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Here are a few recommended readings before getting started with this lesson.
Here are a few practice exercises before getting started with this lesson.
There are different scenarios where not all digits of a number are meaningful. In these cases, the number of decimal places required is usually given. However, there are cases where the number of decimal places needed can be deduced from the problem's context.
In any case, it is important to know how to round a number to a specific number of decimal places.
Rounding a decimal number to $n$ decimal places means to rewrite the number in a simpler and shorter form with exactly $n$ decimals. The final result will be an approximation to the given number.
The number $27.6495183$ rounded to two decimal places is $27.65.$
The following steps should be followed to round a number to $n$ decimal places.
Consider the number $27.64951.$ In each column, the corresponding decimal place is highlighted.
Round to One Decimal Place  Round to Two Decimal Places  Round to Three Decimal Places 

$27.64951$  $27.64951$  $27.64951$ 
Round to One Decimal Place  Round to Two Decimal Places  Round to Three Decimal Places 

$27.6∣4951$  $27.64∣951$  $27.649∣51$ 
Round to One Decimal Place  Round to Two Decimal Places  Round to Three Decimal Places 

$27.6∣4951↓27.6∣4951 $

$27.64∣951↓27.65∣951 $

$27.649∣51↓27.6410 ∣51↓27.650∣51 $

The following table shows the results of rounding $27.64951$ to one, two, and three decimal places.
Round to One Decimal Place  Round to Two Decimal Places  Round to Three Decimal Places 

$27.6$  $27.65$  $27.650$ 
Mrs. Mathematiks asked Izabella to paint the Hubble Telescope on the classroom wall. Izabella thinks that she should draw a grid before doing the actual painting. She is told that squares with a side length of $7.374$ centimeters will work well, but she only has this ruler.
According to this ruler, determine what side length Izabella could use to draw the square accurately. In this situation, it is not possible to measure exactly $7.374$ centimeters.Since the ruler only shows the measurements down to the tenth of a centimeter, round the original length to the tenth value, one decimal place.
The exact side length of the square Izabella wants to draw is given to the thousandth of a centimeter. Therefore, it has a precision of three decimal places.
Because the ruler measures to the tenth of a centimeter, Izabella cannot draw the square using the exact dimensions. To draw the square precisely, she must round the original length to a precise length that the ruler can measure.
The exact length must be rounded to $one$ decimal place.
Next, apply the steps to round a decimal number.
In conclusion, Izabella should use a length of $7.4$ centimeters to precisely draw the square's sides.
Izabella used a pencil to draw an outline of the previously measured square to make a grid layout of many squares. Doing this helped her paint the telescope accurately. The image shows how the grid helped her painting process.
Round the given number to the specified number of decimal places.
When announcing lottery results, an approximation of the winnings is generally used instead of listing the actual amount of money won.
In this case, the actual amount written in the title has been rounded to have only one significant figure. Rounding to significant figures is very similar to rounding to decimal places except for the final step.
Rounding a number to $n$ significant figures means to rewrite the number so that it has exactly $n$ significant figures. The result is an approximation of the number.
The number $6213.4983$ rounded to three significant figures becomes $6.21×10_{3}.$
Consider the number $6213.4983.$ In each column, the corresponding significant figure is highlighted.
Round to Three Significant Figures  Round to Five Significant Figures  Round to Six Significant Figures 

$6213.4983$  $6217.3983$  $6217.3983$ 
Round to Three Significant Figures  Round to Five Significant Figures  Round to Six Significant Figures 

$621∣3.4983$  $6213.4∣983$  $6213.49∣83$ 
Round to Three Significant Figures  Round to Five Significant Figures  Round to Six Significant Figures 

$621∣3.4983↓621∣3.4983 $

$6213.4∣983↓6213.5∣983 $

$6213.49∣83↓6213.410 ∣83↓6213.50∣83 $

The following table shows the results of rounding $6203.4983$ to $3,$ $5,$ and $6$ significant figures.
Round to Three Significant Figures  Round to Five Significant Figures  Round to Six Significant Figures 

$6213.4983↓6210 $

$6213.5983↓6213.5 $

$6213.5083↓6213.50 $

It may happen that rounding a number results in an integer with fewer digits than the number of digits of the integer part of the original number. In this case, a bar is drawn above the last significant digit to indicate that there were more significant digits.
Rounding the Number $43346$ to  

