$(1011)_{10} = (\,)_{2}
Answer (2)
Decompose the decimal number 1011 into a sum of powers of 2.
Express 1011 as 1 ⋅ 2 9 + 1 ⋅ 2 8 + 1 ⋅ 2 7 + 1 ⋅ 2 6 + 1 ⋅ 2 5 + 1 ⋅ 2 4 + 0 ⋅ 2 3 + 0 ⋅ 2 2 + 1 ⋅ 2 1 + 1 ⋅ 2 0 .
Read the coefficients of the powers of 2 to form the binary representation.
The binary representation of ( 1011 ) 10 is ( 1111110011 ) 2 .
Explanation
Problem Analysis We are asked to convert the decimal number ( 1011 ) 10 to its binary equivalent. This means we need to express 1011 as a sum of powers of 2.
Finding the Largest Power of 2 The largest power of 2 less than or equal to 1011 is 2 9 = 512 . So, we can write 1011 = 512 + ( 1011 − 512 ) = 512 + 499 .
Continuing the Decomposition Now, we find the largest power of 2 less than or equal to 499. This is 2 8 = 256 . So, 499 = 256 + ( 499 − 256 ) = 256 + 243 . Thus, 1011 = 512 + 256 + 243 .
Further Decomposition Next, we find the largest power of 2 less than or equal to 243. This is 2 7 = 128 . So, 243 = 128 + ( 243 − 128 ) = 128 + 115 . Thus, 1011 = 512 + 256 + 128 + 115 .
Continuing Decomposition Now, we find the largest power of 2 less than or equal to 115. This is 2 6 = 64 . So, 115 = 64 + ( 115 − 64 ) = 64 + 51 . Thus, 1011 = 512 + 256 + 128 + 64 + 51 .
Continuing Decomposition Now, we find the largest power of 2 less than or equal to 51. This is 2 5 = 32 . So, 51 = 32 + ( 51 − 32 ) = 32 + 19 . Thus, 1011 = 512 + 256 + 128 + 64 + 32 + 19 .
Continuing Decomposition Now, we find the largest power of 2 less than or equal to 19. This is 2 4 = 16 . So, 19 = 16 + ( 19 − 16 ) = 16 + 3 . Thus, 1011 = 512 + 256 + 128 + 64 + 32 + 16 + 3 .
Final Decomposition Now, we find the largest power of 2 less than or equal to 3. This is 2 1 = 2 . So, 3 = 2 + ( 3 − 2 ) = 2 + 1 . Thus, 1011 = 512 + 256 + 128 + 64 + 32 + 16 + 2 + 1 .
Binary Representation We can write this as 1011 = 1 ⋅ 2 9 + 1 ⋅ 2 8 + 1 ⋅ 2 7 + 1 ⋅ 2 6 + 1 ⋅ 2 5 + 1 ⋅ 2 4 + 0 ⋅ 2 3 + 0 ⋅ 2 2 + 1 ⋅ 2 1 + 1 ⋅ 2 0 . Therefore, the binary representation is ( 1111110011 ) 2 .
Final Answer Alternatively, using a python tool, we find that the binary representation of 1011 is ( 1111110011 ) 2 .
Examples
Binary numbers are the language of computers. Every piece of data, from the text you read to the images you see, is stored as a series of 0s and 1s. Converting decimal numbers to binary is essential in computer science for understanding how computers process and store information. For example, if you want to represent the number of students in a class (say, 1011) in a computer program, you would need to convert it to its binary equivalent.
The decimal number (1011) {10} can be converted to binary by expressing it as a sum of powers of 2. The binary representation of (1011) {10} is (1111110011)_{2}. This involves finding the coefficients for each power of 2 that sum up to 1011.
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