Question

88–89. Binary numbersHumans use the ten digits 0 through 9 to form base-10 or decimal numbers, whereas computers calculate and store numbers internally as binary numbers—numbers consisting entirely of 0’s and 1’s. For this exercise, we consider binary numbers that have the form 0.b₁b₂b₃⋯, where each of the digits b₁, b₂, b₃, ⋯ is either 0 or 1. The base-10 representation of the binary number 0.b₁b₂b₃⋯ is the infinite seriesb₁ / 2¹ + b₂ / 2² + b₃ / 2³ + ⋯89. Computers can store only a finite number of digits and therefore numbers with nonterminating digits must be rounded or truncated before they can be used and stored by a computer.a. Find the base-10 representation of the binary number 0.001̅1.

Accepted Answer

Understand that the binary number 0.001\overline{1} means the digits after the decimal point start with 0, 0, 1, followed by an infinite repeating sequence of 1's. Express the number as the sum of two parts: the finite part 0.001 in binary, and the infinite repeating part 0.000\overline{1} starting from the fourth digit. Convert the finite part 0.001 to base-10 by summing the values of each digit: $$b_1/2^1$ + b_2$/2^2$$ + $$b_3/2^3$ where b_1$=0$$, $$b_2=0$, and b_3$=1. $$Represent the infinite repeating part as a geometric series starting at the fourth digit: $$\sum_{n=4}^\infty \frac{1}{2^n}$$, since the repeating digit is 1 at every position from the fourth digit onward. Calculate the sum of the geometric series using the formula for an infinite geometric series $$S = \frac{a}{1 - r}$$, where $$a is $$the first term and $$r$ is the common ratio, then add this to the finite part to get the full base-10 representation.

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Key Concepts

Binary Number Representation

Infinite Geometric Series

Conversion of Repeating Binary Fractions to Decimal