## Digits: Significant And Otherwise

Consider the two statements shown below:

**“The population of our city is 157,872.”****“The number of registered voters as of Jan 1 was 27,833.**

Which of these would you be justified in dismissing immediately? Certainly not the second one, because it probably comes from a database which contains one record for each voter, so the number is found simply by counting the number of records. The first statement cannot possibly be correct. Even if a cityâs population could be defined in a precise way , how can we account for the minute-by minute changes that occur as people are born and die, or move in and move away?

What is the difference between the two population numbers stated above? The first one expresses a quantity that cannot be known exactly â that is, it carries with it a degree of uncertainty. It is quite possible that the last census yielded precisely 157,872 records, and that this might be the âpopulation of the cityâ for legal purposes, but it is surely not the âtrueâ population. To better reflect this fact, one might list the population as **157,900** or even **158,000**. These two quantities have been rounded off to four and three significant figures, respectively, and the have the following meanings:

So, what is a significant digit? According to the usual definition, it is all the numerals in a measured quantity whose values are considered as known exactly, plus one more whose value could be one more or one less:

##### Implied Uncertainty

## Significant Figures And Multiplication Or Division

In multiplication and division the number of significant figures is simply determined by the value of lowest digits. This means that if you multiplied or divided three numbers: 2.1, 4.005 and 4.5654, the value 2.1 which has the fewest number of digits would mandate that the answer be given only to two significant figures.

## Rounding To Significant Figures

Rounding to significant figures is a more general-purpose technique than rounding to *n* digits, since it handles numbers of different scales in a uniform way. For example, the population of a city might only be known to the nearest thousand and be stated as 52,000, while the population of a country might only be known to the nearest million and be stated as 52,000,000. The former might be in error by hundreds, and the latter might be in error by hundreds of thousands, but both have two significant figures . This reflects the fact that the significance of the error is the same in both cases, relative to the size of the quantity being measured.

To round a number to *n* significant figures:

*n*+ 1 digit is greater than 5 or is 5 followed by other non-zero digits, add 1 to the

*n*digit. For example, if we want to round 1.2459 to 3 significant figures, then this step results in 1.25.

*n*+ 1 digit is 5 not followed by other digits or followed by only zeros, then rounding requires a tie-breaking rule. For example, to round 1.25 to 2 significant figures:

In UK personal tax returns, income is rounded down to the nearest pound, whilst tax paid is calculated to the nearest penny.

Precision |
---|

*x*has only one or two significant figures as more precise uncertainty has no meaning.- 1.79 ± 0.06 , 1.79 ± 0.96 , 1.79 ± 1.96 .

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## What Constitutes A Significant Figure

First, lets review these criteria that define sig figs. We can classify numbers as significant figures if they are:

- All digits comprising N are significant in accordance with the rules above
- Neither 10 nor x are significant

Specific amounts of precision, designated by significant figures, must appear in our mathematical calculations. These appropriate degrees of precision vary, corresponding to the type of calculation being completed.

To determine the number of sig figs required in the results of certain calculations, consult the following guidelines.

## Example: Adding And Subtracting With Significant Figures

A step in your “Let’s Make a Latte” chemistry lab assignment requires that you account for the volume of fluids in your latte.

You’re starting with 7 oz. of milk, and your espresso machine uses 2.5 oz. of water to make a 2 oz. espresso shot — the other 0.5 oz. remains in the espresso puck. Finally, your high tech milk steamer tells you how much water is used in the steaming process, out to 3 decimal places.

You make your espresso and see that you’ve pulled the perfect 2 oz. shot. You steam and froth your milk, and the steamer indicator says 0.063 oz. of water was used during the process. You need to add up 2 oz. espresso plus 7 oz. milk plus 0.063 oz. of steam. But because this is a chemistry lab assignment you have to do your math with significant figures.

Reviewing the rules for adding and subtracting with significant figures, **find the place position of the last significant digit of your least certain number**. Your milk and espresso are each one significant digit in volume, in the __ones__ place.

Adding the volumes of fluid in your latte you have:

7 oz. milk + 2 oz. espresso + 0.063 oz. water = 9.063 oz.

9.063 oz. rounded to the ones place = 9 oz.

Although you have a volume of fluids that seems accurate to the thousandths, you have to round to the ones place because that is the least significant place value. So following the rules of addition with significant figures you report that your latte is 9 oz. in volume.

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## Multiplication And Division Of Significant Figures

In multiplication and division, the result should not have more significant figures than the original numbers. Consider an example below.

98.23 × 5.3402 = 524.567846 = 524.6

The least significant figures in the original numbers are 4. So, the result must be limited to 4 significant figures. The same is true in the division.

