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Significant Figures

If you’ve ever studied science, you’ve likely come across the concept of significant figures. Significant figures, or significant digits, are an essential part of scientific measurements and calculations. They play a critical role in ensuring the accuracy and precision of scientific data. In this article, we will explore the importance of significant figures and how they are used in science.

Definition:

These are the digits in a number that contribute to its accuracy. They include all of the digits that are known with certainty, plus one final digit that is estimated or uncertain.

For example, the number 3.14 has three significant figures. The first two digits (3 and 1) are known with certainty, while the final digit (4) is estimated.

They are used to convey the precision of a measurement or calculation. They indicate the degree of certainty in a number and how much confidence can be placed in its accuracy. When performing calculations, the result should be rounded to the appropriate number of significant figures to ensure that the final answer is as accurate and precise as possible.

Why are Significant Figures Important?

They are essential in scientific measurements and calculations because they help to ensure accuracy and precision. Inaccurate measurements can lead to incorrect conclusions and flawed scientific theories. These numbers help to reduce the chances of error by clearly conveying the level of precision in a measurement or calculation.

For example, let’s say that you measure the mass of an object and obtain a value of 25.46 grams. If you round this value to 25 grams, you are losing important information about the precision of the measurement. If the actual mass of the object was 25.455 grams, rounding to 25 grams would result in an error of 0.455 grams. However, if you round to two significant figures and report the value as 25.46 grams, you are conveying the degree of precision in the measurement and minimizing the chances of error.

Books on Significant Figures: Click on the Link Given Below

Significant Figures: The Lives and Work of Great Mathematicians

Rules for Identifying

Identify the certain digits in a number and count them as significant.
Non-zero digits are always significant.
Zeros between two non-zero digits are always significant.
Zeros at the beginning of a number are not significant.
Zeros at the end of a number are significant only if the number contains a decimal point.

Examples:

In the number 4.53, all three digits are significant.
In the number 0.0453, only the digits 4, 5, and 3 are significant.
In the number 400, the zero is not significant because it’s at the beginning of the number.
In the number 500.0, the zero is significant because it’s at the end of the number and after the decimal point.

Rounding Off Numbers with Significant Figures

When rounding off a number to a certain number of significant figures, start from the left and count until you reach the desired number of significant figures.
If the digit to the right of the desired significant figure is less than 5, round down. If it’s greater than or equal to 5, round up.

Examples of Rounding Off Numbers with Significant Figures

Round 7.8731 to three significant figures: The third significant figure is 3, so we look at the fourth digit, which is 1. Since it’s less than 5, we round down and the answer is 7.87.
Round 1.4586 to two significant figures: The second significant figure is 4, so we look at the third digit, which is 5. Since it’s greater than or equal to 5, we round up and the answer is 1.5.

Using Significant Figures in Calculations

When adding or subtracting numbers, the result should be rounded to the same number of decimal places as the number with the fewest decimal places.
When multiplying or dividing numbers, the result should be rounded to the same number of significant figures as the number with the fewest significant figures.

Examples:

Add 2.456 and 5.2: The answer is 7.656, which should be rounded to one decimal place because 2.456 only has one decimal place. The answer is 7.7.
Multiply 6.4 and 3.21: The answer is 20.544, which should be rounded to three significant figures because 3.21 only has three significant figures. The answer is 20.5.

Importance:

Significant figures are important for accurate and precise scientific measurements and calculations, and can help ensure that experimental results are reliable.

How are Significant Figures Used in Calculations?

When performing calculations, the result should be rounded to the appropriate number of significant figures to ensure that the final answer is as accurate and precise as possible. The rules for rounding are as follows:

1. If the digit to be dropped is less than 5, round down.

If the digit to be dropped is 5 or greater, round up.
If the digit to be dropped is 5 and followed by non-zero digits, round up.
If the digit to be dropped is 5 and followed by only zeroes, round to the nearest even digit.

For example, let’s say that you need to add 25.46 grams to 10.3 grams. The sum is 35.76 grams, but you should round this value to two significant figures, which gives you 36 grams.

Exercise for Practice

1. What is the result of adding 2.13 and 4.58, rounded off to two significant figures?

6.7 B. 6.71 C. 6.8 D. 6.9
What is the result of multiplying 3.14 and 2.4, rounded off to three significant figures?

A. 7.536 B. 7.54 C. 7.5 D. 7.53

If the length of a rectangle is 3.26 cm and the width is 2.5 cm, what is the area of the rectangle rounded off to two significant figures?

A. 8.15 cm² B. 8.125 cm² C. 8.2 cm² D. 8.13 cm²

If the mass of an object is 25.234 g and its volume is 12.5 cm³, what is its density rounded off to three significant figures?

