Problem 26 - Thinking Mathematically

Question

Multiply in the indicated base. $$ \begin{array}{r} 34_{\text {five }} \\ \times \quad 3_{\text {five }} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result of \(34_5 \times 3_5\) is \(122_5\).

Step 1: Recognize the Base

Recognize that the numbers are in base-5. Hence, every digit should be between 0 and 4.

Step 2: Carry out the Multiplication

\[3_5 \times 4_5 = 12_{10} = 2_5 \text{ (and 1 to carry)}. \] The base-10 result of \(3 \times 4\) is 12, but in base-5, this translates to 2 with 1 to carry over to the next column.

Step 3: Continue the Multiplication

\[ 3_5 \times 3_5 + 1_{\text{(carried over)}} = 10_{10} = 20_5 \text{ (and nothing to carry)}. \] In base-10, the result is \(3 \times 3 + 1 = 10\). Translating this to base-5, we get \(20_5\) and there is nothing to carry.

Step 4: Write Down the Result

Write down the result under the line, lining up the digits according to their places in the column.

Key Concepts

Number Systems

When solving math problems, it's essential to understand the number system you're working within. We often use the base-10 number system, which uses digits 0 through 9. However, numbers can use different bases. In our exercise, we work in base-5.
The base of a number system defines two things:

• The number of unique digits, including zero, that can be used.
• How place value is determined.

In base-5, each digit represents a power of 5, much like in base-10, where each digit represents a power of 10.
This means a number like \( 34_5 \) is interpreted as:

• 3 multiplied by \(5^1\)
• plus 4 multiplied by \(5^0\)

A key understanding is that each digit can only be 0 through 4, as these are the five digits used in base-5. Recognizing the base ensures accurate computations.

Carry in Arithmetic

Carrying is a concept we often use in arithmetic to handle values exceeding the base during calculations.
When multiplying or adding numbers, if a digit in the result exceeds the largest digit allowed in the base, we "carry over" the excess value to the next column. Here’s the detailed view:

• In base-5, the permissible digit range is 0–4.
• If a multiplication results in a number like 12 (a decimal value), we convert it into base-5; hence, \(12_{10} = 22_5\).
• We keep the rightmost digit (in this case, 2) and carry the next digit (another 2) to the next higher position.

This procedure ensures each column only contains digits permissible in the number system, and the excess is reallocated to the next position, just like in base-10.
This simplifies adding up your results after a multiplication or addition operation.

Base Conversion

Base conversion is the method to translate numbers from one base to another, often to base-10, to simplify calculations. While implementing operations in unfamiliar bases, translating them into base-10 can make processes more intuitive.
For example, converting \(34_5\) into base-10 involves:

• 3 times \(5^1\) = 15.
• 4 times \(5^0\) = 4.
• Adding these values gives us 19 in base-10.

For conversion after calculations, you transform it back into the original base. Understanding this technique aids in translating intermediate results back into the required base, ensuring the overall answer is precise.
This method is essential when working with different numeral systems, as it offers clarity and ensures accurate computation.