23.34 ÷ 2.01 = 11.6119403 = 11.6

885.45 ÷ 2. = 442.725 = 400

Note: When we are working on a long series of calculation, only the final result needed to be round off to desired significant figures, not intermediate ones. To maintain the accuracy, we retain an extra significant digit in intermediate calculations.

## How Do Significant Figures Relate To Accuracy

Accuracy refers to how exactly the calculated value matches the right value. Precision refers to how closely individual measurements are in accordance with each other. The number of significant figures is the number of digits considered to be accurate by the person doing the calculation.

Put your understanding of this concept to test by answering a few MCQs. Click Start Quiz to begin!

Select the correct answer and click on the Finish buttonCheck your score and answers at the end of the quiz

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## What Is The Rule For Adding With Sig Figs

When adding or subtracting decimal places, the answer should have the same number of decimal places as the limit term. A number with the least decimal places must have a limit. When dividing the answer, consider the number of significant figures to be the same as the number of caps.

#### How To Determine The Number Of Significant Figures In A Numbe

To find the number of significant figures in a number, there are three rules. Any digit that does not have a zero on it is always significant. A zero between two significant digits is significant. Only in decimal mode are those final zeros or trailing zeros significant. The three rules for the number 780 are as follows: The number 780 contains two significant figures. When the figs are rounded to 800, the number of sigs equals 780.

## How To Use Significant Figures In Calculations

When **adding or subtracting** numbers, the answer must be expressed with the same number of decimal places as the participating number with the **least number of decimal places**.

When **multiplying or dividing numbers**, the answer must be expressed with the same number of significant figures as the participating number with the **least number of significant figures**.

#### Exercise 15: 2.187 + 10.2 = ?

The calculator gives the answer 12.387. Were adding here, so the rule of least number of decimal places applies. One number has 3 decimal places, and the other has 1 decimal place. The answer should, therefore, have 1 decimal place, so the answer is 12.4.

#### Exercise 16: 178.1 2.08 + 15 = ?

The calculation involves addition/subtraction, so we are applying the rule of least number of decimal places. The three participating numbers, 178.1, 2.08 and 15, have 1, 2 and 0 decimal places, respectively, so the answer should have no decimal place. The calculators answer is 191.02, but were rounding it off to 191.

#### Exercise 17:/31.0= ?

Here, the calculation involves multiplication and division, so the answer has the same sig figs as the number with the least number of sig figs. The numbers 0.081, 1090 and 31.0 have 2, 4 and 3 sig figs, respectively. The answer should have only 2 sig figs. The calculators answer is 2.848064516. We need to round off the answer to 2.8.

#### Exercise 18: What is the perimeter of a square with a side measuring 10.5 inches?

#### Exercise 19: /1.25= ?

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## Scientific Notation And Significant Figures

In the previous example you should have noticed that the answer is presented in what is called scientific notation.

Scientific notation

is a way to express very small or very large numbersis most often used in “scientific” calculations where the analysis must be very preciseconsists of two parts: A Number and a Power of 10. Ex: 1.22 x 103

For a number to be in correct scientific notation only one digit may be to the left of the decimal. So,

\begin1.22 & \times 10^3 \text \\12.2 & \times 10^2 \text\end

How to convert non-exponential numbers to exponential numbers:

Example 1

$$ 234,999 $$

This is a large number and the implied decimal point is at the end of the number.

$$ 234,999. $$

To convert this to an exponential number we need to move the decimal to the left until only one digit resides in front of the decimal point. In this number we move the decimal point 5 times.

$$ 2.34999 \text $$

and thus the exponent we place on the power of 10 is 5. The resulting exponential number is then:

$$2.34999 \times 10^5 $$

Other examples:

\begin21 & \to 2.1 \times 10^1 \\16600.01 & \to 1.660001 \times 10^4 \\455 & \to 4.55 \times 10^2\end

Small numbers can be converted to exponential notation in much the same way. You simply move the decimal to the right until only one non-zero digit is in front of the decimal point. The exponent then equals the number of digits you had to pass along the way.

Example 2

$$ 0.000556 $$

$$ 5.56 \times 10^ $$

Other examples

## Use Of Appropriate Significant Figures

While it is important to know how accurate a measurement is, it is also important to convey one’s degree of confidence in the accuracy of a number when making an observation or carrying out a calculation. This can be accomplished by noting numerical observations or quoting results to the correct number of significant figures.

e.g. weighing the same object with different balances:Analytical balance: Weight = 2.1234 ± 0.0001 gPan balance: Weight = 2.1 ± 0.1 g

Often the ± value is dropped, and it is understood that the number quoted has an uncertainty of at least 1 unit in the last digit. All digits quoted are called **significant figures** . Note that the last digit included is the one with an uncertainty of ±1.

