To analyze the number [tex]\( 7 - \sqrt{3}i \)[/tex] and check each statement, let's break it down step-by-step:
1. Identifying the real part:
- For a complex number in the form [tex]\( a + bi \)[/tex], [tex]\( a \)[/tex] is the real part and [tex]\( b \)[/tex] is the imaginary part (without considering [tex]\( i \)[/tex]).
- Here, the given number is [tex]\( 7 - \sqrt{3}i \)[/tex]. Comparing this with [tex]\( a + bi \)[/tex], we see that [tex]\( a = 7 \)[/tex].
Therefore, the statement "7 is the real part of the number" is true.
2. Identifying the imaginary part:
- The imaginary part of a complex number [tex]\( a + bi \)[/tex] is the coefficient of [tex]\( i \)[/tex].
- In [tex]\( 7 - \sqrt{3}i \)[/tex], the coefficient of [tex]\( i \)[/tex] is [tex]\( -\sqrt{3} \)[/tex].
Therefore, the statement "[tex]\(\sqrt{3}\)[/tex] is the imaginary part of the number" is false because the correct imaginary part here is [tex]\( -\sqrt{3} \)[/tex].
3. Evaluating the coefficient of [tex]\(i\)[/tex]:
- In the complex number [tex]\( 7 - \sqrt{3}i \)[/tex], the coefficient of [tex]\( i \)[/tex] is again [tex]\( -\sqrt{3} \)[/tex].
- The statement [tex]\( 7 - \sqrt{3} \)[/tex] as the coefficient of [tex]\( i \)[/tex] is confusing because [tex]\( 7 - \sqrt{3} \)[/tex] is not solely the coefficient of [tex]\( i \)[/tex].
Therefore, the statement "7 - [tex]\( \sqrt{3} \)[/tex] is the coefficient of [tex]\( i \)[/tex]" is false because the coefficient should be [tex]\( -\sqrt{3} \)[/tex].
4. Sum of a real number and an imaginary number:
- A complex number is always a combination of a real number and an imaginary number.
- Here, the real part is [tex]\( 7 \)[/tex] and the imaginary part is [tex]\( - \sqrt{3} i \)[/tex]. Hence, [tex]\( 7 - \sqrt{3}i \)[/tex] clearly represents the sum of the real number [tex]\( 7 \)[/tex] and the imaginary number [tex]\( - \sqrt{3}i \)[/tex].
Therefore, the statement "This number is the sum of a real number and an imaginary number" is true.
To summarize:
1. The statement "7 is the real part of the number" is true.
2. The statement "[tex]\(\sqrt{3}\)[/tex] is the imaginary part of the number" is false.
3. The statement "7 - [tex]\(\sqrt{3}\)[/tex] is the coefficient of [tex]\(i\)[/tex]" is false.
4. The statement "This number is the sum of a real number and an imaginary number" is true.
