Answer :
To determine which of the given options is an example of a complex number that is not in the set of real numbers, we will analyze each option one by one:
1. Option [tex]$-7$[/tex]:
This is a real number. Real numbers are all the numbers that can be found on the number line, including both positive and negative numbers, as well as zero. [tex]$-7$[/tex] does not have an imaginary part.
2. Option [tex]$2+\sqrt{3}$[/tex]:
Here, [tex]$2$[/tex] and [tex]$\sqrt{3}$[/tex] are both real numbers. This is simply a sum of two real numbers, which results in a real number. Therefore, it is not a complex number with an imaginary component.
3. Option [tex]$4+9i$[/tex]:
This is a complex number, where [tex]$4$[/tex] is the real part and [tex]$9i$[/tex] is the imaginary part. Since it has a non-zero imaginary part ([tex]$9i$[/tex]), it is considered a complex number that is not in the set of real numbers.
4. Option [tex]$\pi$[/tex]:
[tex]$\pi$[/tex] is a real number. It is an irrational number, but still falls within the set of real numbers. It does not have an imaginary part.
Among the options, [tex]$4+9i$[/tex] is the example of a complex number that is not in the set of real numbers because it includes the imaginary unit [tex]$i$[/tex]. Thus, the correct answer is:
[tex]$4 + 9 i$[/tex]
1. Option [tex]$-7$[/tex]:
This is a real number. Real numbers are all the numbers that can be found on the number line, including both positive and negative numbers, as well as zero. [tex]$-7$[/tex] does not have an imaginary part.
2. Option [tex]$2+\sqrt{3}$[/tex]:
Here, [tex]$2$[/tex] and [tex]$\sqrt{3}$[/tex] are both real numbers. This is simply a sum of two real numbers, which results in a real number. Therefore, it is not a complex number with an imaginary component.
3. Option [tex]$4+9i$[/tex]:
This is a complex number, where [tex]$4$[/tex] is the real part and [tex]$9i$[/tex] is the imaginary part. Since it has a non-zero imaginary part ([tex]$9i$[/tex]), it is considered a complex number that is not in the set of real numbers.
4. Option [tex]$\pi$[/tex]:
[tex]$\pi$[/tex] is a real number. It is an irrational number, but still falls within the set of real numbers. It does not have an imaginary part.
Among the options, [tex]$4+9i$[/tex] is the example of a complex number that is not in the set of real numbers because it includes the imaginary unit [tex]$i$[/tex]. Thus, the correct answer is:
[tex]$4 + 9 i$[/tex]
