pixelation - What is the resolution I need to make a picture to make curves look smooth and not pixelated?

I am using Paint to draw some swords for a character I have. I am not very experienced and this is my first time trying to draw swords.

My question is how large (Preferably in pixels) do I need to make the picture to make the curves on the sword to look smooth. I am using current scale of 100 pixels = an inch. For the first sword I made this worked fine as there were few curves and they were small but the one I am working on now have a larger curve. (A good reference is Ichigo's shikai.)

EDIT: I only have access to paint. I would use other programs if I could.

Answer

For screen or print?

For screen, the resolution setting is ignored. What matters is the number of pixels you are using and the resolution of the screen itself. An Apple retina screen is going to look a lot smoother than a non retina screen, as it has a lot more pixels per inch, for example.

In addition, there's the issue of anti-aliasing, which is where individual pixels are 'blended' from one to the next to give the appearance of smoothing from a distance. Simple example:

For print, it's going to depend on what you are printing and how, but typically for photographic/4-color imagery, you want to have images that can be printed at 300dpi. For line art, you typically want 1200dpi.