Express 0.679 as a binary decimal
To express the decimal 0.679 as a binary decimal, we can use the method of multiplying by 2 and extracting the integer part.
1. **Multiply by 2:**
\( 0.679 \times 2 = 1.358 \)
Integer part: **1**
2. **Take the fractional part (0.358) and multiply by 2:**
\( 0.358 \times 2 = 0.716 \)
Integer part: **0**
3. **Take the fractional part (0.716) and multiply by 2:**
\( 0.716 \times 2 = 1.432 \)
Integer part: **1**
4. **Take the fractional part (0.432) and multiply by 2:**
\( 0.432 \times 2 = 0.864 \)
Integer part: **0**
5. **Take the fractional part (0.864) and multiply by 2:**
\( 0.864 \times 2 = 1.728 \)
Integer part: **1**
6. **Take the fractional part (0.728) and multiply by 2:**
\( 0.728 \times 2 = 1.456 \)
Integer part: **1**
7. **Take the fractional part (0.456) and multiply by 2:**
\( 0.456 \times 2 = 0.912 \)
Integer part: **0**
8. **Take the fractional part (0.912) and multiply by 2:**
\( 0.912 \times 2 = 1.824 \)
Integer part: **1**
9. **Take the fractional part (0.824) and multiply by 2:**
\( 0.824 \times 2 = 1.648 \)
Integer part: **1**
10. **Take the fractional part (0.648) and multiply by 2:**
\( 0.648 \times 2 = 1.296 \)
Integer part: **1**
11. **Take the fractional part (0.296) and multiply by 2:**
\( 0.296 \times 2 = 0.592 \)
Integer part: **0**
12. **Finally, \( 0.592 \times 2 = 1.184 \)**
Integer part: **1**
Gathering the integer parts from each step gives us the binary representation:
**0.679 (decimal) ≈ 0.1010111101 (binary) rounding after 10 bits.**
In summary, the binary decimal approximation of 0.679 is approximately **0.1010111101**.
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