Classify each number below as a rational number or an irrational number.
| | rational | irrational |
| :-------------------- | :------- | :--------- |
| $98.\overline{45}$ | | |
| $\sqrt{23}$ | | |
| $\frac{14}{12}$ | | |
| $-\sqrt{81}$ | | |
| $-9 \pi$ | | |
Answer (1)
98. 45 rational , 23 irrational , 12 14 rational , − 81 rational , − 9 π irrational
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 98. 45 The number 98. 45 is a repeating decimal. Repeating decimals can be expressed as fractions, so 98. 45 is a rational number.
Classifying 23 The number 23 is the square root of 23. Since 23 is not a perfect square, its square root is an irrational number.
Classifying 12 14 The number 12 14 is a fraction where both the numerator and the denominator are integers. Therefore, 12 14 is a rational number.
Classifying − 81 The number − 81 is the negative square root of 81. Since 81 = 9 2 , we have − 81 = − 9 , which is an integer. Integers are rational numbers, so − 81 is a rational number.
Classifying − 9 π The number − 9 π is the product of -9 and π . Since π is an irrational number, and the product of a non-zero rational number and an irrational number is irrational, − 9 π is an irrational number.
Final Classification In summary:
98. 45 is rational.
23 is irrational.
12 14 is rational.
− 81 is rational.
− 9 π is irrational.
Examples
Understanding the difference between rational and irrational numbers is crucial in many real-world applications. For example, when calculating the circumference of a circle ( C = 2 π r ), if the radius r is a rational number, the circumference C will be irrational because it involves π . Similarly, in construction and engineering, knowing whether a measurement will result in a rational or irrational number can affect precision and planning. For instance, if you are building a square frame with an area of 23 square meters, the side length would be 23 meters, an irrational number, which means you can't measure it exactly with standard tools, and you'll have to approximate.
