i - Page 6

Beautiful Involutions

Exploring the most basic propertis of i reveals a remarkable order within the strangest of all possible numbers. We have declared that:

i² = −1

We know that any number to the zero power is by definition 1, and anything raised to the power 1 is just itself, so together we have:

i⁰ = 1

i¹ = i

i² = −1

Following this, we can deduce further values from simple math rules:

i⁰ = 1

i¹ = i

i² = −1

i³ = i² × i

i⁴ = i² × i²

Now we just substitute the known value of i² = −1 :

i⁰ = 1

i¹ = i

i² = −1

i³ = −1 × i

i⁴ = −1 × −1

To arrive at:

i⁰ = 1

i¹ = i

i² = −1

i³ = -i

i⁴ = 1

It's a closed cycle: after four multiplications by i, we return to 1 again. At the same time, each multiplication alternates between a real value (1, −1) and an imaginary one (i, −i).

This is behavior we've seen before. Just as multiplication by −1 alternates between the positive and negative halves of the number line, i serves to alternate between real and imaginary numbers. And like −1, successive multiplication by i eventually returns us to our original value.