Classify each number below as a rational number or an irrational number.
| | rational | irrational |
| :------------ | :------- | :--------- |
| [tex]$\sqrt{36}$[/tex] | | |
| [tex]$29.\overline{75}$[/tex] | | |
| [tex]$-\sqrt{30}$[/tex] | | |
| [tex]$94.66$[/tex] | | |
| [tex]$-13 \pi$[/tex] | | |
Answer (1)
36 = 6 , which is rational.
29. 75 is a repeating decimal and thus rational.
− 30 is the negative square root of a non-perfect square, making it irrational.
94.66 is a terminating decimal, so it is rational.
− 13 π involves π , which is irrational, thus making the entire expression irrational.
The classification is: 36 : rational , 29. 75 : rational , − 30 : irrational , 94.66 : rational , − 13 π : irrational
Explanation
Problem Analysis We are given 5 numbers: 36 , 29. 75 , − 30 , 94.66 , and − 13 π . We need to classify each number as either rational or irrational.
Definitions of Rational and Irrational Numbers A rational number can be expressed as a fraction q p , where p and q are integers and q = 0 . Irrational numbers cannot be expressed in this form.
Classifying 36 36 = 6 , which is an integer and can be written as 1 6 . Therefore, 36 is a rational number.
Classifying 29. 75 The number 29. 75 is a repeating decimal. Repeating decimals can be expressed as fractions. To convert 29. 75 to a fraction, let x = 29. 75 . Then 100 x = 2975. 75 . Subtracting x from 100 x , we get 99 x = 2975.7575... − 29.7575... = 2946 . Thus, x = 99 2946 = 33 982 . Since 29. 75 can be expressed as a fraction, it is a rational number.
Classifying − 30 30 is not a perfect square. The square root of a non-perfect square is an irrational number. Therefore, − 30 is also an irrational number.
Classifying 94.66 The number 94.66 is a terminating decimal. Terminating decimals can be expressed as fractions. 94.66 = 100 9466 = 50 4733 . Since 94.66 can be expressed as a fraction, it is a rational number.
Classifying − 13 π The number π is an irrational number. Multiplying an irrational number by a non-zero integer results in an irrational number. Therefore, − 13 π is also irrational.
Final Classification In summary:
36 is rational.
29. 75 is rational.
− 30 is irrational.
94.66 is rational.
− 13 π is irrational.
Examples
Understanding the difference between rational and irrational numbers is crucial in various real-life applications. For instance, when calculating the dimensions of a room using the Pythagorean theorem, if the sides are integers, the hypotenuse might be an irrational number. Similarly, in engineering, when dealing with circular shapes, the value of π is essential, and since it's irrational, calculations involving circles often yield irrational results. In finance, compound interest calculations can sometimes lead to irrational numbers, especially when dealing with continuous compounding.
