Power of i

When multiplied by themselves, imaginary numbers, generally always denoted as i are unique in that they equal a negative number. You may be asking how this is possible because even negative integers multiplied together equal a positive number. The secret is that when you multiply i = √-1 by itself, it eliminates the radical symbol but leaves the sign of the number inside of the radical sign unchanged.

COMPLEX NUMBERS

With complex numbers, determining the square root of negative values is now easier. In the first century, a Greek mathematician named Hero of Alexandria came and discovered the concept of complex numbers while attempting to find the square root of a negative number. What he all did was turn the negative into a positive and take the numeric root value. Furthermore, Italian mathematician Gerolamo Cardano defined the real identity of a complex number in the 16th century while looking for the negative roots of cubic & quadratic polynomial formulas.

Many scientific fields, such as signal processing, electromagnetism, fluid dynamics, quantum physics, & vibration analysis, use complex numbers. Here we can learn about the definition, vocabulary, visualisation, properties, and operations of complex numbers.

The Powers Of i

As we know i = √-1 and i² = -1

Value of i³ and i⁴:

In reality, we may use the rules of exponents that we know for sure in the real number system to calculate powers of iii, as long as the exponents are integers.

Now i³ = i² x i but we know that i² = -1

i³ = i² x i

   = (-1) x i

   = -i

And, i⁴ = i² x i²

   = (-1) x (-1)

   = 1

And, i⁵ = i² x i² x i

   = (-1) x (-1) x i

   = i

And, i⁶ = i² x i² x i²

   = (-1) x (-1) x (-1)

   = -1

And, i⁷ = i² x i² x i² x i

   = (-1) x (-1) x (-1) x i

   = -i

And, i⁸ = i² x i² x i² x i²

   = (-1) x (-1) x (-1) x (-1)

   = 1

Results

So, from these values we can know the value of i²⁰ which is equal to 1.

 Let’s see if we can use exponents to support this. Remember, just like with real numbers, we can employ exponent qualities here!

 i²⁰ = (i⁴)⁵

   = (1)⁵

   = 1

When the power is a positive whole number, powers of complex numbers are simply special cases of products. We’ve already looked at the powers of imaginary unit I and discovered that they cycle over a period of 4, and so on.

The reasons were that (1) i’s absolute value |i| was one, so all of its powers have absolute value 1 as well and thus lie on the unit circle, and (2) i’s argument arg(i) was 90°, so its nth power will have argument n90°, and all angles will repeat in a period of length 4 because 490° = 360°, a full circle.

In general, zⁿ is defined as a complex number with an absolute value of |z|ⁿ, the nth power of z’s absolute value, and an argument of n times z’s argument.

A complex number z with an argument of 30° and an absolute value of about the 6th root of 1/2, that is, |z| = 0.89. Because the unit circle is coloured black and the area outside it is grey, z is in the black region.

Because |z| is smaller than one, its square is at 60 degrees, bringing it closer to 0. Each increased power moves you 30 degrees closer to zero. As you can see, the first six powers are represented as points on a spiral. A geometric or exponential spiral is the name given to this spiral.

CONCLUSIONS

What’s even more fascinating about imaginary numbers is that increasing their powers results in a predictable, repeatable cycle that allows us to solve issues that would otherwise be disorderly fast. For example, we may use this cycle to solve i³⁴⁷³ rapidly, which would otherwise necessitate a lot of extra effort. Here’s how it works: raising I to powers of 0 through 3 produces varied outputs. However, after that, the outcomes begin to replicate themselves every four digits, indefinitely.

So i⁰ = i⁴ = i⁸ = 1 and i³ = i⁷ = i¹¹ = -I and so forth.

This means that rather of explicitly calculating I raised to every power higher than 4, we can select a number near to that power and simplify it using the technique outlined above, as well as exponent characteristics.