Understanding the Polar Form of a Complex Number: Z3 4i

Let's explore the polar form of the complex number Z3 4i. Understanding the polar form is a fundamental concept in complex analysis, and it involves expressing a complex number in terms of its magnitude and its angle from the positive real axis.

Introduction to Complex Numbers and Polar Form

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, defined by i2 -1. The polar form of a complex number is another way to express a complex number, and it involves expressing the complex number in terms of its magnitude (or modulus) and its argument (or angle).

Calculating the Magnitude (Modulus)

The magnitude (or modulus) of a complex number Z a bi is given by the formula:

r √a2 b2

For the complex number Z 3 4i, we substitute a 3 and b 4 into the formula:

r √(32 42) √(9 16) √25 5

This calculation can be verified as follows:

√(32 42) √(9 16) √25 5

Thus, the magnitude of Z 3 4i is 5.

Calculating the Argument (Angle)

The argument (or angle) of a complex number is found using the arctangent function. The argument θ of a complex number Z a bi is given by:

θ arctan (b/a)

For the complex number Z 3 4i, we substitute a 3 and b 4 into the formula:

θ arctan(4/3)

This can be calculated using a calculator:

θ ≈ 0.93 radians or 53.13°

Alternatively, the argument can be expressed in terms of the arctangent of the rise over run (4/3):

θ arctan(4/3)

Expressing the Complex Number in Polar Form

The polar form of a complex number Z a bi is given by:

Z r(cos θ i sin θ)

Using the values we calculated, we can express the complex number Z 3 4i in polar form:

Z 5(cos 0.93 i sin 0.93)

Additionally, the polar form can also be expressed using Euler's formula:

Z 5eiθ 5ei · 0.93

Thus, the polar form of the complex number Z 3 4i is:

Z 5(cos 0.93 i sin 0.93) or Z 5ei · 0.93

It's worth noting that the argument can also be expressed as:

θ arctan(4/3) ≈ 0.93 radians or 53.13°

This calculation confirms the angle from the positive real axis.

Conclusion

Understanding the polar form of a complex number is crucial for various applications in mathematics, physics, and engineering. The magnitude and argument of a complex number provide valuable information about its geometric representation in the complex plane. By mastering the concepts of modulus and argument, you can effectively navigate and manipulate complex numbers in various contexts.

Key Points:

The polar form of a complex number is given by Z r(cos θ i sin θ). The magnitude (modulus) of Z 3 4i is 5. The argument (angle) of Z 3 4i is approximately 0.93 radians or 53.13°.