Final answer:
To find the sum of the numbers 411, 1101, 431, and 1043 in base-five, we add each place value, adjusting for base-five by carrying over as appropriate. The result is obtained step by step, and the final answer in base-five is 23145.
Step-by-step explanation:
To find the sum using base-five pieces and show your regrouping, first we need to add the given numbers just as we would in the decimal system, but remembering that in base-five, we can only have digits 0-4 in each place value. Therefore, whenever a sum in a column is 5 or more, we need to regroup, or 'carry over', to the next higher place value. Here's how we would add the numbers provided:
- (a) 411
(b) 1101
(c) 431
(d) 1043
Starting from the rightmost digit, we add column by column:
- The ones place: 1 + 1 + 1 + 3 = 6 (base-10), which is 11 in base-five; we write down the 1 and carry over the 1 to the next column.
- The fives place: 1 (carried over) + 1 + 0 + 3 + 4 = 9 (base-10), which is 14 in base-five; again, we write down the 4 and carry over the 1.
- The twenty-fives place: 1 (carried over) + 1 + 1 + 0 + 0 = 3 (base-10); since this is less than 5, we write it as is in base-five.
- The one hundred twenty-fives place: 0 + 1 + 0 + 1 = 2 (base-10); so we write down 2 in base-five.
Placing all the digits together, we get the final answer in base-five:23145