To determine why [tex]\( -\sqrt{10} \)[/tex] is irrational, let's review the properties of irrational numbers and examine the provided options.
Irrational numbers are numbers that cannot be expressed as a ratio of two integers (i.e., they cannot be written in the form [tex]\( \frac{a}{b} \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are integers and [tex]\( b \neq 0 \)[/tex]). Additionally, the decimal expansion of an irrational number is non-terminating (it goes on forever) and non-repeating.
Now, consider the following options:
a. [tex]\( -\sqrt{10} \)[/tex] is irrational because it is not rational.
- This is a true statement, but it is a circular definition. It essentially says that [tex]\( -\sqrt{10} \)[/tex] is irrational because it is irrational, without giving a specific property or reasoning why it is so.
b. [tex]\( -\sqrt{10} \)[/tex] is irrational because it is less than zero.
- This is incorrect. The value being less than zero does not determine whether a number is irrational or not. There are both rational and irrational numbers that are less than zero (e.g., [tex]\(-2\)[/tex] is rational and [tex]\(-\sqrt{2}\)[/tex] is irrational).
c. [tex]\( -\sqrt{10} \)[/tex] is irrational because it is not a whole number.
- This is incorrect. A number not being a whole number does not necessarily mean it is irrational. For example, [tex]\( \frac{1}{2} \)[/tex] is not a whole number but it is rational.
d. [tex]\( -\sqrt{10} \)[/tex] is irrational because if I put [tex]\( -\sqrt{10} \)[/tex] into a calculator, I get -3.16227766 , which does not make a repeating pattern.
- This option correctly identifies a key property of irrational numbers. When [tex]\( -\sqrt{10} \)[/tex] is approximated with a decimal, the decimal expansion does not terminate or repeat. This non-repeating, non-terminating decimal expansion is a clear indication that [tex]\( -\sqrt{10} \)[/tex] is indeed irrational.
Given these evaluations, the best explanation is:
d. [tex]\( -\sqrt{10} \)[/tex] is irrational because if I put [tex]\( -\sqrt{10} \)[/tex] into a calculator, I get -3.16227766 , which does not make a repeating pattern.
This option accurately captures the defining characteristic of irrational numbers, making it the correct choice.
