Question - Base 7 Number Subtraction
Solution:
The image shows a math problem asking to determine the difference between two numbers in base 7. The problem is:$$ 6412_{seven} - 5603_{seven} $$To find the difference, we'll subtract the numbers just like we do in base 10, but we need to remember that we're working in base 7, which means that instead of "carrying over" when we reach 10 as in base 10, we do so when we reach 7. Here's a step-by-step subtraction:Starting from the right (the units place):2 (in base 7) - 3 (in base 7) cannot be done without borrowing since 2 is less than 3. In base 7, borrowing means that we take 1 from the next higher place value, which is normally equivalent to the base we're using, so we take 7 (in base 7) and add it to the 2. Before we do that, we reduce 1 in the next column to the left.So the number becomes:$$ (6402_{seven} + 7_{seven}) - 5603_{seven} $$,which is,$$ 640(2+7)_{seven} - 5603_{seven} $$,or,$$ 6409_{seven} - 5603_{seven} $$.Now, let's try subtracting again:Units place:9 (in base 7) - 3 (in base 7) = 6 (in base 7)Tens place:0 (in base 7) - 0 (in base 7) = 0 (in base 7)Hundreds place:4 (in base 7) - 6 (in base 7) cannot be done without borrowing. Borrowing 1 from the next column, the thousands place, we have:$$ 5(40_{seven} + 7_{seven}) - 5603_{seven} $$,which is,$$ 5(47_{seven}) - 5603_{seven} $$.Now, the subtraction looks like this:Hundreds place:$$ 47_{seven} - 6_{seven} $$ = $$ 41_{seven} $$ (we converted the hundreds place, 4, to 47 by borrowing 1 from the thousands column and reduced the thousands place by 1).Thousands place:5 - 5 (in base 7) = 0 (in base 7)So, adding the results from each place, we get:$$ 6412_{seven} - 5603_{seven} = 406_{seven} $$.The answer is $$ 406_{seven} $$.
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