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http://nbviewer.jupyter.org/gist/leftaroundabout/83df89a7d3bdc24373ea470fb50be629

DFT, size 16

Diagram of the operations in a size-16 naïve DFT

FFT, size 16

Diagram of the operations in a size-16 radix-2 FFT

The difference in complexity is pretty evident from that, isn't it?