Computer BITS are ones and zeros, while BYTES are kind-of whole characters made up of 8 BITS. I cannot fathom how writing all zeros to every storage bit could ever be recovered to any previous information. But normal, every-day programs are not going to do that. I believe that the deep zero writing is what they now call "wiping".
You're talking about "writing zeros to the drive."
When you delete a file, your operating system sets all of the bits in that sector or block of sectors to zero. This is not the same as "writing zeros." If nothing else gets written to that block, most of the readily available data recovery software can recover that data, even if you format the drive and delete the partition from it.
Writing zeros to the drive actively sets the data in all sectors on the drive to zero. This is called a "low level format." Instead of the operating system pretending there's nothing there until you tell it to put something there, it actually puts a null value there.
DoD standards are something like three passes of writing zeros to the drive, or writing zeros, then garbage data, then zeros again. That may or may not stop a company like DriveSavers from recovering some or all of the original data.
I believe that the deep zero writing is what they now call "wiping".
Seems like the middle of a hot camp fire would take care of wiping a hard drive better and faster. Of course, it's a one time deal.
Guess it depends on how sure you want to be of the wipe. 
Possibly not. DriveSavers allegedly can and has recovered data from drives that have gotten flooded and even shot before.