Significant Figures Multiplication And Division

ii.4: Significant Figures in Calculations

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⚙️ Learning Objectives

Use significant figures correctly in arithmetical operations.

Calculators practice simply what is asked of them – no more and no less. Nevertheless, they can sometimes get a little out of hand. If you multiply 2.49 by half dozen.iii, you get an respond of 15.687, a value that ignores the number of significant figures in either number. Division with a computer is even worse. When you divide 12.ii by 1.7, the answer you obtain is seven.176470588. Neither piece of data is authentic to 9 decimal places, but the calculator does not know that. The man being operating the instrument has to make the decision well-nigh how the answer should be reported.

Rounding

Earlier dealing with the specifics of the rules for determining the meaning figures in a calculated upshot, we need to be able to circular numbers correctly. To round a number, start make up one's mind how many significant figures the number should have. Once you know that, circular to that many digits, starting from the left. If the number immediately to the right of the last pregnant digit is less than 5, information technology is dropped and the value of the concluding pregnant digit remains the same. If the number immediately to the right of the concluding significant digit is greater than or equal to 5, the last significant digit is increased by 1.

Consider the measurement 207.518 thou. As it stands, the measurement contains six significant figures. How would this number be rounded to a fewer number of significant figures? Follow the process equally outlined in Table \(\PageIndex{1}\).

Tabular array \(\PageIndex{one}\): Rounding Examples
Number of Significant Figures	Rounded Value	Reasoning
6	207.518 grand ⇒ 207.518 m	All digits are significant.
five	207.518 g ⇒ 207.52 g	Round upward, since the 6^th digit is an "8" (≥v).
4	207.518 g ⇒ 207.5 k	The final two digits are merely dropped, since the 5^th digit is a "ane" (<5).
3	207.518 m ⇒ 208 m	Circular up, since the iv^th digit is a "five" (≥five).
two	207.518 thou ⇒ 210 m ⇒ 2.1×10^two 1000	Round up, since the three^rd digit is a "7" (≥5). However, trailing zeros in a number that has no decimal point are ambiguous, so express in scientific notation.
i	207.518 yard ⇒ two00 m ⇒ 2×10² one thousand	The last five digits are dropped, since the 2^nd digit is a "0" (<5). The ones place and tens place are held with zeros. However, trailing zeros in a number that has no decimal point are ambiguous, so express in scientific notation.

Discover that the further a number is rounded, the less reliable that measurement becomes. An guess value may be sufficient for some purposes, but scientific piece of work requires a much higher level of detail.

Information technology is of import to exist aware of significant figures when mathematically manipulating numbers. For case, dividing 125 by 307 on a reckoner gives 0.4071661238… to an space number of digits. But do the digits in this answer accept any practical meaning, specially when you are starting with numbers that accept simply iii meaning figures each? When performing mathematical operations, at that place are two rules for limiting the number of meaning figures in an respond – i rule is for addition and subtraction, and ane rule is for multiplication and partitioning.

In operations involving significant figures, the reply is reported in such a way that it reflects the reliability of the least precise operation. An answer is no more than precise than the least precise number used to become the answer.

Multiplication and Partition

Suppose yous wanted to find the area of a rectangle and you measured the dimensions as 13.96 cm by 10.77 cm. Multiplying the 2 lengths together on a calculator yields an area of 150.349 cm². Should the answer be reported as 150.349 cm² or should it exist rounded to 150.35 cmⁱⁱ or 150.3 cm² or even more?

Rectangle Sig Figs.png

Recall that all measurements take uncertainty. Assuming the last digit is the uncertain digit and may be estimated to the nearest ±0.01 cm, a second student may mensurate the dimensions of the rectangle as thirteen.95 cm by ten.76 cm. A tertiary student may measure it equally 13.97 cm by 10.78 cm. A quaternary student may come up up with 13.96 cm by 10.78 cm. The measurements are tabulated in the table below, along with the calculated areas.

