Question

Suppose you were to extend the base 5 number system to include decimals. Set up the place value chart and use it to convert the decimal 0.2415 to base 10.

Answer 1

To convert the base 5 decimal 0.2415 to base 10, we calculate the base 10 equivalent for each digit based on its place value in base 5 and sum them up, arriving at 0.576 (base 10).

Step-by-step explanation:

To convert the base 5 decimal 0.2415 to base 10, we need to understand the place values in base 5. Unlike the base 10 system, where place values are powers of 10, in base 5 the place values are powers of 5. Therefore, the place values to the right of the decimal point in base 5 are: 1/5 (tenths), 1/25 (hundredths), 1/125 (thousandths), etc.

Using the place value chart for base 5, we can convert the number:

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• The first digit to the right of the decimal point is 2, which represents 2/5.
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• The second digit is 4, which represents 4/25.
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• The third digit is 1, which represents 1/125.
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• The fourth digit is 5, which represents 5/625 (since 625 is 54).

To find the base 10 equivalent, we multiply each digit by its respective place value and sum them up:

0.24155 = (2/5) + (4/25) + (1/125) + (5/625)

Now, calculating each term:

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• 2/5 = 0.4
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• 4/25 = 0.16
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• 1/125 = 0.008
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• 5/625 = 0.008

Adding these together gives us the base 10 equivalent:

0.24155 = 0.4 + 0.16 + 0.008 + 0.008

= 0.576 (base 10)