When you're solving a problem where an approximation suffices for the answer,
the most commonly-requested approximation looks something like this:
“please round to
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$\,2\,$ decimal places”
or equivalently,
“please round to the hundredths place”.
The purpose of this section is to discuss the concept and the technique of rounding.
Let's start with an example.
Suppose you're asked to round
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$\,x = 2.73845\,$ to two decimal places.
The number $\,x = 2.73845\,$ lies between $\,2.73\,$ and $\,2.74\,$ on the number line.
Indeed, it lies just about here:
If we're rounding to two decimal places,
then the two “candidates” for the desired approximation are
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$\,2.73\,$ and $\,2.74\,$.
Which candidate is $\,x\,$ closest to?
Answer:
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$\,2.74\,$
So we say:
“$\,2.73845\,$, rounded to $\,2\,$ decimal places, is $\,2.74\,$”
or equivalently,
“$\,2.73845\,$, rounded to the hundredths place, is $\,2.74\,$”.
You could now be asked the follow-up question:
“Did you round up, or round down?”
When the candidate to the right of the original number is chosen,
we say that we are rounding up,
because we're moving “up” (to the right) on the number line to get the approximation.
When the candidate to the left of the original number is chosen,
we say that we are rounding down,
because we're moving “down” (to the left) on the number line to get the approximation.
To “round to
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$\,n\,$ decimal places”
means to choose the closest neighbor that uses exactly $\,n\,$ decimal places.
When the two candidates are equidistant, the normal convention is to round UP;
that is, choose the candidate on the right.
The ideas illustrated in the previous example can be efficiently implemented as follows.
ROUNDING TO $\,\,n\,\,$ DECIMAL PLACES | |
GENERAL PROCEDURE | EXAMPLE: Round [beautiful math coming... please be patient] $\,2.73845\,$ to $\,2\,$ decimal places. |
STEP 1: (rounding digit) Go to the digit in the [beautiful math coming... please be patient] $\,n^{\text{th}}\,$ decimal place. This is called the rounding digit. |
Because we're rounding to
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$\,n = 2\,$ decimal places, we want the digit in the second decimal place ($\,2.7\underline{3}845\,$). The rounding digit is $\,3\,$. |
STEP 2: (decider digit) Look at the next digit to the right. This is called the decider digit. It is used to decide whether to round up or down (next step). |
What digit is to the right of the
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$\,3\,$? ($\,2.7\underline{3}{\bf8}45\,$) Answer: $\,8\,$ The decider digit is $\,8\,$. |
STEP 3: (rounding UP or DOWN) If the decider digit is less than [beautiful math coming... please be patient] $\,5\,$, then round DOWN: the rounding digit remains the same, and all the digits to the right are dropped. If the decider digit is [beautiful math coming... please be patient] $\,5\,$ or more, then round UP: the rounding digit increases by $\,1\,$, and all the digits to the right are dropped. If the rounding digit is [beautiful math coming... please be patient] $\,9\,$ and you're rounding up, this is a special case. See the next step! |
Our decider digit is
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$\,8\,$, which is more than $\,5\,$. We need to round UP. The rounding digit increases by $\,1\,$ (from $\,3\,$ to $\,4\,$); all digits to the right are dropped. So, $\,2.73845\,$ rounds to $\,2.74\,$. |
SPECIAL CASE: ROUNDING DIGIT IS [beautiful math coming... please be patient] $\,9\,$, ROUNDING UP If the rounding digit is [beautiful math coming... please be patient] $\,9\,$ and you're rounding up, then the rounding digit becomes a $\,0\,$ (which you keep), and the next digit to the left increases by $\,1\,$. You may need to apply this rule more than once. Whenever a [beautiful math coming... please be patient] $\,9\,$ must be increased by $\,1\,$, then the $\,9\,$ becomes a $\,0\,$ (which you keep), and the next digit to the left increases by $\,1\,$. |
NEW EXAMPLE: Round [beautiful math coming... please be patient] $\,2.79845\,$ to $\,2\,$ decimal places. The rounding digit is $\,9\,$. The decider digit is $\,8\,$, so we round UP. The $\,9\,$ becomes a $\,0\,$ (which we keep); the digit to the left increases by $\,1\,$ (from $\,7\,$ to $\,8\,$). So, [beautiful math coming... please be patient] $\,2.79845\,$ rounds to $\,2.80\,$. Good way to think of this: the ‘$\,79\,$’ in $\,2.\underline{79}845\,$ becomes an ‘$\,80\,$’. |
CAREFUL!
When you are asked to round a number to
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$\,n\,$ decimal places,
then your answer must have $\,n\,$ decimal places,
even if there are zeros in those positions.