To demonstrate my understanding of binary representation, I will express the year 2023 (a significant year) in binary form and explain the conversion process.
Conversion Process from Decimal to Binary
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Understanding Binary System:
- The binary system is base-2, which means it uses only two digits: 0 and 1. Each digit represents a power of 2, starting from \(2^0\) on the right.
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Conversion Method:
- To convert a decimal number (like 2023) to binary, you can use the method of repeated division by 2, keeping track of the remainders.
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Steps:
- Divide the number by 2 and record the quotient and the remainder.
- Continue dividing the quotient by 2 until you get a quotient of 0.
- The binary representation is the string of remainders read from bottom to top (last remainder is the most significant bit).
Example: Converting 2023 to Binary
- Divide 2023 by 2:
- 2023 ÷ 2 = 1011 remainder 1
- Divide 1011 by 2:
- 1011 ÷ 2 = 505 remainder 1
- Divide 505 by 2:
- 505 ÷ 2 = 252 remainder 1
- Divide 252 by 2:
- 252 ÷ 2 = 126 remainder 0
- Divide 126 by 2:
- 126 ÷ 2 = 63 remainder 0
- Divide 63 by 2:
- 63 ÷ 2 = 31 remainder 1
- Divide 31 by 2:
- 31 ÷ 2 = 15 remainder 1
- Divide 15 by 2:
- 15 ÷ 2 = 7 remainder 1
- Divide 7 by 2:
- 7 ÷ 2 = 3 remainder 1
- Divide 3 by 2:
- 3 ÷ 2 = 1 remainder 1
- Divide 1 by 2:
- 1 ÷ 2 = 0 remainder 1
Now, arranging the remainders from bottom to top, we get the binary representation of 2023:
Binary Representation
- The binary representation of the year 2023 is 11111100111.
Verification
To verify, we can convert the binary number back to decimal:
- \(1 \times 2^{10} = 1024\)
- \(1 \times 2^9 = 512\)
- \(1 \times 2^8 = 256\)
- \(1 \times 2^7 = 128\)
- \(1 \times 2^6 = 64\)
- \(1 \times 2^5 = 32\)
- \(0 \times 2^4 = 0\)
- \(0 \times 2^3 = 0\)
- \(1 \times 2^2 = 4\)
- \(1 \times 2^1 = 2\)
- \(1 \times 2^0 = 1\)
Adding these values gives:
- \(1024 + 512 + 256 + 128 + 64 + 32 + 4 + 2 + 1 = 2023\)
Thus, I have confirmed that the binary representation of the year 2023 is 11111100111.
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