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Chapter 4: Problem 55

Describe how to subtract two numbers in a base other than ten. How do you subtract a larger number from a smaller number in the same column?

Short Answer

Expert verified
The final result after subtracting 413 from 625 in base 7 is 1162 in base 7. In general, whenever a subtrahend is larger than the minuend in the same column, borrowing is necessary and done in the form of the base. The borrowing is then subtracted from the minuend to get the difference for that column.

Step by step solution

01

Understanding the Base

Firstly, establish the base of the given numerical system. The base is the number of distinct digits, including zero, used to represent numbers in that system. For instance, if you're operating in a base 7 system, digits would range from 0 to 6.
02

Simple Subtraction

In any base, if the minuend is larger in a given column, perform the subtraction directly. For example, if subtracting 413 from 625 in base 7, you can subtract the first column directly (5-3=2) to get a 2 in the rightmost column.
03

Borrowing When Subtracting

When the subtrahend is larger than the minuend in a given column, borrowing is required. In base 10, you would borrow a '1' from the next column. However, in bases other than 10, borrowing will take the form of the base. For example, in a base 7 system, when subtracting 413 from 625, the second column from the right needs to borrow: 2 (in 625) is less than 1 (in 413). So, 72 is borrowed from the 6 in the third column from the right in 625. This makes the 2 become 72 (in base 7), and 72 - 1 = 61 in base 7.
04

Continuing Subtraction and Borrowing

Continue to the next column and repeat Steps 2 and 3 if necessary. In our base 7 example, the third column from the right now has a 5 (6-72) in 625 and a 4 in 413, which can be subtracted directly. So, the third column from the right in the result is 1.
05

Review and Finish the Subtraction

Proceed to other columns using the same process until subtraction is completed across all the columns. The end result is the difference. For our base 7 example, the leftmost column gives 5 - 4 = 1 in base 7. So, the final result of subtraction is 1162 in base 7.

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