Calculate the value of $321_{\text {five }} \times 42_{\text {five }}$
Answer (1)
Convert 32 1 five and 4 2 five to base ten: 32 1 five = 8 6 ten and 4 2 five = 2 2 ten .
Multiply the base ten numbers: 8 6 ten × 2 2 ten = 189 2 ten .
Convert the result back to base five: 189 2 ten = 3003 2 five .
The final answer is 3003 2 five .
Explanation
Understanding the Problem We need to calculate 32 1 five × 4 2 five . Since these numbers are in base five, we'll perform the multiplication using base five rules. Alternatively, we can convert the numbers to base ten, multiply them, and then convert the result back to base five. Let's use the latter approach.
Converting to Base Ten First, convert 32 1 five to base ten: 32 1 five = ( 3 × 5 2 ) + ( 2 × 5 1 ) + ( 1 × 5 0 ) = ( 3 × 25 ) + ( 2 × 5 ) + ( 1 × 1 ) = 75 + 10 + 1 = 8 6 ten Next, convert 4 2 five to base ten: 4 2 five = ( 4 × 5 1 ) + ( 2 × 5 0 ) = ( 4 × 5 ) + ( 2 × 1 ) = 20 + 2 = 2 2 ten .
Multiplying in Base Ten Now, multiply the base ten numbers: 8 6 ten × 2 2 ten = 189 2 ten .
Converting Back to Base Five Finally, convert 189 2 ten back to base five. We do this by repeatedly dividing by 5 and noting the remainders:
1892 ÷ 5 = 378 remainder 2 378 ÷ 5 = 75 remainder 3 75 ÷ 5 = 15 remainder 0 15 ÷ 5 = 3 remainder 0 3 ÷ 5 = 0 remainder 3
Reading the remainders from bottom to top, we get 3003 2 five .
Final Answer Therefore, 32 1 five × 4 2 five = 3003 2 five .
Examples
Base arithmetic is useful in computer science for understanding how different number systems work. For example, computers use binary (base 2), but sometimes it's easier for humans to read in hexadecimal (base 16). Converting between bases helps in understanding data representation in computers. Imagine you are designing a communication protocol where data needs to be encoded in base 5 for efficient transmission. Multiplying numbers in base 5, as we did here, becomes essential to calculate packet sizes or data throughput.
