Problem 19 - Thinking Mathematically

Question

Subtract in the indicated base. $$ \begin{array}{r} 563_{\text {seven }} \\ -164_{\text {seven }} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result of the subtraction is \( 265_{7} \).

Step 1: Set up the subtraction

The subtraction is set up as follows: \( \begin{array}{r} 563_{7} \ -164_{7} \ \end{array} \)

Step 2: Subtract the rightmost column

In the rightmost column, the computation is \(3_{7} - 4_{7}\). As in normal subtraction, when the top number is smaller than the bottom number, borrowing is needed. In base 7, borrowing will reduce the next left number by 1 but will increase the current top number by 7. Hence, the computation becomes \( (7+3)_{7} - 4_{7} = 6_{7} \).

Step 3: Subtract the middle column with borrowing

In the middle column, the computation is \(5_{7} - 6_{7}\), but we have borrowed 1 from '5_7'. So it now becomes \(4_{7} - 6_{7}\). Again, '4_7' is smaller than '6_7', and we need to borrow 1 from the leftmost column. After the borrowing, the computation becomes \( (7+4)_{7} - 6_{7} = 5_{7} \).

Step 4: Subtract the leftmost column with borrowing

In the leftmost column, the computation is \(4_{7} - 1_{7}\), but we have borrowed 1 from '6_7'. Now it becomes \(3_{7} - 1_{7} = 2_{7}\). This concludes the subtraction.

Key Concepts

Subtraction in Different Bases

Subtraction in different bases, such as base 7, essentially follows the same principles as subtraction in base 10. However, we work with different sets of numbers. In base 10, we have digits ranging from 0 to 9, while in base 7, the digits range from 0 to 6.

• When performing subtraction in any base, align the numbers just like in base 10, starting from the rightmost column and moving to the left.
• If the top digit is smaller than the bottom digit in a column, borrowing from the next column is necessary, just as in regular base 10 subtraction.

To understand base subtraction better, picture it like dealing with different amounts of a currency. Subtracting in base 7 is like dealing with units that 'carry over' after 6 instead of 9.

Number Systems

Number systems are ways of representing numbers. Our usual counting system is decimal or base 10, using the digits 0 through 9. But there are many other systems, such as binary (base 2), octal (base 8), and base 7, as used in this exercise.

• A digit's place in a number determines its actual value in the number based on the system's base. For base 7, each digit is multiplied by powers of 7.
• The number 563 in base 7 is computed as \(5 \times 7^2 + 6 \times 7^1 + 3 \times 7^0\).

Understanding number systems is crucial as it allows us to perform arithmetic operations correctly by grasping what each position in the number contributes to its value. This understanding is key when working with applications in computing and digital circuits, which often use bases like binary.

Borrowing in Arithmetic

Borrowing in arithmetic is a technique used when subtracting a larger digit from a smaller digit in any column. This is commonly seen in base 10 subtraction and is similarly applied in other bases.

• When borrowing in base 7, you take 1 from the next left column, effectively adding 7 (the base) to the current column.
• For instance, if you need to subtract 4 from 3 in base 7, borrow 1 from the next column, turning the 3 into \((7+3)_{7} = 10_{7}\).
• This results in a straightforward subtraction: 10 minus 4 equals 6 in base 7 notation.

Effective borrowing is vital as it ensures accuracy in your calculations and helps to avoid common mistakes during subtraction involving multiple digits and columns.