Remember that pi in MATLAB is not truly pi, any more than pi is 3.14, or even 3.14159. pi is a transcendental number. There are infinitely many digits in pi.
And, yes, that looks vagely like pi. Many digits, but once we get past the first 16 digits or so, we need to recognize that pi there has only the first 16 digits correctly represented. We have an error in the least significant bits, as there must be in any finite representation of a transcendental number.
Double precision numbers are stored in a BINARY form. So they have 52 binary bits. In the case of pi, the binary form would look like
11.001001000011111101101010100010001000010110100011000
So there, the ones represent powers of 2. But the problem is, the true binary form for pi goes on forever, whereas in MATLAB, it stops at that point.
The result of all this is that pi as a double precision number is not truly pi. Close. But not truly pi, any more than pi==3.14. And if we try that simple aproximation for pi, we see the problem far more obviously.
exp(3.14*i*6)
ans =
0.999954342529211 - 0.009555776105247i
Of course you would not expect that result to be exact. But what you did was essentially the same thing.
If you want MATLAB to represent pi as the transcendental number pi, then you need to do something like this
where MATLAB now understands you want to use the actual number pi, instead of a floating point approximation to pi.
