Need help with complex numbers

[abel muroi]

I was given the problem i¹⁰⁰²


Ok so..... since I know that i² = -1

does that mean that i¹⁰⁰² = -1

if that's true then all imaginary numbers that have EVEN exponents will always equal to -1 and the ODD ones will equal to 1?

 

[Ishuda]

I was given the problem i¹⁰⁰²


Ok so..... since I know that i² = -1

does that mean that i¹⁰⁰² = -1

if that's true then all imaginary numbers that have EVEN exponents will always equal to -1 and the ODD ones will equal to 1?

i¹ = i
i² = -1
i³ = i² i¹ = -i
i⁴ = [i²]² = [-1]² = 1
i⁵ = i⁴ i = i
...

 

[Deleted member 4993]

I was given the problem i¹⁰⁰²


Ok so..... since I know that i² = -1

does that mean that i¹⁰⁰² = -1

if that's true then all imaginary numbers that have EVEN exponents will always equal to -1 and the ODD ones will equal to 1?

Not quite! Of the EVEN exponents, those that are divisible by 4 turns (+1) and the rest turns into (-1). In other words i⁴ⁿ = 1 and i^2(2n+1) = -1

 

[abel muroi]

i¹ = i
i² = -1
i³ = i² i¹ = -i
i⁴ = [i²]² = [-1]² = 1
i⁵ = i⁴ i = i
...

What if I divided the exponent with four? and used the remainder as an exponent?

for example...

i¹⁰⁰²

4/ 1002 = 250 with the remainder of 2

so that means i¹⁰⁰² = -1 because i² = -1

do you think this method makes sense?

 

[pka]

What if I divided the exponent with four? and used the remainder as an exponent?
4/ 1002 = 250 with the remainder of 2
do you think this method makes sense?

In fact, that is the only method to make sense of this.

It will also work for negative exponents with one modification.
\(\displaystyle \dfrac{1}{i}=(i)^{-1}=-i\). Thus \(\displaystyle \dfrac{1}{i^{23}}=(i)^{-23}=(-i)^{23}=-i^3=i\)

\(\displaystyle \dfrac{1}{i^{34}}=(i)^{-34}=(-i)^{34}=i^2=-1\)

 

[Steven G]

What if I divided the exponent with four? and used the remainder as an exponent?

for example...

i¹⁰⁰²

4/ 1002 = 250 with the remainder of 2

so that means i¹⁰⁰² = -1 because i² = -1

do you think this method makes sense?

4/1002 is not 250 R 2. Is 4/20 =5? Both of these numbers are less than 1. You need to know this! 20/4 = 5 and 1002/4 =250 R2.