Determine To Which Set Each Of The Following Numbers Belongs:a) \[$ Q = Z_1 \$\], \[$ J - \frac{25}{5} \$\]b) \[$ \frac{13}{6} = Q \$\]c) \[$ \sqrt{6} = I \$\]d) \[$ -\frac{2}{5} = \$\]e) \[$ 0.2166666 = Q

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In mathematics, numbers can be classified into different sets based on their properties and characteristics. The main sets of numbers are Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, and Real Numbers. In this article, we will determine to which set each of the given numbers belongs.

a) { q = z_1 $}$

To determine the set to which the number { q = z_1 $}$ belongs, we need to understand the notation used. The notation { q = z_1 $}$ represents a complex number, where { q $}$ is the imaginary part and { z_1 $}$ is the real part. Complex numbers are of the form { a + bi $}$, where { a $}$ and { b $}$ are real numbers and { i $}$ is the imaginary unit, which satisfies { i^2 = -1 $}$.

Since the given number is a complex number, it belongs to the set of Complex Numbers.

b) { j - \frac{25}{5} $}$

To determine the set to which the number { j - \frac{25}{5} $}$ belongs, we need to simplify the expression. The expression { j - \frac{25}{5} $}$ can be simplified as { j - 5 $}$.

Since { j $}$ is the imaginary unit, the expression { j - 5 $}$ is a complex number. Therefore, it belongs to the set of Complex Numbers.

c) { \frac{13}{6} = Q $}$

To determine the set to which the number { \frac{13}{6} = Q $}$ belongs, we need to understand the notation used. The notation { Q $}$ represents a rational number.

A rational number is a number that can be expressed as the ratio of two integers, i.e., { \frac{p}{q} $}$, where { p $}$ and { q $}$ are integers and { q $}$ is non-zero. Since { \frac{13}{6} $}$ is a ratio of two integers, it is a rational number.

Therefore, the number { \frac{13}{6} = Q $}$ belongs to the set of Rational Numbers.

d) { \sqrt{6} = I $}$

To determine the set to which the number { \sqrt{6} = I $}$ belongs, we need to understand the notation used. The notation { I $}$ represents an irrational number.

An irrational number is a number that cannot be expressed as a ratio of two integers, i.e., it cannot be written in the form { \frac{p}{q} $}$, where { p $}$ and { q $}$ are integers and { q $}$ is non-zero. Since { \sqrt{6} $}$ is a square root and cannot be expressed as a ratio of two integers, it is an irrational number.

Therefore, the number { \sqrt{6} = I $}$ belongs to the set of Irrational Numbers.

e) { 0.2166666 = Q $}$

To determine the set to which the number { 0.2166666 = Q $}$ belongs, we need to understand the notation used. The notation { Q $}$ represents a rational number.

A rational number is a number that can be expressed as the ratio of two integers, i.e., { \frac{p}{q} $}$, where { p $}$ and { q $}$ are integers and { q $}$ is non-zero. Since { 0.2166666 $}$ can be expressed as a ratio of two integers, i.e., { \frac{2}{9} $}$, it is a rational number.

Therefore, the number { 0.2166666 = Q $}$ belongs to the set of Rational Numbers.

Conclusion

In this article, we have determined to which set each of the given numbers belongs. The numbers { q = z_1 $}$ and { j - \frac{25}{5} $}$ belong to the set of Complex Numbers. The number { \frac{13}{6} = Q $}$ belongs to the set of Rational Numbers. The number { \sqrt{6} = I $}$ belongs to the set of Irrational Numbers. The number { 0.2166666 = Q $}$ also belongs to the set of Rational Numbers.

Key Takeaways

  • Complex numbers are of the form { a + bi $}$, where { a $}$ and { b $}$ are real numbers and { i $}$ is the imaginary unit.
  • Rational numbers are numbers that can be expressed as the ratio of two integers, i.e., { \frac{p}{q} $}$, where { p $}$ and { q $}$ are integers and { q $}$ is non-zero.
  • Irrational numbers are numbers that cannot be expressed as a ratio of two integers.
  • Real numbers include all rational and irrational numbers.

References

  • [1] "Complex Numbers" by Math Open Reference
  • [2] "Rational Numbers" by Math Is Fun
  • [3] "Irrational Numbers" by Khan Academy
    Frequently Asked Questions (FAQs) About Sets of Numbers ===========================================================

In the previous article, we discussed the different sets of numbers and determined to which set each of the given numbers belongs. In this article, we will answer some frequently asked questions (FAQs) about sets of numbers.

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., { \frac{p}{q} $}$, where { p $}$ and { q $}$ are integers and { q $}$ is non-zero. An irrational number is a number that cannot be expressed as a ratio of two integers.

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed as a ratio of two integers, i.e., { \frac{p}{q} $}$, where { p $}$ and { q $}$ are integers and { q $}$ is non-zero. A complex number is a number that can be expressed in the form { a + bi $}$, where { a $}$ and { b $}$ are real numbers and { i $}$ is the imaginary unit.

Q: What is the set of all rational numbers called?

A: The set of all rational numbers is called the Rational Numbers or { \mathbb{Q} $}$.

Q: What is the set of all irrational numbers called?

A: The set of all irrational numbers is called the Irrational Numbers or { \mathbb{I} $}$.

Q: What is the set of all real numbers called?

A: The set of all real numbers is called the Real Numbers or { \mathbb{R} $}$.

Q: What is the set of all complex numbers called?

A: The set of all complex numbers is called the Complex Numbers or { \mathbb{C} $}$.

Q: Can a number be both rational and irrational?

A: No, a number cannot be both rational and irrational. A number is either rational or irrational, but not both.

Q: Can a number be both real and complex?

A: Yes, a number can be both real and complex. For example, the number { 3 $}$ is both a real number and a complex number.

Q: What is the relationship between the sets of numbers?

A: The sets of numbers are related as follows:

  • The set of rational numbers is a subset of the set of real numbers.
  • The set of irrational numbers is a subset of the set of real numbers.
  • The set of complex numbers is a superset of the set of real numbers.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about sets of numbers. We have discussed the differences between rational and irrational numbers, real and complex numbers, and the relationships between the sets of numbers.

Key Takeaways

  • Rational numbers are numbers that can be expressed as the ratio of two integers.
  • Irrational numbers are numbers that cannot be expressed as a ratio of two integers.
  • Real numbers include all rational and irrational numbers.
  • Complex numbers are numbers that can be expressed in the form { a + bi $}$, where { a $}$ and { b $}$ are real numbers and { i $}$ is the imaginary unit.
  • The set of rational numbers is a subset of the set of real numbers.
  • The set of irrational numbers is a subset of the set of real numbers.
  • The set of complex numbers is a superset of the set of real numbers.

References

  • [1] "Rational Numbers" by Math Is Fun
  • [2] "Irrational Numbers" by Khan Academy
  • [3] "Complex Numbers" by Math Open Reference