Can you retain a large number in your head for an extended period of time without distortion? If someone where to recite "794 times 869", they might struggle to retain the numbers pitch perfect in their head as I perform the calculations.

Are there any exercises one could perform to improve his/her visual retention of objects and numbers in their heads?

Can you hold the number 7859385 in your head for an extended period without the image of the number becoming distorted at any point?

Note that I have cross-posted on Math SE.
Yes, you can train your memory to be better at certain tasks, such as remembering numbers. 

For example Ericcson et. al. (1980) describe a university student who practiced memorizing numbers several times per week for twenty months and could then memorize and recall more than 70 digits reliably.

I would not recommend such however if you are looking for general purpose memory function. The benefits gained from cognitive training are shallow and do not transfer from one skill to another. For example, while the student was a genius at memorizing numbers, when asked to memorize letters he was not better than a normal person

References

Ericcson KA, Chase WG, Faloon S (June 1980). "Acquisition of a memory skill". Science 208 (4448): 1181–2.