2.3 Scientific Notation and Significant Figures

Scientific Notation and Standard Form

Scientific notation gives you a way to write extremely large or small numbers without all those zeros. It uses powers of 10 to compress numbers into a compact form, which makes calculations and comparisons much more manageable. Significant figures, on the other hand, tell you how precise a measurement actually is. Together, these two tools keep scientific work accurate and honest.

Understanding Scientific Notation

Scientific notation expresses any number as a value between 1 and 10 multiplied by a power of 10:

$a \times 10^n$

where $1 \leq |a| < 10$ and $n$ is an integer.

The coefficient ($a$) captures the meaningful digits, while the exponent ($n$) tells you the scale. A positive exponent means a large number; a negative exponent means a small one.

Why bother? Compare these two ways of writing the speed of light:

• Standard form: 299,792,458 m/s
• Scientific notation: $2.99792458 \times 10^8$ m/s

Or the gravitational constant:

• Standard form: 0.0000000000667 N·m²/kg²
• Scientific notation: $6.67 \times 10^{-11}$ N·m²/kg²

The scientific notation versions are easier to read, compare, and plug into calculations.

Converting Between Standard Form and Scientific Notation

Standard form to scientific notation:

1. Place the decimal point so the number is between 1 and 10.
2. Count how many places you moved the decimal.
3. That count becomes your exponent. Moved left → positive exponent. Moved right → negative exponent.

Example: Convert 45,000,000 to scientific notation. Move the decimal 7 places to the left → $4.5 \times 10^7$

Scientific notation to standard form:

1. Look at the exponent.
2. Positive exponent → move the decimal that many places to the right.
3. Negative exponent → move the decimal that many places to the left.
4. Fill in zeros as needed.

Example: Convert $3.2 \times 10^{-4}$ to standard form. Move the decimal 4 places to the left → 0.00032

Working with Exponents in Scientific Notation

When you calculate with numbers in scientific notation, the exponent rules do the heavy lifting.

Multiplication: Multiply the coefficients, then add the exponents.

$(5 \times 10^3) \times (2 \times 10^4) = 10 \times 10^7 = 1.0 \times 10^8$

Notice that $10 \times 10^7$ isn't proper scientific notation (the coefficient must be between 1 and 10), so you adjust it to $1.0 \times 10^8$.

Division: Divide the coefficients, then subtract the exponents.

$(6 \times 10^5) \div (2 \times 10^2) = 3 \times 10^3$

Addition and Subtraction: You must convert both numbers to the same exponent first, then add or subtract the coefficients.

$(5 \times 10^3) + (3 \times 10^2)$

Rewrite $3 \times 10^2$ as $0.3 \times 10^3$, then add:

$5 \times 10^3 + 0.3 \times 10^3 = 5.3 \times 10^3$

Significant Figures and Precision

Understanding Significant Figures

Significant figures (sig figs) are the digits in a measurement that actually carry meaning about its precision. More sig figs means a more precise measurement.

Rules for counting significant figures:

• All non-zero digits are significant.
• Zeros between non-zero digits are significant (e.g., 1002 has 4 sig figs).
• Leading zeros are never significant. They're just placeholders (e.g., 0.00456 has 3 sig figs).
• Trailing zeros after a decimal point are significant because they show precision (e.g., 1.200 has 4 sig figs).
• Trailing zeros in a whole number with no decimal point are ambiguous. The number 1200 is typically read as having 2 sig figs. Writing it as $1.200 \times 10^3$ makes it clear there are 4.

Rounding and Significant Figures

After a calculation, you'll often need to round your answer so it reflects the right level of precision.

How to round:

1. Decide how many significant figures you need.
2. Look at the digit just after your last significant figure.
3. If that digit is 5 or greater, round up. If it's less than 5, round down.

Example: 3.14159 rounded to 3 significant figures → 3.14

Rounding rules for calculations:

• Addition/Subtraction: Your answer should have the same number of decimal places as the measurement with the fewest decimal places.
  • $45.678 + 1.2 = 46.878$ → round to one decimal place → 46.9
• Multiplication/Division: Your answer should have the same number of significant figures as the measurement with the fewest sig figs.
  • $2.4 \times 3.15 = 7.56$ → 2.4 has 2 sig figs, so round to 2 sig figs → 7.6

The distinction between these two rules trips people up. For adding and subtracting, think about decimal places. For multiplying and dividing, think about sig figs.

Precision in Measurements and Calculations

Precision refers to how close repeated measurements are to each other. It depends on the tool you're using and how carefully you use it.

Different instruments give different levels of precision:

• A standard ruler (1 mm markings) might give you 10.2 cm
• A caliper (0.01 mm markings) might give you 10.23 cm

The caliper reading has more significant figures because the instrument is more precise.

Why this matters for calculations: Your result can never be more precise than your least precise measurement. Reporting extra digits creates false precision, which suggests your answer is more accurate than it really is.

Example: You measure a rectangle as 2.3 cm long and 1.45 cm wide. The area calculation gives $2.3 \times 1.45 = 3.335$ cm². But 2.3 has only 2 sig figs, so you round the area to 3.3 cm². Reporting 3.335 cm² would imply a level of precision your length measurement doesn't support.