Select the irrational numbers. Select all that apply.
[tex]$\sqrt{64}$[/tex]
[tex]$0.49449444944449 \ldots$[/tex]
[tex]$0.4 \overline{9}$[/tex]
[tex]$3.4641 \ldots$[/tex]
0.28125
[tex]$0.12345 \ldots$[/tex]
0
-2.8961
Answer (2)
Identify irrational numbers as non-terminating and non-repeating decimals.
Analyze each number in the list to determine if it can be expressed as a fraction of two integers.
Exclude numbers like 64 = 8 , 0.4 9 = 0.5 , 0.28125 , 0 , and − 2.8961 because they are rational.
Select the remaining numbers that are non-terminating and non-repeating decimals: 0.49449444944449 … , 3.4641 … , 0.12345 … , so the answer is 0.49449444944449 … , 3.4641 … , 0.12345 … .
Explanation
Problem Analysis We are given a list of numbers and asked to identify the irrational numbers among them. An irrational number is a number that cannot be expressed as a fraction q p , where p and q are integers and q = 0 . In other words, irrational numbers are non-terminating and non-repeating decimals. Let's examine each number in the list.
Examining Each Number
64 = 8 . This is an integer, and thus a rational number.
0.49449444944449 … This decimal appears to be non-repeating and non-terminating. Therefore, it is an irrational number.
0.4 9 = 0.49999 … = 0.5 = 2 1 . This is a rational number.
3.4641 … This decimal is non-repeating and non-terminating. Therefore, it is an irrational number.
0.28125 This is a terminating decimal, so it is a rational number.
0.12345 … This decimal is non-repeating and non-terminating. Therefore, it is an irrational number.
0 This is an integer, and thus a rational number.
− 2.8961 This is a terminating decimal, so it is a rational number.
Identifying Irrational Numbers Based on the analysis above, the irrational numbers in the given list are 0.49449444944449 … , 3.4641 … , and 0.12345 … .
Final Answer Therefore, the irrational numbers are 0.49449444944449 … , 3.4641 … , and 0.12345 … .
Examples
Irrational numbers are crucial in various fields such as physics and engineering. For instance, when calculating the precise area of a circle with radius 1, we use π , which is an irrational number. Similarly, in electrical engineering, irrational numbers appear when dealing with impedance calculations in AC circuits. Understanding irrational numbers helps in achieving accurate and reliable results in these applications.
The irrational numbers from the list are 0.49449444944449 … and 3.4641 … because these cannot be expressed as fractions of integers and are non-terminating. Other numbers, such as 64 and others listed, are rational. Thus, the answer is selected accordingly.
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