Discover how IDNLearn.com can help you learn and grow with its extensive Q&A platform. Our community provides accurate and timely answers to help you understand and solve any issue.
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
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Thank you for visiting IDNLearn.com. We’re here to provide clear and concise answers, so visit us again soon.