Write The Complex Number In Polar Form. (θ Should Be In Radians)(Round To The Nearest Hundredth)Given: { 4 + I $}$ { R(\cos \theta + I \sin \theta) \}

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Introduction

In mathematics, complex numbers are a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. 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, which satisfies the equation i^2 = -1. The polar form of a complex number is a way of expressing it in terms of its magnitude (or modulus) and argument (or angle). In this article, we will learn how to write a complex number in polar form.

What is Polar Form?

The polar form of a complex number is a way of expressing it in terms of its magnitude (r) and argument (θ). The magnitude of a complex number is the distance from the origin to the point representing the complex number in the complex plane. The argument of a complex number is the angle between the positive real axis and the line segment connecting the origin to the point representing the complex number in the complex plane.

Converting Complex Numbers to Polar Form

To convert a complex number to polar form, we need to find its magnitude and argument. The magnitude of a complex number a + bi is given by the formula:

r = √(a^2 + b^2)

The argument of a complex number a + bi is given by the formula:

θ = arctan(b/a)

Example

Let's consider the complex number 4 + i. We need to find its magnitude and argument.

Magnitude

The magnitude of the complex number 4 + i is given by:

r = √(4^2 + 1^2) = √(16 + 1) = √17 ≈ 4.12

Argument

The argument of the complex number 4 + i is given by:

θ = arctan(1/4) ≈ 0.25 radians

Writing the Complex Number in Polar Form

Now that we have found the magnitude and argument of the complex number 4 + i, we can write it in polar form. The polar form of a complex number is given by:

r(cos θ + i sin θ)

Substituting the values of r and θ, we get:

4.12(cos 0.25 + i sin 0.25)

Rounding to the Nearest Hundredth

We are asked to round the answer to the nearest hundredth. Therefore, we need to round the values of r and θ to the nearest hundredth.

r ≈ 4.12 θ ≈ 0.25

Conclusion

In this article, we learned how to write a complex number in polar form. We found the magnitude and argument of the complex number 4 + i and used them to write it in polar form. We also rounded the answer to the nearest hundredth.

Key Takeaways

  • The polar form of a complex number is a way of expressing it in terms of its magnitude and argument.
  • The magnitude of a complex number is given by the formula r = √(a^2 + b^2).
  • The argument of a complex number is given by the formula θ = arctan(b/a).
  • To write a complex number in polar form, we need to find its magnitude and argument.
  • We can use the formulas r = √(a^2 + b^2) and θ = arctan(b/a) to find the magnitude and argument of a complex number.

Frequently Asked Questions

Q: What is the polar form of a complex number?

A: The polar form of a complex number is a way of expressing it in terms of its magnitude and argument.

Q: How do I find the magnitude of a complex number?

A: You can find the magnitude of a complex number using the formula r = √(a^2 + b^2).

Q: How do I find the argument of a complex number?

A: You can find the argument of a complex number using the formula θ = arctan(b/a).

Q: Can I use a calculator to find the magnitude and argument of a complex number?

A: Yes, you can use a calculator to find the magnitude and argument of a complex number.

References

Further Reading

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Introduction

Complex numbers are a fundamental concept in mathematics that plays a crucial role in various fields, including algebra, geometry, and calculus. In our previous article, we learned how to write a complex number in polar form. In this article, we will answer some frequently asked questions about complex numbers.

Q&A

Q: What is a complex number?

A: 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, which satisfies the equation i^2 = -1.

Q: What is the imaginary unit i?

A: The imaginary unit i is a number that satisfies the equation i^2 = -1. It is used to extend the real number system to the complex number system.

Q: How do I add complex numbers?

A: To add complex numbers, you need to add the real parts and the imaginary parts separately. For example, if you have two complex numbers 3 + 4i and 2 + 5i, you can add them as follows:

(3 + 4i) + (2 + 5i) = (3 + 2) + (4i + 5i) = 5 + 9i

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you need to subtract the real parts and the imaginary parts separately. For example, if you have two complex numbers 3 + 4i and 2 + 5i, you can subtract them as follows:

(3 + 4i) - (2 + 5i) = (3 - 2) + (4i - 5i) = 1 - i

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you need to use the distributive property and the fact that i^2 = -1. For example, if you have two complex numbers 3 + 4i and 2 + 5i, you can multiply them as follows:

(3 + 4i)(2 + 5i) = (3)(2) + (3)(5i) + (4i)(2) + (4i)(5i) = 6 + 15i + 8i + 20i^2 = 6 + 23i - 20 = -14 + 23i

Q: How do I divide complex numbers?

A: To divide complex numbers, you need to multiply the numerator and the denominator by the conjugate of the denominator. For example, if you have two complex numbers 3 + 4i and 2 + 5i, you can divide them as follows:

(3 + 4i)/(2 + 5i) = ((3 + 4i)(2 - 5i))/((2 + 5i)(2 - 5i)) = ((3)(2) + (3)(-5i) + (4i)(2) + (4i)(-5i))/((2)(2) + (2)(-5i) + (5i)(2) + (5i)(-5i)) = (6 - 15i + 8i - 20i^2)/((4 - 10i + 10i - 25i^2)) = (6 - 7i + 20)/(4 + 25) = (26 - 7i)/29 = 26/29 - 7i/29

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number a + bi is a - bi.

Q: How do I find the magnitude of a complex number?

A: You can find the magnitude of a complex number using the formula r = √(a^2 + b^2).

Q: How do I find the argument of a complex number?

A: You can find the argument of a complex number using the formula θ = arctan(b/a).

Q: Can I use a calculator to find the magnitude and argument of a complex number?

A: Yes, you can use a calculator to find the magnitude and argument of a complex number.

Conclusion

In this article, we answered some frequently asked questions about complex numbers. We covered topics such as adding, subtracting, multiplying, and dividing complex numbers, as well as finding the magnitude and argument of a complex number.

Key Takeaways

  • 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.
  • The imaginary unit i is a number that satisfies the equation i^2 = -1.
  • To add complex numbers, you need to add the real parts and the imaginary parts separately.
  • To subtract complex numbers, you need to subtract the real parts and the imaginary parts separately.
  • To multiply complex numbers, you need to use the distributive property and the fact that i^2 = -1.
  • To divide complex numbers, you need to multiply the numerator and the denominator by the conjugate of the denominator.
  • The conjugate of a complex number a + bi is a - bi.
  • You can find the magnitude of a complex number using the formula r = √(a^2 + b^2).
  • You can find the argument of a complex number using the formula θ = arctan(b/a).

Frequently Asked Questions

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

A: 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. A real number is a number that can be expressed in the form a, where a is a real number.

Q: Can I use complex numbers in real-world applications?

A: Yes, complex numbers are used in various real-world applications, such as electrical engineering, signal processing, and quantum mechanics.

Q: How do I represent complex numbers graphically?

A: You can represent complex numbers graphically using the complex plane, which is a two-dimensional plane with the real axis and the imaginary axis.

References

Further Reading