If it was purchased as an online download, then you should have received an email with the serial number that you used to activate the software. If you had it pre-installed on a Mac that you bought from Apple online, then the serial number should be in the paperwork you received with the Mac. You might also look at your purchase history on
http://store.apple.com for a possible listing of your serial number. If that doesn't help, you might also talk to the Apple sales folks at 800-MY-APPLE. That's less likely to be a resolution for your issue.
Also, if you have a Time Machine backup of your drive, you might restore it to another drive and boot from that drive and launch Aperture. You can then retrieve your serial number from the startup screen or the "About Aperture" window. Hope this helps.
Cheers,
Alex