Convert $12_{10}$ to binary number.
Answer (2)
To convert 1 2 10 to binary, we find the powers of 2 that sum up to 12: 8 ( 2 3 ) + 4 ( 2 2 ) . This gives us the binary representation 110 0 2 .
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Find the largest power of 2 less than or equal to 12, which is 2 3 = 8 .
Subtract 8 from 12, resulting in 4.
Recognize that 4 is 2 2 .
Express 12 as a sum of powers of 2: 12 = 1 ⋅ 2 3 + 1 ⋅ 2 2 + 0 ⋅ 2 1 + 0 ⋅ 2 0 , so the binary representation is 110 0 2 .
Explanation
Understanding the Problem We are asked to convert the decimal number 1 2 10 to its binary equivalent. Binary numbers are base-2 numbers, meaning they are expressed using only 0s and 1s. To convert a decimal number to binary, we need to express it as a sum of powers of 2.
Finding the Largest Power of 2 We need to find the largest power of 2 that is less than or equal to 12. Let's list some powers of 2: 2 0 = 1 2 1 = 2 2 2 = 4 2 3 = 8 2 4 = 16 Since 16 is greater than 12, the largest power of 2 that is less than or equal to 12 is 2 3 = 8 .
First Subtraction Now, we subtract 8 from 12: 12 − 8 = 4 . So, we have 12 = 8 + 4 .
Finding the Next Power of 2 Next, we find the largest power of 2 that is less than or equal to 4. From our list above, we see that 2 2 = 4 .
Second Subtraction Subtract 4 from 4: 4 − 4 = 0 . So, we have 12 = 8 + 4 + 0 .
Expressing as Sum of Powers of 2 Now we express 12 as a sum of powers of 2: 12 = 1 ⋅ 2 3 + 1 ⋅ 2 2 + 0 ⋅ 2 1 + 0 ⋅ 2 0 .
Writing the Binary Representation Finally, we write the coefficients of the powers of 2 as the binary representation: 110 0 2 .
Final Answer Therefore, the binary representation of 1 2 10 is 110 0 2 .
Examples
Binary numbers are fundamental in computer science. For example, when you save a document on your computer, the text, images, and formatting are all converted into binary code (0s and 1s). The computer uses these binary digits to store and process the information. Converting decimal numbers to binary is essential for understanding how computers handle data internally.
