The Need For Significant Figures In Physics
Physics calculation relies heavily on the accuracy and precision of the measurements under the same conditions. Significant figures in physics help you do just that.
When you have to determine the accuracy and precision of measurement in physics, Significant Figures enable you to get to that. Besides, when you reduce a value to its significant digits, you basically declutter it.
SO, when you have to do the further calculation, you can use only the significant figures of the value and still get the same level of accuracy and precision with your measurement.
Other Numbers Having Significant Figures
Direct measurement is not the only way a number may contain significant digits. The number may be an Exact or Defined Number, it may be an integer, or the number could have been computed from numbers that have significant digits.
Defined numbers: The base of the natural logarithms is e = 2.781828 . This number has a mathematical definition and is exact. Every digit you choose to display from this number is significant. = 3.14159, the square root of 2 and similar numbers are also exact. Defined unit conversion values are also exact. For example there are exactly 2.54 centimeters to the inch. This is how the inch is defined. So this number is exact.
Integers: When you count, the result is exact . If you tell a friend that you have paid $2000 dollars for a computer, there is only one significant figure in this number. Most people will not trouble their friends with the price One thousand nine hundred eighty seven dollars and thirty six cents . On the other hand the year 2000 computer problem that received so much press is a number with four significant figures. The count of years is exact. The problem did not occur when computer date counters flipped to 99, and is a dead issue when the flip to 01.
When we convert rational numbers to decimal fractions they always produce a set of repeating digits. For example = 3.2121212 .
What Are The Significant Figures Rules
To determine what numbers are significant and which aren’t, use the following rules:
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Significant Figures Solved Examples
1. The result of an experimental calculation corrected up to seven significant figures is 7.363573. Round it off to six, five, four, three and two significant figures.
The given quantity P = 7.363573When rounding off to 6 significant figures, P6 = 7.36357 .When rounding off to 5 significant figures, P5 = 7.3636 .Similarly, When rounding off to 4 significant figures, P4 = 7.364When rounding off to 3 significant figures, P3 = 7.36When rounding off to 2 significant figures, P2= 7.4
2. The length, breadth and thickness of a rectangular sheet of metal are 2.324m, 2.005m and 1.01 cm respectively. Find the surface area and volume of the sheet to the correct significant figures.
Given, Length, l = 2.324 mbreadth, b = 2.005 mthickness, t = 1.01 cm = 0.0101 m
As thickness has least number of significant figures , the area should be rounded off to three significant figures. Also, the fourth significant figure in the answer is greater than 5, so 1 should be added to the last digit retained. Area corrected to significant figres = 9.41 m2
Similarly, the volume should be rounded off to three significant figures. Also, the fourth significant figure in the answer is greater than 5, so 1 should be added to the last digit retained. Volume corrected to significant figures = 0.0471 m3
3. 6.84 g of a substance occupies 1.3cc. Express its density by keeping the significant figures in view.
Rules To Identify Significant Figures In A Number
Note that identifying the significant figures in a number requires knowing which digits are reliable since only reliable digits can be significant e.g., 3 and 4 in 0.00234 g are not significant if the measurable smallest weight is 0.001 g.
- Non-zero digits within the given measurement or reporting resolution are significant.
- 91 has two significant figures if they are measurement-allowed digits.
- 123.45 has five significant digits if they are within the measurement resolution. If the resolution is 0.1, then the last digit 5 is not significant.
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Significant Figures In This Text
In this text, most numbers are assumed to have three significant figures. Furthermore, consistent numbers of significant figures are used in all worked examples. You will note that an answer given to three digits is based on input good to at least three digits, for example. If the input has fewer significant figures, the answer will also have fewer significant figures. Care is also taken that the number of significant figures is reasonable for the situation posed. In some topics, particularly in optics, more accurate numbers are needed and more than three significant figures will be used. Finally, if a number is exact, such as the two in the formula for the circumference of a circle, c = 2r, it does not affect the number of significant figures in a calculation.
How To Count Significant Figures
Here are the rules you need to know for identifying significant figures.
All of the following are significant figures
And these are not significant figures
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What Is The Relationship Between The Tolerance And The Number Ofsig Figs
The “tolerance” is the range of numbers that WebAssign will markas correct. The tolerance can be set by the problem coder to justabout anything. But by default, the tolerance is 2% of the “key”. The”key” is the exactly correct value. For example, if the key were”9.81 m/s^2″, with a tolerance of 2%, then any entry between”9.6138 m/s^2″ and “10.0062 m/s^2” would be counted ascorrect. This is if there is no sig figs checking.
If sig figs are being checked, then in the question coding thereis a variable called $SIGFIGS that is set to a number greater than 0.That number determines how many significant figures there must be inorder for the question to be marked correct. By default, any entrythat does not have exactly that number of sig figs will be markedtotally wrong , regardless of the value of thenumber.
If sig figs checking is used, the default tolerance is not acertain percent of the key but is instead usually 1 of the leastsignificant digits. For example, if, in the previous problem,$SIGFIGS was set to 3, then the tolerance would be set to allowanswers between 9.80 m/s^2 and 9.83 m/s^2.
Sig Figs In Numbers With Uncertainty
When a value is reported with an uncertainty, special rules apply. Theuncertainty can affect the required number of significant figures in thevalue. A large uncertainty may require that the value be rounded to fewerfigures, and a very small uncertainty may imply that the value be known to agreater number of figures. In addition, the uncertainty itself cannot havean arbitrary number of significant figures.
