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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.

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Retaining a large number

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

Long term memory of a single long number: Obviously a large proportion of people remember a few phone numbers that are at least as long as the number you mention. So for long term recall there are many strategies for retaining long sequences of digits:

  • practice seeing the number and trying to recall the number
  • try deeper encoding strategies that try to give some structure or meaning to the number
  • A focussed desire to retain the number

Sustained short term memory of a single long number: However, it seems like you are talking about scenarios where you are trying to hold some arbitrary large number in your head and that number is different from occasion to occasion. In general, there are many studies on digit span which look at the length of numbers that can be reliably retained in working memory by normal untrained participants. The traditional claim is that people can hold sequences of digits between about 5 and 9 in their head, but presumably this would depend on a number of factors.

In particular, distraction can often destroy the contents of working memory. Distraction can often come from external sources, but could even be internally driven.

In the case of mathematics, the need to perform other steps in a calculation are a good reason to store intermediate steps externally (e.g., on paper, whiteboard, computer).

Training

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

Training of of short term memory of single long numbers: The Ericcson et al study mentioned by John Henry is a classic study in the expertise literature. The participant was a runner and used a mnemonic strategy based on running times to be able to increase their digit span to well above normal levels. As @John mentions, this skill did not transfer to memory for other objects. It was not a fundamental change to intelligence or working memory capacity. Rather it was a modification to the way that numbers were processed that enabled greater meaning to be given to numbers.

General thoughts: I would not specifically try to learn how to retain numbers in your head, unless that's a party trick that you think would be fun. If you want to get better at a particular form of mathematics, then practice doing problems in that area. In that case, the problem space itself is likely to suggest what you should retain in long term memory, and what can be computed. Or there may be particular problems where you make a deliberate effort to retain in long term memory (e.g., the first few digits of $\pi$ or $e$). Similarly, the more time you spend doing mathematics, the more meaning that will be associated with numbers. In general, richly interconnected mental objects are easier to recall and manipulate.

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Jeromy and John thank you very much for your response. This is a phenomenal study. Are there any more like it?? From the paper the test patient seemed to use a couple of techniques. First chunking the large numbers into 3-4 digits. This chucking was repetitious. 3 4 3 4 3 4 etc. He associated the numbers with TIME. WOW!!!! Why couldn't I think of that. A Much simpler association and one that everyone can relate too. One technique I tried to employ to remember numbers was to associate each digit with a noun/verbs (eg. 2=twins 3=triangle, 4=glasses 5="a high five".. etc.) –  jessica Sep 2 '13 at 17:19
    
However, while this can work for memorizing text, words, speechs, lists. For numbers its a lot harder, tiring, and cumbersone. How many associate stories can you make with the nouns/verbs for numbers with repeated digits (eg. 2235544) Difficult to make a story for -Twins and another set of twins lived in a triangle house with .. etc) –  jessica Sep 2 '13 at 17:21
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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.

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That is a very interesting study actually. Sadly, you must also must consider a bit that that case report happened back in 1980 - which doesn't mean it's less valid it just means that there have probably been improvements to "brain training" type techniques. For instance, is there any evidence to support the contrary nowadays? I'm sure those sites like luminosity have at least one good study behind them. Although this contradicts what I just said, in doing intense research into the physiology behind muscle growth & specific sports the best way was always to practice exactly how you perform. –  user3433 Aug 31 '13 at 21:40
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