Convert $10111_{\text {rmo }}$ to base ten
A. $15_{\text {ten }}$
B. $22_{\text {ten }}$
C. $23_{\text {ten }}$
D. 24
Answer (2)
Express 1011 1 r in expanded form: r 4 + r 2 + r + 1 .
Assume r = 2 since the digits are 0 and 1.
Calculate 2 4 + 2 2 + 2 + 1 = 16 + 4 + 2 + 1 = 23 .
Therefore, 1011 1 2 = 23 .
Explanation
Understanding the Problem We are asked to convert the number 1011 1 r from base r to base 10. The digits used in the number are 0 and 1, so the base r must be greater than 1. We will express the number in expanded form using powers of r .
Converting to Base 10 The number 1011 1 r in base r can be written in expanded form as: ( 1 × r 4 ) + ( 0 × r 3 ) + ( 1 × r 2 ) + ( 1 × r 1 ) + ( 1 × r 0 ) Simplifying this expression, we get: r 4 + 0 + r 2 + r + 1 = r 4 + r 2 + r + 1
Finding the Value of r Now, we need to determine the value of r . Since the problem does not explicitly state the value of r , and the answer choices are all integers, we can assume that r is an integer. Also, since the digits in the number 1011 1 r are 0 and 1, the smallest possible value for r is 2. Let's assume r = 2 and see if it matches any of the answer choices.
Calculating with r=2 If r = 2 , then the base 10 equivalent is: 2 4 + 2 2 + 2 + 1 = 16 + 4 + 2 + 1 = 23 Since 23 is one of the answer choices, we can conclude that r = 2 .
Final Conversion Therefore, 1011 1 2 = 2 3 10 .
Final Answer The base ten equivalent of 1011 1 rmo is 23.
Examples
Base conversions are used in computer science to represent numbers in different formats. For example, computers use binary (base 2) to store and process data, while humans often use decimal (base 10) for everyday calculations. Converting between bases allows us to understand how computers represent numbers and perform calculations.
After converting 1011 1 r m o using base 2, we find that it equals 2 3 t e n . Hence, the correct choice is option C, 23. This process showcases how to translate numbers between numeral systems effectively.
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