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Sagot :
To determine to which set(s) of numbers the number [tex]\(\sqrt{-16}\)[/tex] belongs, let's analyze its properties:
1. Real Numbers:
Real numbers include all rational and irrational numbers and do not include the square root of negative numbers. Since [tex]\(\sqrt{-16}\)[/tex] involves the square root of a negative number, it is not a real number.
- Conclusion: [tex]\(\sqrt{-16}\)[/tex] does NOT belong to real numbers.
2. Complex Numbers:
Complex numbers include all numbers that can be written in the form [tex]\(a + bi\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are real numbers and [tex]\(i\)[/tex] is the imaginary unit ([tex]\(i = \sqrt{-1}\)[/tex]). The square root of a negative number is a complex number.
- [tex]\(\sqrt{-16} = 4i\)[/tex], which fits the form [tex]\(a + bi\)[/tex] where [tex]\(a = 0\)[/tex] and [tex]\(b = 4\)[/tex].
- Conclusion: [tex]\(\sqrt{-16}\)[/tex] DOES belong to complex numbers.
3. Rational Numbers:
Rational numbers are numbers that can be expressed as a ratio of two integers, [tex]\( \frac{p}{q} \)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]. The result of a square root of a negative number cannot be expressed in this form.
- Conclusion: [tex]\(\sqrt{-16}\)[/tex] does NOT belong to rational numbers.
4. Imaginary Numbers:
Imaginary numbers are a subset of complex numbers where [tex]\(a = 0\)[/tex]. They can be written as [tex]\(bi\)[/tex] where [tex]\(b\)[/tex] is a real number and [tex]\(i\)[/tex] is the imaginary unit ([tex]\(i = \sqrt{-1}\)[/tex]). Since [tex]\(\sqrt{-16} = 4i\)[/tex], it fits this definition.
- Conclusion: [tex]\(\sqrt{-16}\)[/tex] DOES belong to imaginary numbers.
5. Irrational Numbers:
Irrational numbers are real numbers that cannot be expressed as a simple fraction, i.e., their decimal expansion is non-terminating and non-repeating. Since [tex]\(\sqrt{-16}\)[/tex] is not a real number at all, it cannot be classified as irrational.
- Conclusion: [tex]\(\sqrt{-16}\)[/tex] does NOT belong to irrational numbers.
In summary, [tex]\(\sqrt{-16}\)[/tex] belongs to the sets:
- Complex Numbers
- Imaginary Numbers
Therefore, the correct sets of numbers to which [tex]\(\sqrt{-16}\)[/tex] belongs are:
- Complex numbers
- Imaginary numbers
1. Real Numbers:
Real numbers include all rational and irrational numbers and do not include the square root of negative numbers. Since [tex]\(\sqrt{-16}\)[/tex] involves the square root of a negative number, it is not a real number.
- Conclusion: [tex]\(\sqrt{-16}\)[/tex] does NOT belong to real numbers.
2. Complex Numbers:
Complex numbers include all numbers that can be written in the form [tex]\(a + bi\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are real numbers and [tex]\(i\)[/tex] is the imaginary unit ([tex]\(i = \sqrt{-1}\)[/tex]). The square root of a negative number is a complex number.
- [tex]\(\sqrt{-16} = 4i\)[/tex], which fits the form [tex]\(a + bi\)[/tex] where [tex]\(a = 0\)[/tex] and [tex]\(b = 4\)[/tex].
- Conclusion: [tex]\(\sqrt{-16}\)[/tex] DOES belong to complex numbers.
3. Rational Numbers:
Rational numbers are numbers that can be expressed as a ratio of two integers, [tex]\( \frac{p}{q} \)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]. The result of a square root of a negative number cannot be expressed in this form.
- Conclusion: [tex]\(\sqrt{-16}\)[/tex] does NOT belong to rational numbers.
4. Imaginary Numbers:
Imaginary numbers are a subset of complex numbers where [tex]\(a = 0\)[/tex]. They can be written as [tex]\(bi\)[/tex] where [tex]\(b\)[/tex] is a real number and [tex]\(i\)[/tex] is the imaginary unit ([tex]\(i = \sqrt{-1}\)[/tex]). Since [tex]\(\sqrt{-16} = 4i\)[/tex], it fits this definition.
- Conclusion: [tex]\(\sqrt{-16}\)[/tex] DOES belong to imaginary numbers.
5. Irrational Numbers:
Irrational numbers are real numbers that cannot be expressed as a simple fraction, i.e., their decimal expansion is non-terminating and non-repeating. Since [tex]\(\sqrt{-16}\)[/tex] is not a real number at all, it cannot be classified as irrational.
- Conclusion: [tex]\(\sqrt{-16}\)[/tex] does NOT belong to irrational numbers.
In summary, [tex]\(\sqrt{-16}\)[/tex] belongs to the sets:
- Complex Numbers
- Imaginary Numbers
Therefore, the correct sets of numbers to which [tex]\(\sqrt{-16}\)[/tex] belongs are:
- Complex numbers
- Imaginary numbers
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