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## Definition of Significant Figures

In chemistry, Significant figures are the digits of value which carry meaning towards the resolution of the measurement. They are also called significant digits in chemistry.

### Significant Digits

Number of digits in a figure that expresses the precision of measurement instead of its magnitude are called significant digits.

- The easiest method to determine significant digits is done by first determining whether or not a number has a decimal point. This rule is known as the Atlantic-Pacific Rule. The rule states that if a decimal point is Absent, then the zeroes on the Atlantic/right side are insignificant. If a decimal point is present, then the zeroes on the Pacific/left side are insignificant.
- Significant figures are any non-zero digits or trapped zeros. They do not include leading or trailing zeros.
- When going between decimal and scientific notation, maintain the same number of significant figures.
- The final answer in a multiplication or division problem should contain the same number of significant figures as the original number with the fewest significant figures.
- In addition and subtraction, the final answer should contain the same number of decimal places as the original number with the fewest number of decimal places.
- Significant figures of a number in positional notation are digits in the number that are reliable and absolutely necessary to indicate the quantity of something. Suppose a number expressing the result of measurement of something (e.g., length, pressure, volume, or mass) has more digits than the digits allowed by the measurement resolution. In that case, only the digits allowed by the measurement resolution are reliable. So only these can be significant figures.

#### For Examples

- If a length measurement gives 114.8 mm while the smallest interval between marks on the ruler used in the measurement is 1 mm, then the first three digits (1, 1, and 4, and these show 114 mm) are only reliable, so can be significant figures. There is uncertainty in the last digit (8, to add 0.8 mm). But it is also considered a significant figure since digits that are uncertain but reliable are considered significant figures.
- Another example is a volume measurement of 2.98 L with the uncertainty of ± 0.05 L. The actual volume is somewhere between 2.93 L and 3.03 L. Even if all three digits are not certain (e.g., the actual volume can be 2.94 L but also can be 3.02 L.) b ut reliable as these indicate to the actual volume with the acceptable uncertainty. So, these are significant figures.

## Examples of Significant Figures

Some examples of significant figures are-

- 4308 – 4 is a significant figure in this number
- 40.05 – 4 is a significant figure in this number
- 470,000 – 2 is a significant figure in this number
- 4.00 – 3 is a significant figure in this number
- 0.00500 – 3 is a significant figure in this number

## Digits which are not the Significant Figure

- All leading zeros.

**For example**, 013 kg has two significant figures, 1 and 3, and the leading zero is not significant. Since it is not necessary to indicate the mass; 013 kg = 13 kg so 0 is not necessary.

Also, 0.056 m has two insignificant leading zeros since 0.056 m = 56 mm so the leading zeros are not absolutely necessary to indicate the length.

- Trailing zeros when they are merely placeholders.

**For example**, the trailing zeros in 1500 m as a length measurement are not significant if they are just placeholders for ones and tens places as the measurement resolution is 100 m. In this case, 1500 m means the length to measure is close to 1500 m rather than saying that the length is exactly 1500 m.

- Spurious digits, introduced by calculations resulting in a number with greater precision. Then the precision of the used data in the calculations or in a measurement reported to a greater precision than the measurement resolution.

## Rules of Significant Figures

There are some rules to determine the significant figure

### 1. Rules For Determining If a Number Is Significant or Not

- All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five signific ant figures (1, 2, 3,4, and 5).
- Zeros appearing between two non-zero digits (trapped zeros) are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1, and 2.
- Leading zeros (zeros before non-zero numbers) are not significant. For example, 0.00052 has two significant figures: 5 and 2.
- Trailing zeros (zeros after non-zero numbers) in a number without a decimal are generally not significant (see below for more details). For example, 400 has only one significant figure (4). The trailing zeros do not count as significant.
- Trailing zeros in a number containing a decimal point are significant.
**For example**, 12.2300 has six significant figures: 1, 2 , 2, 3, 0, and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 h as five significant figures since it has three trailing zeros. This convention clarifies the precision of such numbers.**For example**, if a measurement tha t is precise to four decimal places (0.0001) is given as 12.23, then the measurement might be understood as having only two decimal places of prec ision available. Stating the result as 12.2300 makes it clear that the measurement is precise to four decimal places (in this case, six significant figures). - The number 0 has one significant figure. Therefore, any zeros after the decimal point are also significant. Example: 0.00 has three significant figures.
- Any numbers in scientific notation are considered significant. For example, 4.300 x 10 -4 has 4 significant figure.

### 2. Rules for Numbers without a Decimal Point

- START counting for sig. figs. On the FIRST non-zero digit.
- STOP counting for sig. figs. On the LAST non-zero digit.
- Non-zero digits are ALWAYS significant.
- Zeroes in between two non-zero digits are significant. All other zeroes are insignificant.

### 3. Addition and Subtraction with Significant Figures

When combining measurements with different degrees of accuracy and precision, the accuracy of the final answer can be no greater than the least accurate measurement.

