Significant Figures in Chemistry

Master the Art of Precision and Accuracy

What Are Significant Figures?

Significant figures (also called significant digits or "sig figs") tell us how precise a measurement is. They show which digits in a number are meaningful and which are just placeholders.

This lesson is divided into three parts:

  • Part 1: Counting Significant Digits
  • Part 2: Finding the Limiting Factor in Calculations
  • Part 3: Scientific Notation and Sig Figs

Part 1: Counting Significant Digits

The Rules

✅ SIGNIFICANT (They Count!)
  1. All non-zero digits
    Example: 245 has 3 sig figs
  2. Zeros between non-zero digits
    Example: 1005 has 4 sig figs
  3. Trailing zeros AFTER a decimal point
    Example: 2.500 has 4 sig figs
  4. Trailing zeros in whole numbers WITH a decimal point
    Example: 500. has 3 sig figs
❌ NOT SIGNIFICANT (They Don't Count!)
  1. Leading zeros (they're just placeholders)
    Example: 0.0045 has 2 sig figs (only the 4 and 5 count)
  2. Trailing zeros WITHOUT a decimal point (they're just placeholders)
    Example: 500 has 1 sig fig (only the 5 counts)

Study Guide - Part 1: Count the Significant Digits

Practice Problems

How many significant figures are in each number?

  1. 0.00234 → ____
  2. 0.09080 → ____
  3. 3.14159 → ____
  4. 0.00100 → ____
  5. 500 → ____
  6. 1234.56 → ____
  7. 123456 → ____
  8. 4.00 → ____
  9. 50 → ____
  10. 500. → ____

Answers:

  1. 2 sig figs (2 and 3 are significant; leading zeros don't count)
  2. 4 sig figs (9, 0, 8, 0 all count; the trailing zero after decimal counts)
  3. 6 sig figs (all non-zero digits count)
  4. 3 sig figs (1, 0, 0 count; trailing zeros after decimal count)
  5. 1 sig fig (only 5 counts; trailing zeros without decimal don't count)
  6. 6 sig figs (all digits count)
  7. 6 sig figs (all non-zero digits count)
  8. 3 sig figs (4, 0, 0 all count; trailing zeros after decimal count)
  9. 1 sig fig (only 5 counts; trailing zero without decimal doesn't count)
  10. 3 sig figs (5, 0, 0 all count; decimal point indicates zeros are significant)

Part 2: Finding the Limiting Factor in Calculations

⭐ The Golden Rule ⭐

When multiplying or dividing:
Your answer can only be as precise as your LEAST precise measurement.

How to Approach Problems

STEP 1: Count sig figs in each given measurement

STEP 2: Find the SMALLEST number of sig figs (this is your limit!)

STEP 3: Round your final answer to that many sig figs

Study Guide - Part 2: Find the Limiting Factor

Problem 1:

\(2.5 \text{ cm} \times 3.14 \text{ cm} \times 0.125 \text{ cm} = \text{?} \text{ cm}^3\)

  • Given: 2.5 (__ sig figs), 3.14 (__ sig figs), 0.125 (__ sig figs)
  • Answer should have: __ sig figs

Answer:

  • 2.5 has 2 sig figs
  • 3.14 has 3 sig figs
  • 0.125 has 3 sig figs
  • The smallest is 2 sig figs, so answer should have 2 sig figs
  • Calculator gives: 0.98125 cm³
  • Rounded to 2 sig figs: 0.98 cm³

Problem 2:

\(45.0 \text{ g} \div 15 \text{ mL} = \text{?} \text{ g/mL}\)

  • Given: 45.0 (__ sig figs), 15 (__ sig figs)
  • Answer should have: __ sig figs

Answer:

  • 45.0 has 3 sig figs
  • 15 has 2 sig figs
  • The smallest is 2 sig figs, so answer should have 2 sig figs
  • Calculator gives: 3 g/mL
  • Rounded to 2 sig figs: 3.0 g/mL

Problem 3:

\(0.0250 \text{ L} \times 1.5 \text{ mol/L} \times 98.08 \text{ g/mol} = \text{?} \text{ g}\)

  • Given: 0.0250 (__ sig figs), 1.5 (__ sig figs), 98.08 (__ sig figs)
  • Answer should have: __ sig figs

Answer:

  • 0.0250 has 3 sig figs
  • 1.5 has 2 sig figs
  • 98.08 has 4 sig figs
  • The smallest is 2 sig figs, so answer should have 2 sig figs
  • Calculator gives: 3.6765 g
  • Rounded to 2 sig figs: 3.7 g

Problem 4:

\(8.314 \text{ J/(mol·K)} \times 298 \text{ K} \div 2.00 \text{ atm} = \text{?} \text{ J/mol}\)

  • Given: 8.314 (__ sig figs), 298 (__ sig figs), 2.00 (__ sig figs)
  • Answer should have: __ sig figs

Answer:

  • 8.314 has 4 sig figs
  • 298 has 3 sig figs
  • 2.00 has 3 sig figs
  • The smallest is 3 sig figs, so answer should have 3 sig figs
  • Calculator gives: 1238.786 J/mol
  • Rounded to 3 sig figs: 1240 J/mol or 1.24 × 10³ J/mol

Problem 5:

\(750 \text{ mL} \times 1.20 \text{ g/mL} \div 6.022 \times 10^{23} = \text{?}\)

(The number \(6.022 \times 10^{23}\) is Avogadro's constant)

  • Given: 750 (__ sig figs), 1.20 (__ sig figs), \(6.022 \times 10^{23}\) (__ sig figs)
  • Answer should have: __ sig figs

Answer:

  • 750 has 2 sig figs
  • 1.20 has 3 sig figs
  • \(6.022 \times 10^{23}\) has 4 sig figs
  • The smallest is 2 sig figs, so answer should have 2 sig figs
  • Calculator gives: \(1.495 \times 10^{-21}\)
  • Rounded to 2 sig figs: \(1.5 \times 10^{-21}\)

Part 3: Scientific Notation and Significant Figures

What is Scientific Notation?

