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Chapter 4: Problem 69

Explain how to determine the place values for a four-digit numeral in base six.

Short Answer

Expert verified
The place values for the base six numeral 2145 are 5, 24, 36 and 432, greatest to smallest from left to right. 5 is the value in the exact place, 24 in the units place, 36 in the squares place and 432 in the cubics place. Summing these values gives 497 in base ten.

Step by step solution

01

Understand the Base Six Numeral System

In the base-six numeral system, each place in a four-digit numeral is a power of six. Going from right to left, the 'exact place' represents \(6^0\), the 'units place' represents \(6^1\), the 'squares place' represents \(6^2\), and the 'cubics place' represents \(6^3\).
02

Assign Place Values to a Given Four-Digit Numeral

Let's say the given four-digit numeral is 2145 (base 6). To assign place values, map each digit to the corresponding power of six. Thus, the four digits are:5 (the exact place) - this represents 5*\(6^0\) which equals 5. 4 (the units place) - this represents 4*\(6^1\) which equals 24. 1 (the squares place) - this represents 1*\(6^2\) which equals 36. 2 (the cubics place) - this represents 2*\(6^3\) which equals 432. So, 2145 (base 6) = 432+36+24+5 (base 10).
03

Validate the Results

Finally, it's important to validate the results. Add together the base ten equivalents found in the previous step to see if they make sense within the base six numeral system. In this case, adding together 5, 24, 36, and 432 in base ten equals 497. So, 2145 in base six equals 497 in base ten.

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