If the calculated areas are examined more closely, information technology is easy to see that the hundreds identify (the 1), tens place (the 5), and ones place (the 0) are all certain. They take no variation. We begin to see variation or uncertainty in the tenths place. Since significant figures are divers as all of the sure digits in a measurement plus one uncertain digit, the calculated areas should exist rounded and reported to the nearest ±0.1 cm².

Pupil	Length	Width	Calculated Surface area	Reported Area
i	13.96 cm	10.77 cm	150.three49 cm²	150.iii cm²
2	xiii.95 cm	10.76 cm	150.102 cm²	150.i cm²
3	13.97 cm	10.78 cm	150.597 cmⁱⁱ	150.6 cm²
four	xiii.96 cm	10.78 cm	150.489 cmⁱⁱ	150.five cm^two

This is because the length and width each have four pregnant figures. Multiplying them together results in the area having iv meaning figures, i.e. three certain digits plus one uncertain digit. When the multiplied values have a different number of significant figures, the answer should exist limited to the factor that has the lowest count of significant figures. The same rule applies to segmentation.

⚡️ Rounding Rule for Multiplication and Division

The answer should be rounded then it contains the aforementioned number of significant figures equally the measurement having the fewest number of significant figures.

✅ Example \(\PageIndex{1}\)

Write the answer for each expression using the appropriate number of significant figures.

\(\dfrac{346.ane\;\mathrm{mi}}{five.3\;\mathrm h}=\)
\(fourteen.58\;\mathrm{ft}\;\times\;5.73\;\mathrm{ft}\)
\(36\;\mathrm{in}\;\times26\;\mathrm{in}\;\times16\;\mathrm{in}\)

Solution

Explanation	Answer
The calculated answer is 65.301887 mi/h. The reported answer should accept two significant figures, since 5.3 h has the fewest significant figures (two) between the two measurements.	65 mi/h

Caption	Answer
The calculated answer is 83.5434 ft². The reported reply should have 3 significant figures, since five.73 ft has the fewest significant figures (three) betwixt the ii measurements.	83.five ft²

Caption	Answer
The calculated answer is 14,976 in³. The reported answer should accept ii significant figures, since all three measurements take two significant figures. Since an answer of 15,000 in³ would represent an ambiguous number of significant figures, the respond should be reported in scientific notation with two significant figures.	1.5×10^iv inⁱⁱⁱ

Addition and Subtraction

At present that we know how to report answers that involved multiplication and division, y'all may wonder if answers resulting from improver and subtraction are handled in the same manner. What if nosotros wanted to know the combined distance of 385.17 thou, 6.2 m, and 45.86 m? Notice that the first and last lengths were measured to the nearest ±0.01 thousand, while the second length was measured to the nearest ±0.1 m.

Original Measurements	Change 1st Measurement	Change 2nd Measurement	Change tertiary Measurement

The tabular array above shows what happens to the sum when the uncertain digit in each measurement is changed, i by ane. Notice when the second measurement is inverse from 6.two m to 6.3 m that the sum also changes in the tenths place, going from 437.23 m to 437.iii3 m. In other words, the 2d measurement has the greatest impact on the calculated result since it is the to the lowest degree precise of the three measurements, ±0.1 k vs. ±0.01 m.

Therefore, the sum of 385.17 1000, 6.2 chiliad, and 45.86 yard should be reported as 437.2 m.

This leads united states to the rounding rule for addition and subtraction. While different, it is worded similarly to the rule for multiplication and sectionalisation. Can you lot spot the difference? It is probable the rule with which you are most familiar.

⚡️ Rounding Dominion for Improver and Subtraction

The answer should be rounded so it contains the same number of decimal places every bit the measurement having the fewest number of decimal places.

✅ Example \(\PageIndex{2}\)

1,027 mL + 611 mL + 363.06 mL

Solution

Explanation	Answer
A calculator provides an answer of 267.31 kg, simply considering 8.4 kg is known to the nearest ±0.1 kg, the terminal reply should be expressed to the nearest ±0.ane kg.	267.3 kg

Explanation	Respond
A calculator provides an reply of 2,001.06 mL, but because 1027 mL and 611 mL are both known to the nearest ±1 mL, the final answer should be expressed to the nearest ±1 mL.	2001 mL

✏️ Exercise \(\PageIndex{ane}\)

Write the reply for each expression using the correct number of pregnant figures. As a reminder, calculators do not empathise meaning figures. You are the one who must apply the rules for calculated answers to a result obtained from a calculator.

