Yes, if the rhizome is long enough you can cut into it and have two rooted rhizomes which will look the same, you can call this 'making an offset' if you want to think of it that way. Lleaves should root and make a plant, but you might want to look into that type of propagation a bit. My method would be to take leaf, soak it until it is turgid, then , with a bit of rooting hormone on top or on the stem, place it on damp perlite or damp sphagnum. This time of year it would be well to make sure the rooting material won't dry out, I would put a clear container over material, such as a clear plastic cup, there are lots of methods for propagation and it is really quite easy after you learn.

If you take the cutting and place it in a humid environment as soon as possible, your fortune will be multiplied greatly. As the plant ages and the rhizomes grow, it should (operative word - should) make some offsets but often it may just produce new leaves.

To insure more plants do try the leaf propagation again but use the right environment for nearly 100% success. (rooting hormone may help but is not necessary). There are many methods to make more but one thing seem to make the most of it is a humid enclosure at least indoors. Outdoors you can usually get by without an enclosure except to exclude bugs, debris, and slugs and snails.

A lettuce container from the grocery store is a very good enclosure. A shallow layer of good potting mix (DO NOT WET) that has some moisture to it (not totally dried out in other words) is all you really need. Place your cuttings inside barely making contact with the soil. A much better method is the Jiffy peat pellets with the Jiffy greenhouse - while not fool proof, it yields very good results.

Here is a photo of one of the lettuce containers I used recently. Not the best example but I don't want to look through tons of photos right now. The small green leaves are from one large leaf I got from a friend a couple of months ago. I've added the other leaves over time (I have no hesitation about overcrowding).

Here is a partial tray of Jiffy cube propagation from leaves pinched from other plants.

And for a large leaf (you can use the entire leaf with as long a stem as you can squeeze in the container or cut it into smaller wedges around the major veins and stick those vertically). Here is a large leaf for example I put down early this month. I added the heracleifolia hybrid leaf a week or so ago.

And the roots of the larger leaf (Island Magic)

By simply placing leaves gently on a moistened surface is often all you need to do.

when the begonia looks like it is going to die, then DO NOT water but DO place a humidity cover over the struggling plant. I bought a beautiful rex in May but since bringing it indoors it has gone downhill to one small rhizome. I placed a drinking cup over it and it has sent up two new shoots since then. I have to check regularly though since one of the dying rhizomes was getting mold on it and I did not want it to spread to the sole survivor.

Here is what the plant can look like when grown by the commercial grower (Firestorm) shown at the convention in Tampa (March 21, 2014)

The plant as I bought it when I picked up plants in May for our club's annual sale.

A simple drinking cup over the soil surface makes an effective humidity tent.