**Rules**

**All non-zero digits are significant.**For example:

**Zeros between non-zero digits are significant.**For example:

**Zeros to the left of the first non-zero digit are not significant.**For example: 0.12 0.012

**If a number ends in zeros to the right of the decimal point, those zeros are significant.**For example: 2.0 2.00

These rules apply to measured quantities which are non-exact. *If you are told that a number is exact then there is no uncertainty * i.e., the number is good to an infinite number of sig figs. Exact integers fall in this category.

**Calculations with numbers having different accuracies****Multiplication or Division:** the result can have no more sig figs than the least accurate number. For example:

__143 g__ = 1.164572 g cm-3 = 1.16 g cm-3

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## Note: Doing Math With Significant Figures

If you are entering a constant or exact value as you might find in a formula, be sure to include the proper number of significant figures.

For example, consider the formula for diameter of a circle, d = 2r, where diameter is twice the length of the radius. If you measure a radius of 2.35, multiply by 2 to find the diameter of the circle: 2 * 2.35 = 4.70

If you use this calculator for the calculation and you enter only “2” for the multiplier constant, the calculator will read the 2 as one significant figure. Your resulting calculation will be rounded from 4.70 to 5, which is clearly not the correct answer to the diameter calculation d=2r.

You can think of constants or exact values as having infinitely many significant figures, or at least as many significant figures as the the least precise number in your calculation. In this example you would want to enter 2.00 for the multiplier constant so that it has the same number of significant figures as the radius entry. The resulting answer would be 4.70 which has 3 significant figures.

## Origination Of Significant Figures

We can trace the first usage of significant figures to a few hundred years after Arabic numerals entered Europe, around 1400 BCE. At this time, the term described the nonzero digits positioned to the left of a given values rightmost zeros.

Only in modern times did we implement sig figs in accuracy measurements. The degree of accuracy, or precision, within a number affects our perception of that value. For instance, the number 1200 exhibits accuracy to the nearest 100 digits, while 1200.15 measures to the nearest one hundredth of a digit. These values thus differ in the accuracies that they display. Their amounts of sig figs2 and 6, respectivelydetermine these accuracies.

Scientists began exploring the effects of rounding errors on calculations in the 18th century. Specifically, German mathematician Carl Friedrich Gauss studied how limiting sig figs could affect the accuracy of different computation methods. His explorations prompted the creation of our current checklist and associated rules.

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## Rules For Multiplication And Division Calculations:

## How Many Significant Figures Does A Measurement Have

When you take a measurement, you write down the correct number of digits. You write down the significant digits. That is, the way you write a number conveys some information about how accurate it is. It is up to you to determine how many digits are worth writing down. It is important that you do so, since what you write conveys not only the measurement but something about its quality. For many common lab instruments, the proper procedure is to estimate one digit beyond those shown directly by the measurement scale. If that one estimated digit seems meaningful, then it is indeed a significant digit.

##### Example 1: Reading a typical scale

The scale shown here is a “typical” measurement scale. The specific scale is from a 10 mL graduated cylinder — shown horizontally here for convenience. The arrow marks the position of a measurement.Glossary entry: Scale.

Our goal is to read the scale at the position of the arrow. Let’s go through this in detail.

- The numbered lines are 1 mL apart.
- The little lines are 0.1 mL apart.
- The arrow is clearly between 4.7 and 4.8 mL.
- We will estimate the position of the arrow to 1/10 the distance between the little lines, that is, to the nearest 0.01 mL.
- A reasonable estimate is 4.78 mL. Some people might say 4.77 mL or 4.79 mL. No one should say 4.72 mL! That is, the estimate is 4.78 mL to about +/- 0.01 mL. 4.78 mL is 3 Significant Digits the last Significant Digits is not certain, but is “close”.

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## The Significance Of Significant Figures In Chemistry

If you had two sigs, you would get the concentration you wanted. If you want to keep your pH constant, you should have a rounded pH. = 1.34 x 10*-5, for example, would result in three sig figs. If the pH was 3.22, would we apply the sig fig to the and not to the 3?

When calculating **molar masses**, use the same number of sig figs as in the other measurements. The last rule you should follow is rule number 6. Keep decimal places on your calculator in between your calculations to keep them accurate.

Is the **molar mass** of a sig fig calculated? The value determined by the given answers will ultimately determine the number of sig figs.

## Significant Figures In Operations

There are additional rules regarding the operations – addition, subtraction, multiplication, and division.

To use an exact value in the calculator, give the value to the greatest number of significant figures in the calculation. So for this example, you would enter 15.23 * 3.600 into the calculator.

Since we are talking about basic arithmetic operations, how about checking our distributive property calculator to learn how to handle complex mathematical problems that involve more than one arithmetic operation?

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