# How many significant digits should be retained mid-calculation?

I always feel paranoid when dealing with long series of calculations on paper. How many significant digits should one use to reduce to negligible the probability of getting a wrong final result in a modestly long series of calculations? (I know that the answer to my question depends on the setting. If it helps, I'm a college student; but I would be interested in learning anything on the matter.)

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Are you using a calculator? It is rare nowadays for one to be asked to do anything complicated without one. – André Nicolas Jul 7 '12 at 3:41
Right, so I just keep all digits that the calculator stores. I was teaching a financial math class and students would round to 2 decimal places at an intermediate step and end up off by several dollars. If you're using a calculator, there are keys like "Ans" which bring up the last answer and you can store intermediate results. So, just use all the digits the calculator saves (which is more than what it shows I believe). – Graphth Jul 7 '12 at 3:45
Do you have some specific computations in mind? The number of digits you would want to keep depends on your application. You would probably take many more digits for engineering applications than, say, trying to calculate sales tax on a small purchase. – roninpro Jul 7 '12 at 7:02

If on a test you are not allowed to use a calculator, the calculations, if done properly, do not involve much work with decimals. You just have to resist the urge to convert fractions to decimals until, possibly, the end.

If you are using a calculator, learn how to use the memory feature present on most scientific and financial calculators. This has many advantages. For one thing, it means you do not have to rekey intermediate results. Keying mistakes are a frequent source of error. You are also likely to save a fair bit of time.

It is particularly dangerous to round off interest rates, when the period you are talking about involves several years. Even a smallish error in the interest rate, when compounded for a long enough period, can lead to substantial dollar differences. Another example of the magic of compound interest!

Remark: An additional subtlety is that calculators calculate internally to higher precision than the precision they display. So even if you key in exactly what the calculator shows, you are potentially losing some of the accuracy that the calculator is capable of giving.

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Another subtlety for financial calculations involving time involves the use of functions such as "=NOW()" in a spreadsheet. If you are dealing with daily interest you need to round this. i.e. you need to round some things early.

I discovered this when auditing a colleague's complex spreadsheet for a discounted cash flow - it would never give the same answer - we discovered it was calculating interest by the second (or fraction of a second), and this was making a material difference to the answer.

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