Can you Express the Solution in Base 7?

Number Theory Level 3

You are given that $x_1=0.333\ldots$ is a number in base 9. Similarly, $x_2= 0.333\ldots$ is a number in base 10. Then what is $x_1 - x_2$ in base 7?

Give your answer to 2 decimal places.

The answer is 0.02.

1 solution

Drex Beckman
Dec 27, 2015

We first need to convert 0.333... from base 9 to 10: $\sum_{i=1}^{\infty}\frac{3}{9^{n}} = \frac{3}{9^{1}}+\frac{3}{9^{2}}+\frac{3}{9^{3}}+\frac{3}{9^{4}}+...=0.375$ So now we have $0.375-0.333=0.0416666...$ , which we need to convert from base 10 to base 7: $0.041666*7*7=2.041666-2 = 0.04166*7*7 = 2.041666...$ This algorithm gives a result of 0.020202... rounded to two places: 0.02.

Can you Express the Solution in Base 7?

The answer is 0.02.

1 solution

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