Convert the numeral to a numeral in base 10. B8312
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Convert the Numeral to a Numeral in Base 10**
**Given: \( B83_{12} \)**
To convert the numeral \( B83_{12} \) to a base 10 numeral, follow these steps:
1. **Understand the Base 12 System:**
- In base 12, the digits range from 0 to B, where A represents 10 and B represents 11 in base 10.
2. **Break Down the Numeral:**
- The numeral \( B83_{12} \) consists of the digits B, 8, and 3.
3. **Convert Each Digit:**
- Convert B to 11 in base 10.
- 8 remains 8 in base 10.
- 3 remains 3 in base 10.
4. **Calculate the Base 10 Value:**
- Multiply each digit by 12 raised to the power of its position, starting from right (position 0):
\[
B83_{12} = (11 \times 12^2) + (8 \times 12^1) + (3 \times 12^0)
\]
- Calculate each term:
- \( 11 \times 12^2 = 11 \times 144 = 1584 \)
- \( 8 \times 12^1 = 8 \times 12 = 96 \)
- \( 3 \times 12^0 = 3 \times 1 = 3 \)
5. **Sum the Values:**
\[
1584 + 96 + 3 = 1683
\]
Therefore, \( B83_{12} \) is \( 1683 \) in base 10.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6727d57a-eab8-4a5b-bd1a-5eddc663e9c0%2F34a11af0-fbf4-4a62-b161-1b3a123063ce%2Fzuzaixo_processed.png&w=3840&q=75)
Transcribed Image Text:**Convert the Numeral to a Numeral in Base 10**
**Given: \( B83_{12} \)**
To convert the numeral \( B83_{12} \) to a base 10 numeral, follow these steps:
1. **Understand the Base 12 System:**
- In base 12, the digits range from 0 to B, where A represents 10 and B represents 11 in base 10.
2. **Break Down the Numeral:**
- The numeral \( B83_{12} \) consists of the digits B, 8, and 3.
3. **Convert Each Digit:**
- Convert B to 11 in base 10.
- 8 remains 8 in base 10.
- 3 remains 3 in base 10.
4. **Calculate the Base 10 Value:**
- Multiply each digit by 12 raised to the power of its position, starting from right (position 0):
\[
B83_{12} = (11 \times 12^2) + (8 \times 12^1) + (3 \times 12^0)
\]
- Calculate each term:
- \( 11 \times 12^2 = 11 \times 144 = 1584 \)
- \( 8 \times 12^1 = 8 \times 12 = 96 \)
- \( 3 \times 12^0 = 3 \times 1 = 3 \)
5. **Sum the Values:**
\[
1584 + 96 + 3 = 1683
\]
Therefore, \( B83_{12} \) is \( 1683 \) in base 10.
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