With respect to the integers, 2 is prime. But with respect to the Gaussian integers, it’s not: it has factorization 2=(1−i)(1+i). Here’s what’s happening.
You can view complex multiplication as scaling and rotating the complex plane. So, when we take our unit vector 1 and multiply by (1+i), we’re scaling it by |1+i|=√2 and rotating it counterclockwise by 45∘:
This gets us to the purple vector. Now, we multiply by (1−i), scaling it up by √2 again (in green), and rotating it clockwise again by the same amount. You can even deal with the scaling and rotations separately (scale twice by √2, with zero net rotation).
With respect to the integers, 2 is prime. But with respect to the Gaussian integers, it’s not: it has factorization 2=(1−i)(1+i). Here’s what’s happening.
You can view complex multiplication as scaling and rotating the complex plane. So, when we take our unit vector 1 and multiply by (1+i), we’re scaling it by |1+i|=√2 and rotating it counterclockwise by 45∘:
This gets us to the purple vector. Now, we multiply by (1−i), scaling it up by √2 again (in green), and rotating it clockwise again by the same amount. You can even deal with the scaling and rotations separately (scale twice by √2, with zero net rotation).