4.5.3 Repeating Decimals in Other Bases - Convert to Base 10

Converting repeating decimals in other bases to base-10 fractions.

Simple Form: $.xxx\dots_b$

Use the infinite geometric series formula:

$$.555\dots_8 = \frac{5}{8} + \frac{5}{64} + \dots = \frac{5}{8(1 - 1/8)} = \frac{5}{7}$$

Form: $.xyxyxy\dots_b$

1. Numerator: Convert two-digit number $xy$ to base-10
2. Denominator: $b^2 - 1$

Example: $.353535\dots_8$ → Numerator: $35_8 = 29_{10}$; Denominator: $64 - 1 = 63$ → 29/63

Form: $.xyyy\dots_b$

1. Numerator: Convert $xy$ to base-10 and subtract $x$
2. Denominator: $b(b - 1)$

Example: $.3555\dots_8$ → Numerator: $35_8 - 3_8 = 26_{10}$; Denominator: $8 \times 7 = 56$ → 13/28

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Problem Set 4.5.3

Practice these problems. Type your answer and press Enter to check: