) |
| \(58 \div 2\) | \(29\) | \(0\) |
| \(29 \div 2\) | \(14\) | \(1\) |
| \(14 \div 2\) | \(7\) | \(0\) |
| \(7 \div 2\) | \(3\) | \(1\) |
| \(3 \div 2\) | \(1\) | \(1\) |
| \(1 \div 2\) | \(0\) | \(1\) |
Reading the remainders from the bottom up, we get the binary number \(11101011\).
Therefore, the decimal number 235 is equal to \(11101011\) in binary.
We can verify this by converting the binary number \(11101011_2\) back to decimal:
\[ \begin{align*} (11101011)_2 &= 1 \times 2^7 + 1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 \\ &= 1 \times 128 + 1 \times 64 + 1 \times 32 + 0 \times 16 + 1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1 \\ &= 128 + 64 + 32 + 0 + 8 + 0 + 2 + 1 \\ &= 235 \end{align*} \]The conversion back to decimal 235 confirms our binary result is correct.
Revision Table: Decimal to Binary Conversion
| Concept | Description | Method for Conversion |
|---|---|---|
| Decimal System (Base-10) | Uses digits 0-9. Each position represents a power of 10. | Repeated division by the target base (2 for binary), recording remainders. |
| Binary System (Base-2) | Uses digits 0 and 1. Each position represents a power of 2. |
Additional Information on Number Systems
Number systems are ways of representing numbers. The most common are decimal (base-10), binary (base-2), octal (base-8), and hexadecimal (base-16). Computers internally use the binary system because it is easily implemented using electronic switches that are either on (representing 1) or off (representing 0).
Converting between different number bases is a crucial skill in computer science and digital electronics. While decimal to binary uses repeated division by 2, converting from binary back to decimal involves summing the products of each binary digit and the corresponding power of 2 based on its position. Other conversions, like decimal to hexadecimal or binary to octal/hexadecimal, use similar principles of base representation and division/multiplication.
Important Questions from Binary Operations
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What is (1000000001) 2– (0.0101) 2equal to?
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If (11101011) 2is converted to decimal system, then the resulting number is
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