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Significant Figures

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Questions are often raised on the precision of measured quantities. If thickness of a wire is measured using screw gauge as 1.5 mm, the possibility of the measure being 1.6 mm or 1.4 mm is not ruled out. Hence, with the reading 1.5 mm is said to have a nearest precision of 0.1 mm. The digits like 1 and 5 in 1.5 are significant figures which give reliability to the extent required on the reading 1.5 mm.

We understand the digits other than 0 contribute to the accuracy of a measured value. But, the 0 present in certain places are considered significant in readings and measurements. Let us learn more about significant figures and the arithmetic associated with them.

Define Significant Figures

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Significant figures are defined as digits in numbers which are important to attach a required precision on the number.
Significant figures are also known as significant digits.
The significant digits do not include zeros present in a number, as place holders or zeros appended to present a greater precision than that is actually present. Numbers with 100% reliability are known as Exact numbers, like the counted number of people or objects and the whole number conversion factors. An exact number is considered to have infinite number of significant digits.

Significant Figures Rules

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Given below are some of the rules of significant figures:
  1. All non zero digits are significant. For example, the number 82 has two significant digits and 4.35 has three significant figures.
  2. Zeros placed between non zero digits are significant. Each of the Numbers 9.02 and 803 have three significant digits.
  3. Zeros to the left of first non zero digit in a number are not significant. These zeros function only as place holders. For example, both the numbers 0.24 and 0015 have only two significant digits.
  4. Trailing zeros of a number with a decimal point are significant as the presence of these zeros indicate rounded measures. For example, the number 1.20 means that it is rounded to the hundredth and hence has three significant digits.
  5. The significance of trailing zeros in a number not containing a decimal point cannot be determined by mere observation. For example, The number 23,400 may be given precise to a unit or rounded to the nearest hundred. In the first case, the the number has three significant digits while the rounded number has 5 significant figures.

In the last case, different notations are used to indicate whether the trailing zero is significant or not.

  • A bar is placed above the last significant digit or it is underlined.
  • If all the trailing zeros are significant, a decimal point is placed after the last zero in the number.
  • Writing the number in scientific notation clears the ambiguity.

Rounding Significant Figures

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The rules applied to round off a number to a given significant figures is similar to the rounding rules applied with decimals.
  • Counting for significant digits is done from left to right. If the given number is a decimal less than 1.0, the counting of significant digits start from the first non zero digit.
  • Round up. That is, add 1 to the last digit counted, if the immediate digit to its right is greater than or equal to 5 and not all the other digits following are zero. Round down (retain the number) if the digit to the right is less than 5.
  • If the immediate digit to the right of number to be rounded is 5 followed by all zeros, the digit is rounded up to the next even number if odd, and the digit is retained if even.
  • Ignore all the other digits. Append enough zeros to keep the number to the right size.

The following table demonstrates the rounding done on numbers to different significant numbers.

Number
Rounded to
1 SF
Rounded to
2 SF
Rounded to
3 SF
36,281 40,000 36,000 36,300
625.76 600
630
626
4,972 5,000
5,000 4, 970
0.006358 0.006 0.0064 0.00636
208.05 200
210
208
0.7028 0.7
0.70
0.703
7.0500 7
7.0
7.05
8.13500 8 8.1
8.14

In the last two case, the digit is rounded up or down to an even digit, if it is followed by 5 with trailing zeros.
7.0500 is rounded down to two significant digits as 7.0 as the digit rounded 0 is even. On the other hand, 8.13500 is rounded up to three significant digits as 8.14, as the digit rounded 3 is odd.

Adding Significant Figures

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The arithmetic of adding significant figures consists mainly of two steps, adding the numbers and rounding the sum suitably.
  1. The numbers are added just as two decimal numbers are added.
  2. Round the answer to the least number of decimal places in the decimal portion of any number.

Solved Example

Question: Add the measures 50.85 gms and 100.280 gms.
Solution:
The first measure 50.85 gms has four significant figures with two decimal places.
The second number has 6 significant digits with three decimal place.
50.85 + 100.280 = 151.130 gms
The measure 50.85 has the least number of decimal places.
Hence, the sum of two measures = 151.13 gms.

Subtracting Significant Figures

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The rule applied for subtraction done between measures for significant figures is same as that applicable for addition. The difference found is rounded to the least number of decimal places present in the two measures.

Solved Example

Question: Subtract the measure 6.865 seconds from 14.38 seconds.
Solution:
14.38 - 6.865 = 7.515.
The lesser number of decimal places 2 is observed in the measure 14.38 seconds.
Hence, the difference is rounded as 7.52 seconds.

Multiplying Significant Figures

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While performing multiplication with measured quantities, the product is rounded to the same significant digits as the number with smallest significant digits.

Solved Example

Question: Find the area of the rectangular sheet whose dimensions are measured as 6.52 ft and 0.65 ft.
Solution:
Of the two measures, 6.52 ft has three significant digits and 0.65 has 2.
The area which is the product of these two measures is to be rounded to two significant digits.
6.52 x 0.65 = 4.238 = 4.2 sq.ft when rounded to two significant digits.

Dividing Significant Figures

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The rule applied to division is the same as that followed for multiplication of measured quantities.

Solved Example

Question: Divide 15.8 meters by 0.25 meters.
Solution:
The quotient is to be rounded to two significant digits as the measure 0.25 meter has the smaller significant figures.
15.8 $\div$ 0.25 = 63.2 = 63

Significant Figures Problems

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Given below are some of the problems on significant figures.

Solved Examples

Question 1: The reading on a burette containing a solvent was 8.9 ml. After taking out some solvent, the reading was found to be 4.38 ml. Find the amount of solvent taken out observing the standard rules for significant digits.
Solution:
We need to find the difference between initial and final reading on the burette.
Quantity of solvent released = 8.9 - 4.38 = 4.52
Out of the two reading, 8.9 ml has lesser number of decimal places which is one. Hence, rounding 4.38 to one decimal place.
The quantity of solvent released = 4.5 ml

Question 2: The population of a small town is given rounded to two significant digits as 2,400. What are the least and the greatest number of people expected to live in the town?
Solution:
We get the least number, if we assume that the population is rounded up to an even digit. This is the case, when the actual population was 2,350. Note that the immediate digit to the right of the rounding place is 5 followed by 0. Hence, the digit 3 is rounded up an even digit 4 and the number is rounded to two significant figures as 2,400. Same way, the greatest possible number of people got if rounding was down to even is assumed. That is, for the actual number of 2,450. Hence, we can expect that the town has at least 2,350 and at the maximum 2,450 residents.