Scientific notation expresses numbers as:

\[N \times 10^n\]

  • N = a number between 1 and 10
  • n = the power of 10 (positive for large numbers, negative for small numbers)

Why Use It?

1. Makes sig figs CLEAR

  • Is 5000 one sig fig or four?
  • 5000. would be 4 sig figs
  • 5000 would be 1 sig fig
  • But \(5.000 \times 10^3\) is clearly 4 sig figs
  • And \(5 \times 10^3\) is clearly 1 sig fig

2. Easier to write huge/tiny numbers

Compare:

  • 602,200,000,000,000,000,000,000
  • vs \(6.022 \times 10^{23}\)

3. Easier to multiply/divide

Just add or subtract exponents!

How to Convert

TO Scientific Notation:
  1. Move decimal until you have one non-zero digit to the left
  2. Count how many places you moved (that's your exponent)
  3. If you moved LEFT, exponent is positive; if RIGHT, exponent is negative
FROM Scientific Notation:
  1. Positive exponent = move decimal RIGHT
  2. Negative exponent = move decimal LEFT

Examples (Converting with Correct Sig Figs)

Large Numbers:

  1. 5,400,000 (2 sig figs) → \(5.4 \times 10^6\)
  2. 8,729,000 (4 sig figs) → \(8.729 \times 10^6\)
  3. 120,000. (6 sig figs) → \(1.20000 \times 10^5\)
  4. 67,000 (2 sig figs) → \(6.7 \times 10^4\)
  5. 3,000,000,000 (1 sig fig) → \(3 \times 10^9\)

Small Numbers:

  1. 0.00056 (2 sig figs) → \(5.6 \times 10^{-4}\)
  2. 0.000340 (3 sig figs) → \(3.40 \times 10^{-4}\) (note the trailing zero!)
  3. 0.0100 (3 sig figs) → \(1.00 \times 10^{-2}\)
  4. 0.000009 (1 sig fig) → \(9 \times 10^{-6}\)
  5. 0.070800 (5 sig figs) → \(7.0800 \times 10^{-2}\)

Two Ways to Write Scientific Notation

They Mean the EXACT Same Thing!

  • Standard Scientific Notation: \(6.022 \times 10^{23}\)
  • E Notation (Calculator/Computer): 6.022E23

Use × 10ⁿ when:

  • Writing by hand
  • Formal lab reports
  • Tests (unless told otherwise)
  • Scientific papers

Use E notation when:

  • Using a calculator
  • Typing in spreadsheets
  • Computer programming
  • When superscripts aren't available

How to Read E Notation

The "E" means "times ten to the power of"

  1. 2.5E4 = \(2.5 \times 10^4\) = 25,000
  2. 3.4E-3 = \(3.4 \times 10^{-3}\) = 0.0034
  3. 9.7E8 = \(9.7 \times 10^8\) = 970,000,000
  4. 1.15E0 = \(1.15 \times 10^0\) = 1.15
Quick Conversion Practice

Convert these E notation answers to standard scientific notation:

  1. 2.26E10 = ______
  2. 9.7E8 = ______
  3. 6.77E14 = ______
  4. 5.54E6 = ______
  5. 1.02E4 = ______

Answers:

  1. \(2.26 \times 10^{10}\)
  2. \(9.7 \times 10^8\)
  3. \(6.77 \times 10^{14}\)
  4. \(5.54 \times 10^6\)
  5. \(1.02 \times 10^4\)

Study Guide - Part 3: Practice Conversions

Convert TO scientific notation (keep correct sig figs):

  1. 45,000 (2 sig figs) → ______
  2. 0.000782 (3 sig figs) → ______
  3. 930,000,000 (2 sig figs) → ______
  4. 0.00650 (3 sig figs) → ______
  5. 1,234,000 (4 sig figs) → ______

Convert FROM scientific notation to standard form:

  1. \(3.5 \times 10^4\) → ______
  2. \(2.00 \times 10^{-3}\) → ______
  3. \(7.8 \times 10^8\) → ______
  4. \(4.500 \times 10^{-5}\) → ______
  5. \(1.2 \times 10^6\) → ______

Answers:

  1. \(4.5 \times 10^4\) (2 sig figs)
  2. \(7.82 \times 10^{-4}\) (3 sig figs)
  3. \(9.3 \times 10^8\) (2 sig figs)
  4. \(6.50 \times 10^{-3}\) (3 sig figs - note the trailing zero!)
  5. \(1.234 \times 10^6\) (4 sig figs)
  6. 35,000
  7. 0.00200
  8. 780,000,000
  9. 0.00004500
  10. 1,200,000

Summary: Key Takeaways

Counting Sig Figs:
  • All non-zero digits are significant
  • Zeros between non-zero digits are significant
  • Trailing zeros after a decimal are significant
  • Leading zeros are NOT significant
  • Trailing zeros without a decimal are NOT significant
Calculations:
  • Your answer can only be as precise as your least precise measurement
  • Always count sig figs BEFORE calculating
  • Round your final answer to the appropriate number of sig figs
Scientific Notation:
  • Makes sig figs crystal clear
  • Format: \(N \times 10^n\) where 1 ≤ N < 10
  • E notation means the same thing (2.5E4 = \(2.5 \times 10^4\))
  • Keep all significant figures in the N value

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