To determine the type of number resulting from the expression [tex]\(\sqrt{19} - \frac{3}{2}\)[/tex], let's proceed step-by-step through the calculations and analysis:
1. Calculate [tex]\(\sqrt{19}\)[/tex]:
[tex]\[\sqrt{19} \approx 4.358898943540674\][/tex]
2. Calculate [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[\frac{3}{2} = 1.5\][/tex]
3. Subtraction of the two values:
[tex]\[\sqrt{19} - \frac{3}{2} \approx 4.358898943540674 - 1.5 = 2.858898943540674\][/tex]
4. Determine the type of number:
Next we need to figure out if this result [tex]\(2.858898943540674\)[/tex] is a rational or an irrational number.
- A rational number can be expressed as the fraction of two integers.
- An irrational number cannot be expressed as the fraction of two integers and has a decimal expansion that neither terminates nor becomes periodic.
The resulting value [tex]\(2.858898943540674\)[/tex] is not terminating and does not repeat periodically, which classifies it as an irrational number.
Summarizing the analysis, the type of number resulting from the expression [tex]\(\sqrt{19} - \frac{3}{2}\)[/tex] is an irrational number. Therefore, the correct answer is:
an irrational number
