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Significant Digits and Multi-Step Calculations

Expressing a result with the correct significant digits often requires rounding.  But using rounded results in calculations can sometimes cause errors in the result.  Consider the case of a college student, Simon, who goes to an ATM to withdraw money for a trip to the grocery store.  He wants to get milk ($3.19), cereal ($3.38), chips ($1.99), and salsa ($2.29).  Simon can't get coins from the ATM, so he estimates the number of dollars he'll need.  The milk and the cereal are $3 each, the chips are $2, and the salsa is $2.  Adding these up gives $10.  When Simon takes his $10 to the store, however, the total rings up as $10.85, and Simon does not have enough money.  He followed all the rules for significant figures, but his calculation yielded an amount in error.  This example may seem trivial; after all, Simon could have gotten an extra dollar to cover the change.  But if the result of this calculation were repeated many times, the error becomes significant.  Simon buys the same groceries twice a week, and he wants to figure out how much to budget.  (Do not try Simon's diet; he will eventually die of scurvy.)  If Simon budgets $10 per trip, that becomes $1040 a year.  But he'd spend $1128.40 a year.  That's $88 over budget.  For a college student like Simon, $88 is hard to come by.  Calculations in physics often involve multiplying by large numbers and raising to powers.  These operations can quickly magnify a rounding error.  To avoid rounding mistakes, you should follow the final rule for significant figures:
When using numbers in calculations, it's a good idea to keep one digit beyond the significant digits. Once the final answer is calculated, it may be expressed with the correct number of significant digits.


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