How to Calculate the Modulus of 1i^6 and Its Implications

When dealing with complex numbers, it is essential to understand the various operations and properties that govern their behavior. One such operation is the modulus of a complex number. In this article, we will explore how to calculate the modulus of 1i6 and the implications of this calculation. We will also delve into the use of exponential form for complex numbers and their applications.

Understanding Complex Numbers

A complex number can be represented as (a bi), where (a) is the real part and (b) is the imaginary part. The modulus (or magnitude) of a complex number (z a bi) is given by the formula:

[|z| sqrt{a^2 b^2}]

The modulus of a complex number provides a measure of its distance from the origin in the complex plane.

Calculating the Modulus of 1i

To start, let's consider the complex number 1i. Notice that 1i can be considered as (0 1i), making its real part 0 and its imaginary part 1. Using the modulus formula, we find:

[|1i| sqrt{0^2 1^2} sqrt{1} 1]

This means the modulus of 1i is 1. However, in the original problem, we have a more complex expression: the modulus of 1i6.

Modulus of 1i^6

Let's calculate the modulus of 1i6. First, we need to express 1i in its polar form. The polar form of a complex number is given by:

[r(costheta isintheta)]

For 1i, the modulus (r) is 1 and the argument (theta) is (frac{pi}{2}) (since it lies on the positive imaginary axis). Therefore, 1i can be written as:

[1i 1left(cosfrac{pi}{2} isinfrac{pi}{2}right)]

Now, to find 1i6, we raise the complex number to the power of 6. Using De Moivre's theorem, which states that ((r(costheta isintheta))^n r^n(cos(ntheta) isin(ntheta))), we get:

[1i^6 1^6left(cosleft(6cdotfrac{pi}{2}right) isinleft(6cdotfrac{pi}{2}right)right) cos(3pi) isin(3pi)]

Since (cos(3pi) -1) and (sin(3pi) 0), we have:

[1i^6 -1]

The modulus of -1 is:

[|-1| 1]

However, this confusion might arise due to the misunderstanding of the modulus of a complex number raised to a power. Let's revisit the modulus calculation properly:

[|1i^6| |1i|^6 1^6 1]

This indicates that the modulus of 1i6 is 1, not 8 as initially stated. It is important to clearly distinguish between the modulus of the complex number and the modulus of its power.

Exponential Form of Complex Numbers

The exponential form of a complex number is another useful representation. Euler's formula states:

[e^{itheta} costheta isintheta]

Using this, we can express 1i as:

[1i e^{ipi/2}]

Raising this to the sixth power gives:

[1i^6 (e^{ipi/2})^6 e^{i3pi} cos(3pi) isin(3pi) -1]

The modulus of -1 is still 1, as expected.

Conclusion

In conclusion, the modulus of 1i6 is 1, not 8. Understanding the correct use of the modulus calculation and the application of complex numbers in both Cartesian and exponential forms is crucial in advanced mathematics. This knowledge is particularly useful in fields such as signal processing, quantum mechanics, and electrical engineering, where complex numbers and their properties play a significant role.

By mastering these concepts, one can handle more complex problems and applications in various scientific and engineering domains.