This principle can be translated into a simple rule for addition and subtraction. When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement.

150.0 g H_{2}O (using significant figures)+0.507 g salt

= 150.5 g solution

### 4. Multiplication and Division With Significant Figures

The same principle governs the use of significant figures in multiplication and division: the final result can be no more accurate than the least accurate measurement.

In this case, however, we count the significant figures in each measurement, not the number of decimal places. When measurements are multiplied or divided, the answer can contain no more significant figures than the least accurate measurement.

**Example :** To illustrate this rule, let’s calculate the cost of the copper in an old penny that is pure copper. Let’s assume that the penny has a mass of 2.531 grams, that it is essentially pure copper, and that the price of copper is 67 cents per pound.

We can start from grams to pounds.

2.531 g x 1 lb / 453.6 g = 0.005580 lb

We then use the price of a pound of copper to calculate the cost of the copper metal.

0.005580 lb x 67 / 1 lb = 0.3749 g.

### 5. Rounding Off

- When rounding numbers to a significant digit, keep the number of significant digits wished to be kept, and replace the other numbers with insignificant zeroes.
- The reason for rounding a number to a particular amount of significant digits is that some values have less significant digits than other values in a calculation, and the answer to a calculation is only accurate to the number of significant digits of the value with the least amount.
- NOTE: be careful when rounding numbers with a decimal point. Any zeroes added after the first non -zero digits is considered to be a significant zero. TIP: When doing calculations for quizzes/tests/midterms/finals, it would be best to not round in the middle of your calculations and round to the significant digit only at the end of your calculations.
- When the answer to a calculation contains too many significant figures, it must be rounded off.
- There are 10 digits that can occur in the last decimal place in a calculation. One way of rounding off involves underestimating the answer for five of these digits (0, 1, 2, 3, and 4) and overestimating the answer for the other five (5, 6, 7, 8, and 9). This approach to rounding off is summarized as follows.
- If the digit is smaller than 5, drop this digit and leave the remaining number unchanged. Thus, 1.684 becomes 1.68.
- If the digit is 5 or larger, drop this digit and add 1 to the preceding digit. Thus, 1.247 becomes 1.25.

### 6. Exact Numbers

The exact numbers, such as the number of people in a room, have an infinite number of significant figures. Exact numbers are counting up how many of something is present, and they are not measurements made with instruments.

Another example of this is defined numbers, such as 1 foot = 12 inches.

**Examples of exact numbers include**

- Conversions within the American system (such as pounds to ounces, the number of feet in a mile, the number of inches in a foot, etc.).
- Conversions with the metric system (such as kilograms to grams, the number of meters in a kilometre, the number of centimetres in a meter).
- The words per and per cent mean exactly out of one or one hundred, respectively.
- Counted numbers are exact: there are two chairs in the photograph. There are fifteen books on the shelf. Eighty -seven people attended the lecture.

### 7. Measured Numbers

In contrast, measured numbers always have a limited number of significant digits. A mass reported as 0.5 grams is implied to be known to the nearest tenth of a gram and not to the hundredth of a gram.

There is a degree of uncertainty any time you measure something.

**For example**, the weight of a particular sample is 0.825 g, but it may actually be 0.828 g or 0.821 g because there is inherent uncertainty involved. On the other hand, because exact numbers are not measured, they have no uncertainty and infinite numbers of significant figures.

Mass is an example of a measured number. When a mass is reported as 0.5237 g, as shown on this scale, it is more precise than a mass reported as 0.5 g.

**Examples of measured numbers**

- The diameter of a coin, such as 10.2 mm.
- The weight of an object, such as 8.887 grams.
- The length of a pen, such as 12 cm.

## Overall Examples of Significant Figures

Based on these rules here are some examples of significant figures

### Example-1

The scientific notation for 4548 is 4.548 x 103.

**Solution**

Disregard the “10b,” and determine the significant digits in “a.”

4.548 x 103 has 4 significant digits.

### Example-2

How many significant digits are in 1.52 x 106 ?

NOTE: Only determine the amount of significant digits in the “1.52” part of the scientific notation form.

**Answer-** 3 significant digits.

### Example-3

How many significant digits are in 0.70620?

**Solution :**

Start counting for significant digits On the first non-zero digit (7). Stop counting for significant digits On the last digit (0).

0 . 7 0 6 2 0 Key : 0 = significant zero.0 = insignificant zero.

5 significant digits.

### Example-4

Round 32445.34 to 2 significant digits.

**Answer-:** 32000 (NOT 32000.00, which has 7 significant digits. Due to the decimal point, the zeroes after the first non-zero digit become significant).

### Example-5

Y = 28 x 47.3 Find Y

**Answer-:** Y = 1300

NOTE: 28 has 2 significant digits and 47.3 has 3 significant digits. The least amount of significant digits is 2. Thus, the answer must me rounded to 2 significant digits (which is done by keeping 2 significant digits and replacing the rest of the digits with insignificant zeroes).

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