\(\dfrac{165.110\;\mathrm g}{eight.35\;\mathrm{mL}}=\)
\(viii.6\;\mathrm thou\;+\;32.06\;\mathrm yard\;+\;88.vii\;\mathrm thou\)
\(255.0\;\mathrm{km}\;-\;99\;\mathrm{km}\)
\(44\;\mathrm{cm}\;\times\;43\;\mathrm{cm}\)

Answer A: \(19.viii\;\mathrm g/\mathrm{mL}\)

Answer B: \(129.4\;\mathrm g\)

Answer C: \(156\;\mathrm{km}\)

Answer D: \(1.9\times10^iii\;\mathrm{cm}^2\)

Calculations Involving Mixed Operations

In practice, chemists generally work with a calculator and carry all digits forwards through subsequent calculations. When working on paper, notwithstanding, we oft want to minimize the number of digits we have to write out. Considering successive rounding can compound inaccuracies, intermediate rounding needs to exist handled correctly. When working on newspaper, always round an intermediate result so as to retain at least one more digit than can be justified and acquit this number into the next step in the adding. The final answer is then rounded to the correct number of pregnant figures at the very end.

✅ Example \(\PageIndex{iii}\)

A cylinder filled with water was combined with v.2 cm^three of water that had been drawn into a syringe. If the cylinder had an inner top of 16.eight cm and an inner radius of i.86 cm, what is the combined book of water? Report the answer with the right number of significant figures. (Notation: For a cylinder, \(5\mathit=\pi r^{\mathit2}h\). You may use the \(\pi\) part on your calculator or estimate the value of \(\pi\) as 3.1416.)

Solution

Volume	Caption	Outcome
Syringe	The volume of water in the syringe is 5.2 cm³, known to the nearest ±0.1 cm³.	five.2 cm^three
Cylinder	The book of water in the cylinder = \(\pi r^{\mathit2}h\) = \(iii.141\underline6(1.eight\underline6\;\mathrm{cm})^two(16.\underline8\;\mathrm{cm})\) = \(xviii\underline2.594\;\mathrm{cm}^3\). The rules for multiplication and sectionalisation apply. Since xvi.8 cm and 1.86 cm both accept three significant figures, the calculated volume has three significant figures, or 183 cmⁱⁱⁱ. Compounding of rounding errors in the last respond may be avoided by carrying extra digits forth in the intermediate results. The uncertain digit is underlined (the ones identify).	eighteenii.594 cm³
Combined	The sum of 5.2 cm³ and 182.594 cm³ is 187.794 cm³. The rules for addition and subtraction apply. Since 5. 2 cm³ is uncertain in the tenths identify (±0.i cm³) and 18ii.594 cm³ is uncertain in the ones place (±1 cm³), the sum is uncertain in the ones place (±1 cm³), or 18seven.794 cm³.	188 cm^three

The volume should be reported equally 188 cm³. Check out the video below for additional examples.

Video \(\PageIndex{1}\): Significant figures in mixed operations (https://www.youtube.com/watch?five=yBntMndXQWA).

Summary

Rounding
- If the number to exist dropped is greater than or equal to 5, increment the number to its left by ane.
- If the number to be dropped is less than 5, there is no change.
The dominion in multiplication and division is that the final answer should have the same number of significant figures as the measurement having the fewest significant figures.
The dominion in addition and subtraction is that the last answer is should have the same number of decimal places as the measurement having the fewest decimal places.

Significant Figures Multiplication And Division,

Source: https://chem.libretexts.org/Courses/Anoka-Ramsey_Community_College/Introduction_to_Chemistry/02%3A_Measurements/2.04%3A_Significant_Figures_in_Calculations

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