To determine the type of number resulting from the expression [tex]\(\sqrt{19} - \frac{3}{2}\)[/tex], let's follow these steps:
1. Calculate the square root of 19:
The square root of 19 is approximately [tex]\(4.358898943540674\)[/tex].
2. Calculate the value of [tex]\(\frac{3}{2}\)[/tex]:
[tex]\(\frac{3}{2} = 1.5\)[/tex].
3. Subtract [tex]\(\frac{3}{2}\)[/tex] from [tex]\(\sqrt{19}\)[/tex]:
[tex]\[
\sqrt{19} - \frac{3}{2} \approx 4.358898943540674 - 1.5 = 2.858898943540674.
\][/tex]
4. Determine the type of the result:
- The result [tex]\(2.858898943540674\)[/tex] is not an integer.
- It's also not a whole number since whole numbers are non-negative integers.
- Since this number cannot be expressed exactly as a fraction of two integers (its decimal representation does not terminate or repeat), it is not a rational number.
- Thus, [tex]\(2.858898943540674\)[/tex] is an irrational number.
Therefore, the resulting number from the expression [tex]\(\sqrt{19} - \frac{3}{2}\)[/tex] is an irrational number.
