Classify each number below as a rational number or an irrational number.
| | rational | Irrational |
| :-------------- | :------- | :--------- |
| [tex]$\sqrt{25}$[/tex] | | |
| [tex]$\sqrt{23}$[/tex] | | |
| [tex]$-17 \pi$[/tex] | | |
| -45.77 | | |
| [tex]$-73.\overline{18}$[/tex] | | |
Answer (2)
25 simplifies to 5, which is a rational number.
23 is not a perfect square, thus it's an irrational number.
− 17 π is a product of a rational and irrational number, making it irrational.
− 45.77 can be expressed as a fraction, so it is rational.
− 73. 18 is a repeating decimal and therefore rational.
The final classification is:
Rational
Irrational
25
X
23
X
− 17 π
X
− 45.77
X
− 73. 18
X
Explanation
Understanding Rational and Irrational Numbers We are asked to classify the given numbers as either rational or irrational. A rational number can be expressed as a fraction q p , where p and q are integers and q = 0 . An irrational number cannot be expressed in this form.
Classifying 25 25 = 5 , which is an integer and can be written as 1 5 . Therefore, 25 is a rational number.
Classifying 23 23 is not a perfect square. Its decimal representation is non-terminating and non-repeating. Therefore, 23 is an irrational number.
Classifying − 17 π − 17 π is a product of a rational number − 17 and an irrational number π . The product of a non-zero rational number and an irrational number is always irrational. Therefore, − 17 π is an irrational number.
Classifying − 45.77 − 45.77 can be written as − 100 4577 . Since it can be expressed as a fraction of two integers, it is a rational number.
Classifying − 73. 18 − 73. 18 is a repeating decimal. Repeating decimals can be expressed as fractions. Therefore, − 73. 18 is a rational number.
Final Classification Here's the classification:
Rational
Irrational
25
X
23
X
− 17 π
X
− 45.77
X
− 73. 18
X
Examples
Understanding the difference between rational and irrational numbers is crucial in many real-world applications. For instance, when calculating the dimensions of a circular garden, if the radius involves an irrational number like π or 2 , the circumference and area will also be irrational. This knowledge helps in accurately estimating the amount of fencing or soil needed, preventing waste and ensuring precise planning. Similarly, in financial calculations involving compound interest or amortization, irrational numbers often appear, requiring careful handling to avoid significant errors in long-term projections.
In summary, 25 , -45.77, and − 73. 18 are rational numbers, while 23 and − 17 π are irrational numbers. This classification is based on whether the numbers can be expressed as fractions or not. It's essential to understand these definitions to categorize numbers correctly.
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