Question
Demonstrate your understanding of the binary representation of data by expressing the year of your birth (or another significant year) in binary form. Explain how you arrived at your answer.
Answers
GPT-4o mini
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
1. **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.
2. **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.
3. **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
1. Divide 2023 by 2:
- 2023 ÷ 2 = 1011 remainder 1
2. Divide 1011 by 2:
- 1011 ÷ 2 = 505 remainder 1
3. Divide 505 by 2:
- 505 ÷ 2 = 252 remainder 1
4. Divide 252 by 2:
- 252 ÷ 2 = 126 remainder 0
5. Divide 126 by 2:
- 126 ÷ 2 = 63 remainder 0
6. Divide 63 by 2:
- 63 ÷ 2 = 31 remainder 1
7. Divide 31 by 2:
- 31 ÷ 2 = 15 remainder 1
8. Divide 15 by 2:
- 15 ÷ 2 = 7 remainder 1
9. Divide 7 by 2:
- 7 ÷ 2 = 3 remainder 1
10. Divide 3 by 2:
- 3 ÷ 2 = 1 remainder 1
11. 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**.
### Conversion Process from Decimal to Binary
1. **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.
2. **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.
3. **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
1. Divide 2023 by 2:
- 2023 ÷ 2 = 1011 remainder 1
2. Divide 1011 by 2:
- 1011 ÷ 2 = 505 remainder 1
3. Divide 505 by 2:
- 505 ÷ 2 = 252 remainder 1
4. Divide 252 by 2:
- 252 ÷ 2 = 126 remainder 0
5. Divide 126 by 2:
- 126 ÷ 2 = 63 remainder 0
6. Divide 63 by 2:
- 63 ÷ 2 = 31 remainder 1
7. Divide 31 by 2:
- 31 ÷ 2 = 15 remainder 1
8. Divide 15 by 2:
- 15 ÷ 2 = 7 remainder 1
9. Divide 7 by 2:
- 7 ÷ 2 = 3 remainder 1
10. Divide 3 by 2:
- 3 ÷ 2 = 1 remainder 1
11. 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**.