Problem 54 Describe how to add two numbers ... [FREE SOLUTION]

Chapter 4: Problem 54

Describe how to add two numbers in a base other than ten. How do you express and record the sum of numbers in a column if that sum exceeds the base?

Short Answer

Expert verified

To add two numbers in a base other than ten, perform the addition column by column from right to left (as in base-ten addition) and carry forward the 'excess' every time the sum exceeds the base. In our example where we added 2312 base 5 and 1234 base 5, we obtained 3541 in base 5.

Step by step solution

Comprehending the Base Number System

In the base-ten system used in everyday arithmetic, an excess of ten in any column is carried over to the next higher column. Similarly in any base-b system, where b is an integer greater than one, an excess of b from the addition of numbers in any column will need to be carried over to the next higher column.

Executing the Addition Process

Let's take an example. Suppose we want to add 2312 and 1234 in base 5. Write these two numbers down under each other, aligning each column. The numbers are written down as if you were performing addition in base ten (i.e., column by column from right to left). The number in the rightmost column (units) will be added first, proceeding towards the left. When the sum of the numbers exceeds the base, the 'excess' is carried forward to the next column.

Calculating and Carrying Over the Excess

Now, calculate the sum of the two rightmost digits. In this case, it is 2+4 = 6 (in base ten). Since 6 is more than 5 (our base), subtract 5 from 6, which gives 1. So, 1 will be the rightmost digit in the final sum. The 'excess' (1, because 6 = 5*1 + 1) is carried forward to the next column. Repeat this process for each column until the leftmost column.

Finalizing the Sum

Continue adding the numbers column by column, carrying over the excess every time the sum of numbers exceeds the base. In the case of our example (2312 base 5 + 1234 base 5), the final sum would be 3541 in the base-5 system.

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91影视

Describe how to add two numbers in a base other than ten. How do you express and record the sum of numbers in a column if that sum exceeds the base?

Short Answer

Step by step solution

Comprehending the Base Number System

Executing the Addition Process

Calculating and Carrying Over the Excess

Finalizing the Sum

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