Which of the following numbers are irrational?
A. 0.3
B. $0.777 \ldots$
C. $\sqrt{5}$
D. $\pi$
Answer (2)
Identify rational numbers as those expressible as a fraction q p , where p and q are integers.
Recognize that 0.3 can be written as 10 3 , thus it is rational.
Convert the repeating decimal 0.777 … to the fraction 9 7 , indicating it is rational.
Identify 5 and π as irrational numbers because they cannot be expressed as fractions of integers.
The irrational numbers are 5 , π .
Explanation
Analyzing each option We need to identify which of the given numbers are irrational. Let's analyze each option:
A. 0.3: This is a decimal number. We can express it as a fraction 10 3 . Since it can be written as a fraction of two integers, it is a rational number.
B. 0.777 … : This is a repeating decimal. Repeating decimals can be expressed as fractions. In this case, 0.777 … = 9 7 . Since it can be written as a fraction of two integers, it is a rational number.
C. 5 : This is the square root of 5. The square root of a non-perfect square is an irrational number. 5 is not a perfect square (1, 4, 9, 16, etc.), so 5 is irrational. It cannot be expressed as a fraction of two integers.
D. π : This is the ratio of a circle's circumference to its diameter. π is a well-known irrational number. It cannot be expressed as a fraction of two integers. Its decimal representation is non-repeating and non-terminating.
Identifying irrational numbers From the analysis above, we can conclude that 5 and π are irrational numbers.
Final Answer Therefore, the irrational numbers from the given options are C and D.
Examples
Irrational numbers are crucial in various fields, such as engineering and physics, where precise measurements are required. For instance, when calculating the area of a circle using the formula A = π r 2 , if the radius r is a rational number, the area A will still be irrational due to the presence of π . Similarly, in electrical engineering, the impedance of a circuit can involve irrational numbers, especially when dealing with complex waveforms. Understanding irrational numbers ensures accurate calculations and reliable results in these applications.
The irrational numbers among the options are 5 and π . Both cannot be expressed as fractions of integers. The rational numbers are 0.3 and 0.777... .
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