A. 2.019 g/cm³ B. 2.02 g/cm³ C. 2.02 g/cm² D. 2.02 g/mL

If a car travels 98.6 miles in 2.3 hours, what is its average speed rounded off to two significant figures?

A. 43 mi/hr B. 42.96 mi/hr C. 43.0 mi/hr D. 43.00 mi/hr

If a solution contains 0.025 moles of solute in 250 mL of solution, what is its concentration in moles per liter rounded off to two significant figures?

A. 0.10 M B. 0.100 M C. 0.025 M D. 10.0 M

If the pressure in a gas cylinder is 124.5 kPa and the volume is 0.5 L, what is the number of moles of gas in the cylinder rounded off to three significant figures? (Assume standard temperature and pressure.)

A. 0.002 M B. 0.020 M C. 0.023 M D. 0.025 M

If the length of a wire is measured as 0.9545 m, what is the length of the wire rounded off to four significant figures?

A. 0.954 m B. 0.955 m C. 0.9545 m D. 0.9546 m

If the temperature of a liquid is measured as 24.5 °C, what is the temperature of the liquid in Fahrenheit rounded off to three significant figures? (Use the formula F = (9/5)C + 32.)

A. 76.1 °F B. 76.15 °F C. 76.2 °F D. 76.16 °F

If the time taken for a pendulum to swing back and forth is measured as 1.26 s, what is the period of the pendulum rounded off to two significant figures? (Assume small angle approximation.)

A. 0.79 s B. 0.79 ms C. 0.80 s D. 0.80 ms

Solution or Hints:

What is the number of significant figures in 0.0250?

Answer: 3 significant figures

Explanation: The zeroes between the non-zero digits (2 and 5) are significant, and the final zero at the end is also significant because it’s to the right of the decimal point.

What is the product of 2.50 and 3.1, expressed to the correct number of significant figures?

Answer: 7.75

Explanation: When multiplying numbers with significant figures, the answer should have the same number of significant figures as the number with the fewest significant figures. In this case, 2.50 has three significant figures and 3.1 has two significant figures, so the answer should be expressed with two significant figures. Multiplying 2.50 and 3.1 gives 7.75.

What is the sum of 14.12 and 0.3, expressed to the correct number of significant figures? Answer: 14.4

Explanation: When adding or subtracting numbers with significant figures, the answer should have the same number of decimal places as the number with the fewest decimal places. In this case, 0.3 has one decimal place, so the answer should be expressed with one decimal place. Adding 14.12 and 0.3 gives 14.4.

What is the number of significant figures in 4001?

Answer: 4 significant figures

Explanation: All non-zero digits are significant. In this case, 4, 0, 0, and 1 are all non-zero, so there are four significant figures.

What is the difference between 3.502 and 2.65, expressed to the correct number of significant figures?

Answer: 0.85

Explanation: When subtracting numbers with significant figures, the answer should have the same number of decimal places as the number with the fewest decimal places. In this case, both numbers have three decimal places, so the answer should be expressed with three decimal places. Subtracting 2.65 from 3.502 gives 0.852, which should be rounded to 0.85.

What is the number of significant figures in 0.00234?

Answer: 3 significant figures

Explanation: All non-zero digits are significant, and the zeroes to the right of the last non-zero digit are significant if they’re to the right of the decimal point. In this case, there are three non-zero digits and two zeroes to the right of the last non-zero digit, so there are three significant figures.

What is the quotient of 7.50 and 0.02, expressed to the correct number of significant figures?

Answer: 375

Explanation: When dividing numbers with significant figures, the answer should have the same number of significant figures as the number with the fewest significant figures. In this case, 0.02 has one significant figure, so the answer should be expressed with one significant figure. Dividing 7.50 by 0.02 gives 375.

What is the number of significant figures in 0.0012?

Answer: 2

Explanation: All non-zero digits are significant, and the zeroes to the right of the last non-zero digit are significant if they’re to the right of the decimal point. In this case, there are two non-zero digits and one zero to the right of the last non-zero digit, so there are two significant figures.

Answer: The temperature in Celsius is 24.5 °C. Using the formula F = (9/5)C + 32, we can convert this to Fahrenheit: F = (9/5)(24.5) + 32 = 76.1 °F Rounding to three significant figures, we get the final answer as 76.1 °F.
Answer: The time taken for a pendulum to swing back and forth is called the period. The period is defined as the time taken for one complete swing. In this case, the time taken for half a swing is 0.63 s (half of 1.26 s).

The formula for the period of a pendulum is T = 2π√(L/g), where T is the period, L is the length

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Significant Figures

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