To determine the type of number that results from the expression [tex]\(\sqrt{19} - \frac{3}{2}\)[/tex], we need to follow these steps:
1. Understand the Components:
- [tex]\(\sqrt{19}\)[/tex]: This is the square root of 19, which is an irrational number since 19 is not a perfect square.
- [tex]\(\frac{3}{2}\)[/tex]: This is a rational number because it can be expressed as a ratio of two integers (3 and 2).
2. Perform the Calculation:
- First, we approximate [tex]\(\sqrt{19}\)[/tex]. The value of [tex]\(\sqrt{19}\)[/tex] is approximately 4.358.
- Then, we subtract [tex]\(\frac{3}{2}\)[/tex]. The value of [tex]\(\frac{3}{2}\)[/tex] is 1.5.
So, the calculation becomes:
[tex]\[
\sqrt{19} - \frac{3}{2} \approx 4.358 - 1.5 = 2.858
\][/tex]
3. Assess the Type of Resulting Number:
- The value we get from [tex]\(\sqrt{19} - \frac{3}{2}\)[/tex] is approximately 2.858.
Now, let’s classify this number:
- Integer: An integer is a whole number that can be positive, negative, or zero, like -1, 0, 1, etc. Clearly, 2.858 is not an integer.
- Whole Number: Whole numbers are non-negative integers (0, 1, 2, ...). Again, 2.858 is not a whole number.
- Rational Number: A rational number is defined as any number that can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] of two integers, where [tex]\(p\)[/tex] is an integer and [tex]\(q\)[/tex] is a non-zero integer. The number 2.858 can indeed be expressed as a fraction (assuming more precise values, it can be approxiamted closer to a fraction, but for simplicity, the decimal itself is rational).
Given this analysis, 2.858 can be classified as a rational number.
Therefore, the type of number resulting from the expression [tex]\(\sqrt{19} - \frac{3}{2}\)[/tex] is a rational number.