Two Significant Figures  Three Significant Figures  Four Significant Figures 
$43ˉ000$ or $4.3×10_{4}$  $433ˉ00$ or $4.33×10_{4}$  $4335ˉ0$ or $4.335×10_{4}$ 
Alternatively, writing the number using scientific notation helps identify the number of significant figures.
In the last rounding step, the numbers to the right of the second significant figure and the left of the decimal point must be replaced with zeros.
According to her boss, Paulina must round the number $21392$ to $two$ significant figures. This can be done by following the steps below.
Paulina will announce on the radio that about $21000$ people have signed up for the marathon.
Round the given number to the specified number of significant figures.
When carrying out computations with significant figures, scientists usually use two different methods for rounding off a final answer. One method is used for addition and subtraction, while the other is used for multiplication and division.
Addition or Subtraction  Multiplication or Division 

Round the result to the least number of decimal places in the numbers involved in the calculation.  Round the result to the least number of significant figures in the numbers involved in the calculation. 
The following table shows how to round the sum and the product of the numbers $34.78$ and $156.294.$ Notice that the least number of decimal places is two and the least number of significant figures is four.
Computation  Rounded Result 

$34.78+156.294=191.074$  $191.07$ ($2$ decimal places) 
$34.78⋅156.294=5435.90532$  $5436$ ($4$ significant figures) 
The words accuracy and precision play an important role when doing measurements and rounding. Accuracy refers to the closeness of the measurements to the exact, known, or acceptable value. Precision is the closeness of the measurements to each other.
The word precision is also used to indicate the number of decimal places of a number. Moreover, it can be used to determine which of two numbers is given with more detail.
The absolute error is the absolute value of the difference between a measured value and the exact value. The units of the absolute error correspond to the units of the measures. By definition, the absolute error is always greater than or equal to $0.$
$Absolute Error$ $=$ $∣∣∣ Measured Value−Exact Value∣∣∣ $
Eager to dish out another challenge, Mrs. Mathematikz gave Izabella, Paulina, and another friend Jordan the following problem.
The three friends solved the problem separately and came up with the following results.$Result_{Izabella}=15.6164$, $Exact=15.6$
Subtract terms
$∣0.0164∣=0.0164$
Absolute Error  

Izabella  $∣15.6164−15.6∣=0.0164$ 
Paulina  $∣15.62−15.6∣=0.02$ 
Jordan  $∣15.616−15.6∣=0.016$ 
The relative error is the ratio of the absolute error of a measurement to the exact value. The relative error tells how good a measurement is relative to the size of the object being measured. In other words, the relative error indicates how significant the absolute error is.
Relative Error $=Exact ValueAbsolute Error $
The Relative Error Formula can be rewritten by substituting the Absolute Error Formula.
Relative Error $=Exact Value∣Measured Value−Exact Value∣ $
The percent error is the product between the relative error and $100%.$ It represents the relative error as a percentage.
Percent Error $=Relative Error⋅100%$
Paulina did such a great job as a radio host that she is invited to announce another smaller race.
She notices something interesting about the runners at this smaller race — some are wearing headbands, and some are wearing sunglasses. She decides to take note of her observation to share it on the radio as a fun fact.
Headband  Sunglasses  

Number of Counted Runners With an Accessory  $26$  $111$ 
Actual Number of Runners With an Accessory  $27$  $115$ 
$Measured Value=26$, $Exact Value=27$
Subtract terms
$∣1∣=1$
$Measured Value=111$, $Exact Value=115$
Subtract terms
$∣4∣=4$
The Hubble Space Telescope is a telescope the size of a school bus that travels around Earth. Scientists and astronomers have learned a lot about space from Hubble's pictures of stars, planets, and galaxies.
After its launch in $1990,$ NASA found an error in the primary mirror that made the first few images appear blurrier than expected. It turned out that the curvature of the mirror was deflected by less than onemillionth of a meter — just onefiftieth the width of a human hair! The following photo, taken in November $1993,$ shows the core of the galaxy $M100.$
NASA's first solution was to build replacement instruments — similar to a pair of prescription lenses — to fix the flaw. These repairs were done in December of $1993.$ The picture below, taken after the repairs, is the same galaxy shown in the first photo. What a difference!
This case shows that even though the error was a mere onefiftieth of the width of a human hair, the final result was significantly impacted by the error.