SwiftConvert is a nice simple app for people who need to swiftly convert between different number bases, as programmers often do.

It supports bases of 2, 8, 10 and 16.
Enter a number in any of those bases, and simultaneously see the number in every base.
Just touch the display for the base you want to use for entry - the keyboard will adjust to fit that base.

The binary keyboard even has a 64-bit grid display which lets you toggle any bit directly with your finger, giving you a key for each bit.

You can copy the resulting number in any base, just hold your finger on the display you want and the "copy" menu item will appear.