A number that is to be used as an uncertainty itself isfundamentally an estimate: uncertainty cannot be known precisely. Thus, by convention, you must never use more than 2 significantfigures to state it.
Remember these two rules for stating a number with an uncertainty: The uncertainty should be stated with 1 or 2 significantfigures. A value and its uncertainty should be stated to the sameprecision: they should have the same number of digits past thedecimal point. Usually, if the leading nonzero digit in theuncertainty is greater than one, there should be only onesignificant digit in the uncertainty. But, if the leading digit is a1, there may be 2 significant digits in the uncertainty. Unitsshould always be written after the uncertainty. Note thefollowing examples:
|Not OK, too few digits in value.|
|2.6000e2±0.02 m/s2||OK, both numbers stated to same precision.|
How To Use The Sig Fig Calculator
Our significant figures calculator works in two modes – it performs arithmetic operations on multiple numbers or simply rounds a number to your desired number of sig figs.
Following the rules noted above, we can calculate sig figs by hand or by using the significant figures counter. Suppose we have the number 0.004562 and want 2 significant figures. The trailing zeros are placeholders, so we do not count them. Next, we round 4562 to 2 digits, leaving us with 0.0046.
Now we’ll consider an example that is not a decimal. Suppose we want 3,453,528 to 4 significant figures. We simply round the entire number to the nearest thousand, giving us 3,454,000.
What if a number is in scientific notation? In such cases the same rules apply. To enter scientific notation into the sig fig calculator, use E notation, which replaces x 10 with either a lower or upper case letter ‘e’. For example, the number 5.033 x 1023 is equivalent to 5.033E23 . For a very small number such as 6.674 x 10-11 the E notation representation is 6.674E-11 .
When dealing with estimation, the number of significant digits should be no more than the log base 10 of the sample size and rounding to the nearest integer. For example, if the sample size is 150, the log of 150 is approximately 2.18, so we use 2 significant figures.
Another Way To Determine Sig Figs: The Pacific Rule & The Atlantic Rule
It can be challenging to remember all the rules about significant figures and whether each zero is significant or not significant. Hereâs another way to determine significant figures : the Pacific and Atlantic Rule.
If a number has a decimal Present, use the Pacific rule . The Pacific Ocean is on the left side of the United States so start at the left side of the number. Start counting sig figs at the first non-zero number and continue to the end of the number. For example, since there is a decimal present in 0.000560 start from the left side of the number. Donât start counting sig figs until the first non-zero number , then count all the way to the end of the number. Therefore, there are 3 sig figs in this number .
If a number has no decimal use the Atlantic rule . Since the Atlantic Ocean is on the right side of the United States, start on the right side of the number and start counting sig figs at the first non-zero number. For example, since there is no decimal in 2900 start from the right side of the number and start counting sig figs at the first non zero number . So there are two sig figs in this number .
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Rules For Multiplication And Division With Significant Figures
In multiplication and division, the computed result should retain the significant digits, equal to those present in the least significant number involved in the calculation.
- 63 × 3.041 = 191.583Since 63 has the least significant figures among the quantities multiplied, the final result must be rounded to 2 significant figures. Also, the third significant figure in the answer is less than 5, so the third figure can be dropped. So the answer after rounding off is 1.9 X 102
- 4800 ÷ 10.5 = 457.1428Since 4800 has the least significant figures among the quantities divided, the final result must be rounded to 2 significant figures. Also, the third significant figure in the answer is greater than 5, so 1 should be added to the last digit retained. So the answer after rounding off is 4.6 X 102
How To Count Significant Figures/digits In Physics Problems
When working on physics problems, you have to deal with a lot of measurements. You need to process these measurements accurately and precisely, to get to your desired results.
Accuracy and Precision matter a lot when we talk about measurements in physics. Accuracy refers to how close a measurement is to the actual value of the measurement.
Precision tells how close the values repeated under the same conditions, are to one another.
When determining both these factors, the use of significant figures or digits is of great importance. Physics problems often require you to express your answer as the significant digits in that answer.
So, you need to know how to count and identify the significant digits that represent the accuracy of your answer.
In this article, we are going to take a deep dive into significant figures and how to count them for physics problems. So, lets get started and talk about the method for counting significant figures in physics problems.
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Rounding Off The Significant Figures
To get better precision we need to rounding off the numbers up to some significant figures. There are also some rules for rounding off the significant figures. Let we need to rounding off a number up to three significant figures.
Homework Problems: Find the rounding of values up to two significant digits for the numbers i) 2.04 ii) 2.27 iii) 0.589
Rules To Find Significant Figures In Multiplication
Rules: Do the usual multiplication and then write the result in the form the smallest number of significant figures that the given numbers have.
Example-1: Multiply 3 with 2.5 in significant figure.
Multiplication between 3 and 2.5 is 7.5.
Now the number 3 has smallest significant figure after decimal. So the result will not contain the digit after decimal. So, answer is 8 as 7 is odd any after 7 there is 5.
Emaple-2: Multiply 5 with 2.5 in significant figure.
Multiplication of 5 with 2.5 is 12.5. But among the given numbers 5 has no digit after decimal. So, answer will be 12.
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Significant Digits In Multiplication Division Trig Functions Etc
In a calculation involving multiplication, division, trigonometric functions, etc., the number of significant digits in an answer should equal the least number of significant digits in any one of the numbers being multiplied, divided etc.
Thus in evaluating \\), where \ and \ , the answer should have two significant digits.
Note that whole numbers have essentially an unlimited number of significant digits. As an example, if a hair dryer uses \ of power, then 2 identical hairdryers use \:
\ x \ = \