Final Answer:
To construct an addition table for base eight, you can create a table with rows and columns representing the digits 0 through 7. Each cell in the table will contain the sum of the corresponding row and column digits, following the rules of addition in base eight. Here's a simplified version:
Step-by-step explanation:
To perform subtraction with base eight (octal) numbers, we first need an addition table for octal, which involves the digits 0 through 7 only. Since we don't need to construct a complete addition table for these particular questions, I’ll address the subtractions directly, using the rules of octal subtraction.
Firstly, here's a quick reminder of how subtraction works in base eight (octal system):
1. In the octal system, each digit represents a power of 8.
2. If the digit in the minuend (the number being subtracted from) is less than the digit in the subtrahend (the number being subtracted), you need to borrow from the next higher position (which is equivalent to borrowing 8 in decimal terms).
Now, let us solve each of the given computations:
a) \( 573_8 - 77_8 \)
To subtract these octal numbers, we write them one on top of the other, aligned to the right:
573
- 77
Starting with the rightmost digit:
3 - 7 cannot be done because 3 is less than 7. We need to borrow from the next column (5 becomes 4, and we add 8 to the 3, making it 11).
4(13)
- 77
Now we perform the subtraction digit by digit:
5th column (13 - 7): 13 - 7 = 6
4th column (4 - 7): We can't subtract 7 from 4, so we need to borrow again. However, since there is no digit to borrow from in the next column (because this is the first column), we consider it as 0 and borrow from it, turning 0 into 7 and decreasing the 4 to 3.
7(11)
- 77
Now, continue with the subtraction:
4th column (3 - 0): 3 - 0 = 3
5th column: Already calculated as 6
Now we have:
573
- 77
-----
364
So, \( 573_8 - 77_8 = 364_8 \)
b) \( 765_8 - 76_8 \)
Similar to the previous computation, we write down the numbers:
765
-76
Starting from the rightmost digit:
5 - 6 cannot be done because 5 is less than 6. Borrow from the next column (6 becomes 5 and we add 8 to the 5, making it 13).
5(13)
-76
Now we perform the subtraction digit by digit:
5th column (13 - 6): 13 - 6 = 7
4th column (5 - 7): We can't subtract 7 from 5, so we need to borrow again from the leftmost column (7 becomes 6, and we add 8 to the 5, making it 13).
6(13)
- 76
Now, continue with the subtraction:
4th column (13 - 7): 13 - 7 = 6
6th column (6 - 0): 6 - 0 = 6
Our final result is:
765
-76
-----
667
So, \( 765_8 - 76_8 = 667_8 \)
The final answer is \( 765_8 - 76_8 = 667_8 \)
These are the results of the octal subtraction problems using the base eight arithmetic system.