Scientific Notation, Measurement, and Significant Figures in Chemistry

Scientific Notation: Writing Large and Small Numbers

Units and Measurement

In chemistry, measurements are made using standard units agreed upon by the scientific community. These units allow scientists to communicate quantities precisely and consistently.

• Unit: A standard quantity by which other quantities are measured (e.g., meter, second).

• Modern measurements can express extremely large or small values, such as 0.000000000001 seconds or distances as great as 400,000,000,000 kilometers.

Scientific Notation

Scientific notation is a method used to write very large or very small numbers in a more compact and manageable form. It expresses numbers as a product of a decimal part and an exponential part.

• Format: $a \times 10^n$ where a is the decimal part (between 1 and 10), and n is the exponent (an integer).

• Example: $0.000000000001 = 1.0 \times 10^{-12}$; $140,000,000,000 = 1.4 \times 10^{11}$

Parts of Scientific Notation

• Decimal part: A number between 1 and 10.

• Exponential part: 10 raised to an integer power.

Exponents

• Positive exponent: Indicates multiplication by 10 for each increase in exponent. $10^3 = 10 \times 10 \times 10 = 1000$

• Negative exponent: Indicates division by 10 for each decrease in exponent. $10^{-3} = \frac{1}{10 \times 10 \times 10} = 0.001$

Writing Numbers in Scientific Notation

1. Move the decimal point to obtain a number between 1 and 10.

2. Count the number of places the decimal was moved; this is the exponent.

3. If the decimal is moved to the left, the exponent is positive; if to the right, the exponent is negative.

• Example: 1,387,000,000 people = $1.387 \times 10^9$ people

Reporting Measurements and Significant Figures

Precision and Significant Figures

Scientists report measurements using significant figures, which reflect the precision of the measurement. The more digits reported, the greater the precision.

• Numbers are usually written so that the uncertainty is indicated by the last reported digit.

• Example: Reporting a temperature as 0.7 °C implies an uncertainty of ±0.1 °C.

Reading Measuring Devices

• Estimate one digit beyond the smallest marking on the device.

• Example: If a balance has markings every 1 g, estimate to the tenths place (e.g., 1.3 g).

• If a balance has markings every 0.1 g, estimate to the hundredths place (e.g., 1.26 g).

Significant Figures in Measurement

• Non-place-holding digits in a measurement are significant figures.

• The greater the number of significant figures, the greater the precision.

• Leading zeros are not significant; trailing zeros after a decimal point are significant.

How to Determine the Number of Significant Figures

• All nonzero digits are significant.

• Interior zeros (zeros between two numbers) are significant.

• Trailing zeros after a decimal point are significant.

• Leading zeros are not significant; they only serve to locate the decimal point.

• Trailing zeros at the end of a number but before an implied decimal point are ambiguous and should be avoided by using scientific notation.

Number	Significant Figures
1.05	3
0.010	2
4.0208	5
5.10	3
0.00050	2
2100	Ambiguous (unless written as 2.1 × 103 or 2.10 × 103)

Exact Numbers

• Exact numbers have an unlimited number of significant figures.

• Examples: Counting discrete objects (10 pencils), defined quantities (100 cm = 1 m), numbers in equations.

Rounding and Calculations with Significant Figures

Rounding Rules

• When rounding, if the digit dropped is less than 5, round down; if 5 or more, round up.

• Only the last digit being dropped determines rounding.

• Example: 2.349 rounded to two significant figures is 2.3 (since the 4 is less than 5).

Addition and Subtraction

• The result has the same number of decimal places as the quantity with the fewest decimal places.

• Example: $5.74 + 0.823 + 2.651 = 9.214$ rounds to $9.21$ (two decimal places).

Multiplication and Division

• The result has the same number of significant figures as the quantity with the fewest significant figures.

• Example: $4.56 \times 1.4 = 6.4$ (two significant figures).

Summary Table: Significant Figures in Different Contexts

Context	Rule	Example
Nonzero digits	Always significant	123 (3 sig figs)
Interior zeros	Always significant	101 (3 sig figs)
Leading zeros	Not significant	0.0025 (2 sig figs)
Trailing zeros after decimal	Significant	2.300 (4 sig figs)
Trailing zeros before decimal	Ambiguous	2100 (ambiguous)
Exact numbers	Unlimited	100 pencils

Key Terms

• Scientific notation: A method of expressing numbers as a product of a decimal and a power of ten.

• Significant figures: The digits in a measurement that are known with certainty plus one estimated digit.

• Precision: The degree to which repeated measurements under unchanged conditions show the same results.

• Exact number: A value known with complete certainty, often from counting or definitions.

Additional info: Some examples and explanations have been expanded for clarity and completeness, following standard introductory chemistry textbook conventions.