Significant digits We use currency conversion rates for financial calculations. Our currency conversion table stores conversion rates to and from each currency (about 150 world currencies) for each day, going back 20 years. The table has millions of rows. Each conversion rate is stored with 4 decimal places (e.g. USD to EUR $1.5105$).
I was wondering if it's possible to only store rates from other currencies to USD with increased precision and then divide rates for other currencies. For example, if the rate for USD to EUR is $1.5105$ and the rate for USD to Pound is $2.2324$, then the rate  from Pound to Euro would be $1.5101 / 2.2324$. 
If I want the result of division to have 4 significant decimal places, unaffected by rounding, how many decimal places should dividend and divisor have?
Thanks.

First, we must understand the nature of inaccuracy. As for exchange rates, there are random error caused by fluctuations at the market, and measurement error caused by finite number of digits after point in values. Random error is much greater than measurement at almost all liquid financial markets.
Considering random error we can notice, that it has relative nature, i.e. absolute error is  is proportional to the value:
$$\varepsilon_x = \frac{\Delta x}{x}$$
Mean relative error of product of statistically independent values is a sum of relative errors of multipliers:
$$ \varepsilon_x = \sqrt{\sum \varepsilon_{x_i}^2} \qquad for\quad x = \prod x_i$$
Particularly, in your case relative variances are summed: both measurement error and random, and random is much more than measurement. This is answer to your question. But notice, quotes are often statistically dependend, this way you get biased (usually overestimated) value of relative random error. 
In this case is more appropriate to consider relative variances, or stochastic volatility. There are different numerical methods to estimate volatilities.
Another approach is to operate in logarithm values $y_i=\ln x_i$
and calculate their volatilities. Using logarithms you can rewrite you product of quotes as a sum, and their absolute volatilities can be summed. Moreover for for small volatilities:
$$\varepsilon_{x_i} \approx \Delta y_i$$
All above is applicable to shares or currencies (which actually have lognormal distribution), but not to interest rates or discount factors. Also one should keep in mind, that almost all quotes are dependend each other, and it is more proper to use covariance matrices.
