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Sagot :
Let's examine each of the statements related to the complex number [tex]\( 7 - \sqrt{3}i \)[/tex]:
1. 7 is the real part of the number.
- Explanation: In a complex number of the form [tex]\( a + bi \)[/tex], [tex]\( a \)[/tex] represents the real part. For the complex number [tex]\( 7 - \sqrt{3}i \)[/tex], the real part is [tex]\( 7 \)[/tex]. Therefore, the statement "7 is the real part of the number" is true.
2. [tex]\(\sqrt{3}\)[/tex] is the imaginary part of the number.
- Explanation: In a complex number of the form [tex]\( a + bi \)[/tex], [tex]\( b \)[/tex] (the coefficient of [tex]\( i \)[/tex]) represents the imaginary part. For the complex number [tex]\( 7 - \sqrt{3}i \)[/tex], the imaginary part is [tex]\(-\sqrt{3}\)[/tex] because 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 is [tex]\(-\sqrt{3}\)[/tex].
3. [tex]\( 7 - \sqrt{3} \)[/tex] is the coefficient of [tex]\( i \)[/tex].
- Explanation: In a complex number [tex]\( a + bi \)[/tex], the coefficient of [tex]\( i \)[/tex] is [tex]\( b \)[/tex]. For the complex number [tex]\( 7 - \sqrt{3}i \)[/tex], the coefficient of [tex]\( i \)[/tex] is [tex]\(-\sqrt{3}\)[/tex]. The statement implies that [tex]\( 7 - \sqrt{3} \)[/tex] is the coefficient of [tex]\( i \)[/tex], which is incorrect. Therefore, this statement is false.
4. This number is the sum of a real number and an imaginary number.
- Explanation: Any complex number can be written as the sum of a real part [tex]\( a \)[/tex] and an imaginary part [tex]\( bi \)[/tex]. Here the real part is [tex]\( 7 \)[/tex] and the imaginary part is [tex]\( -\sqrt{3}i \)[/tex]. Hence, we have [tex]\( 7 + (-\sqrt{3}i) \)[/tex], which is indeed the sum of a real number and an imaginary number. Therefore, the statement "This number is the sum of a real number and an imaginary number" is true.
Based on the explanations for each statement, the truthful evaluation of the statements is as follows:
1. True
2. False
3. False
4. True
1. 7 is the real part of the number.
- Explanation: In a complex number of the form [tex]\( a + bi \)[/tex], [tex]\( a \)[/tex] represents the real part. For the complex number [tex]\( 7 - \sqrt{3}i \)[/tex], the real part is [tex]\( 7 \)[/tex]. Therefore, the statement "7 is the real part of the number" is true.
2. [tex]\(\sqrt{3}\)[/tex] is the imaginary part of the number.
- Explanation: In a complex number of the form [tex]\( a + bi \)[/tex], [tex]\( b \)[/tex] (the coefficient of [tex]\( i \)[/tex]) represents the imaginary part. For the complex number [tex]\( 7 - \sqrt{3}i \)[/tex], the imaginary part is [tex]\(-\sqrt{3}\)[/tex] because 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 is [tex]\(-\sqrt{3}\)[/tex].
3. [tex]\( 7 - \sqrt{3} \)[/tex] is the coefficient of [tex]\( i \)[/tex].
- Explanation: In a complex number [tex]\( a + bi \)[/tex], the coefficient of [tex]\( i \)[/tex] is [tex]\( b \)[/tex]. For the complex number [tex]\( 7 - \sqrt{3}i \)[/tex], the coefficient of [tex]\( i \)[/tex] is [tex]\(-\sqrt{3}\)[/tex]. The statement implies that [tex]\( 7 - \sqrt{3} \)[/tex] is the coefficient of [tex]\( i \)[/tex], which is incorrect. Therefore, this statement is false.
4. This number is the sum of a real number and an imaginary number.
- Explanation: Any complex number can be written as the sum of a real part [tex]\( a \)[/tex] and an imaginary part [tex]\( bi \)[/tex]. Here the real part is [tex]\( 7 \)[/tex] and the imaginary part is [tex]\( -\sqrt{3}i \)[/tex]. Hence, we have [tex]\( 7 + (-\sqrt{3}i) \)[/tex], which is indeed the sum of a real number and an imaginary number. Therefore, the statement "This number is the sum of a real number and an imaginary number" is true.
Based on the explanations for each statement, the truthful evaluation of the statements is as follows:
1. True
2. False
3. False
4. True
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