Significant Digits

Accuracy and precision are very important in chemistry. However, the laboratory equipment and machines used in labs are limited in such a way that they can only determine a certain amount of data. For example, a scale can only mass an object up until a certain decimal place, because no machine is advanced enough to determine an infinite amount of digits. Machines are only able to determine a certain amount of digits precisely. These numbers that are determined precisely are called significant digits. Thus, a scale that could only mass until 99.999 mg, could only measure up to 5 figures of accuracy (5 significant digits). Furthermore, in order to have accurate calculations, the end calculation should not have more significant digits than the original set of data.

Introduction

Significant Digits - Number of digits in a figure that express the precision of a measurement instead of its magnitude. 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.

General Rules for Determining Number of Significant Figures

1. All nonzero digits are significant.
2. Zeros are also significant with two exceptions:
   1. zeros preceding the decimal point.
   2. zeros following the decimal point and preceding the first nonzero digit.
3. Terminal zeros preceding the decimal point in amounts greater than one is an ambiguous case.

Rules for Numbers WITHOUT a Decimal Point

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

Example \(\PageIndex{1}\):

The first two zeroes in 200500 (four significant digits) are significant because they are between two non-zero digits, and the last two zeroes are insignificant because they are after the last non-zero digit.

It should be noted that both constants and quantities of real world objects have an infinite number of significant figures. For example if you were to count three oranges, a real world object, the value three would be considered to have an infinite number of significant figures in this context.

Example \(\PageIndex{1}\)

How many significant digits are in 5010?

Solution

1. Start counting for significant digits On the first non-zero digit (5).
2. Stop counting for significant digits On the last non-zero digit (1).

5 0 1 0 Key: 0 = significant zero. 0 = insignificant zero.

3 significant digits.

Rules for Numbers WITH a Decimal Point

1. START counting for sig. figs. On the FIRST non-zero digit.
2. STOP counting for sig. figs. On the VERY LAST digit (regardless whether or not the last digit is a zero or non-zero number).
3. Non-zero digits are ALWAYS significant.
4. Any zero AFTER the first non-zero digit is STILL significant. The zeroes BEFORE the first non-zero digit are insignificant.​​

Example \(\PageIndex{3}\)

The first two zeroes in 0.058000 (five significant digits) are insignificant because they are before the first non-zero digit, and the last three zeroes are significant because they are after the first non-zero digit.

Example \(\PageIndex{4}\)

How many significant digits are in 0.70620?

Solution

1. Start counting for significant digits On the first non-zero digit (7).
2. 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.

Scientific Notation

Scientific notation form: a x 10^b, where “a” and “b” are integers, and "a" has to be between 1 and 10.

Example \(\PageIndex{5}\)

The scientific notation for 4548 is 4.548 x 10³.

Solution

• Disregard the “10^b,” and determine the significant digits in “a.”
• 4.548 x 10³ has 4 significant digits.

Example \(\PageIndex{6}\)

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.

Rounding Significant Digits

When rounding numbers to a significant digit, keep the amount 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 because in a calculation, some values have less significant digits than other values, and the answer to a calculation is only accurate to the amount 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 digit 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.

Example \(\PageIndex{7}\)

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).

Rules for Addition and Subtraction

When adding or subtracting numbers, the end result should have the same amount of decimal places as the number with the least amount of decimal places.

Example \(\PageIndex{8}\)

Y = 232.234 + 0.27 Find Y.

Answer

Y = 232.50

NOTE: 232.234 has 3 decimal places and 0.27 has 2 decimal places. The least amount of decimal places is 2. Thus, the answer must be rounded to the 2^nd decimal place (thousandth).

Rules for Multiplication and Division

When multiplying or dividing numbers, the end result should have the same amount of significant digits as the number with the least amount of significant digits.

Example \(\PageIndex{9}\)

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).

Exact Numbers

Exact numbers can be considered to have an unlimited number of significant figures, as such calculations are not subject to errors in measurement. This may occur:

1. By definition (1 minute = 60 seconds, 1 inch = 2.54 cm, 12 inches = 1 foot, etc.)
2. As a result of counting (6 faces on a cube or dice, two hydrogen atoms in a water molecule, 3 peas in a pod, etc.)

References

1. Brown, Theodore E., H. Eugene LeMay, and Bruce E. Bursten. Chemistry: The Central Science, Tenth Edition. Pearson Education Inc. Upper Saddle River, New Jersey: 2005.
2. Petrucci, Ralph H., William S. Harwood, F. Geoffrey Herring, and Jeffry D. Madura. General Chemistry: Principles and Modern Applications, Ninth Edition. Pearson Education Inc. Upper Saddle River, New Jersey: 2007.
3. Petrucci, Ralph H., William S. Harwood, F. Geoffrey Herring, and Jeffry D. Madura. General Chemistry: Principles and Modern Applications, Tenth Edition. Pearson Education Inc. Upper Saddle River, New Jersey: 2011. Custom Edition for Chem 2, University of California, Davis

Additional Problems

1. 1. a) How many significant digits Are in 50?
   2. b) How many significant digits Are in 50.0?
2. How many significant digits Are in \(3.670 \times 10^{35}\)?
3. Round 4279852.243 to 3 significant digits.
4. Round 0.0573000 to 1 significant digit.
5. Y = 45.2 + 16.730 Find Y.
6. Y = 23 – 26.2 Find Y.
7. Y = 16.7 x 33.2 x 16.72 Find Y.
8. Y = 346 ÷ 22 Find Y.
9. Y = (23.2 + 16.723) x 28 Find Y
10. Y = (16.7 x 23) – (23.2 ÷ 2.13) Find Y

Solutions

1. a) 1 significant digit.

b) 2 significant digits.

2. 4 significant digits.

3. 4280000

4. 0.06

5. Y = 61.9

6. Y = -3

7. Y = 9270

8. Y = 16

9. Y = (23.2 + 16.723) x 28

Y = 39.923 x 28 (TIP: Do not round until the end of calculations.)

Y = 1100 (NOTE: 28 has the least amount of significant digits (2 sig. figs.) Thus, answer must be rounded to 2 sig. figs.)

10. Y = (16.7 x 23) – (23.2 ÷ 2.13)

Y = 384.1 – 10.89201878 (TIP: Do not round until the end of calculations.)

Y = 373.2 (NOTE: 384.1 has the least amount of decimal point (tenth). Thus, answer must be rounded to the tenth.)

Contributors and Attributions

• Jeffrey Susila (UCD), Neema Shah (UCD)