Significant Figures and Scientific Notation

By the end of this lesson, you will be able to count significant figures including the awkward zeros, apply the different rules for multiplying and adding, round correctly, and say why the rules exist at all.

Digits You Actually Measured

$$25.4 \text{ mL} \qquad \text{three significant figures}$$

Significant figures are the digits a measurement genuinely supports: the ones you read from the scale, plus the one you estimated. Writing more claims precision the instrument never had, and that is a false statement about the experiment.

Why the rules exist. A calculator dividing $10.1$ by $3$ returns $3.366666\ldots$, which asserts seven trustworthy digits from data that had three. The rules are how you stop a calculator from lying on your behalf.

The counting rules:

All non-zero digits count. $4.72$ has three
Zeros BETWEEN non-zeros count. $4.02$ has three
Leading zeros never count. $0.0034$ has two, because those zeros only place the decimal point
Trailing zeros count only if there is a decimal point. $4.50$ has three; $450$ is ambiguous and probably has two

Exact numbers have infinite significant figures and never limit a result: a count of $12$ beakers, or the $1000$ in $1$ km $= 1000$ m, are definitions rather than measurements.

Sort It: How Many Significant Figures

The Ambiguous Trailing Zero

$$450$$

Two significant figures, or three? The notation cannot say. Perhaps it was measured to the nearest ten, perhaps to the nearest one.

Scientific notation removes the ambiguity completely, which is one of its main jobs:

$$4.5 \times 10^2 \quad \text{two sig figs}$$

$$4.50 \times 10^2 \quad \text{three sig figs}$$

The exponent handles the size and the front handles the precision, so the two questions stop interfering with each other. That separation is why every scientific paper reports numbers this way.

A decimal point at the end also works in some conventions: $450.$ means three. It is easy to miss when reading, so scientific notation is the safer habit.

Example 1: Counting Them

Here is every step, worked all the way through. $$0.004070$$

By the end of this lesson, you will be able to count significant figures including the awkward zeros, apply the different rules for multiplying and adding, round correctly, and say why the rules exist at all.

Digits You Actually Measured

$$25.4 \text{ mL} \qquad \text{three significant figures}$$

The counting rules:

All non-zero digits count. $4.72$ has three
Zeros BETWEEN non-zeros count. $4.02$ has three
Leading zeros never count. $0.0034$ has two, because those zeros only place the decimal point
Trailing zeros count only if there is a decimal point. $4.50$ has three; $450$ is ambiguous and probably has two

Exact numbers have infinite significant figures and never limit a result: a count of $12$ beakers, or the $1000$ in $1$ km $= 1000$ m, are definitions rather than measurements.

Sort It: How Many Significant Figures

The Ambiguous Trailing Zero

$$450$$

Two significant figures, or three? The notation cannot say. Perhaps it was measured to the nearest ten, perhaps to the nearest one.

Scientific notation removes the ambiguity completely, which is one of its main jobs:

$$4.5 \times 10^2 \quad \text{two sig figs}$$

$$4.50 \times 10^2 \quad \text{three sig figs}$$

The exponent handles the size and the front handles the precision, so the two questions stop interfering with each other. That separation is why every scientific paper reports numbers this way.

A decimal point at the end also works in some conventions: $450.$ means three. It is easy to miss when reading, so scientific notation is the safer habit.

Example 1: Counting Them

Here is every step, worked all the way through. $$0.004070$$

Digits You Actually Measured

Sort It: How Many Significant Figures

The Ambiguous Trailing Zero

Example 1: Counting Them

Significant Figures and Scientific Notation

Digits You Actually Measured

Sort It: How Many Significant Figures

The Ambiguous Trailing Zero

Example 1: Counting Them

Two Different Rules

Sort It: Which Rule Applies

Worked Example 1: Multiplying

Worked Example 2: Adding

Rounding, and the Digit That Decides

Worked Example 3: A Multi-Step Calculation

Common Mistakes

Where This Goes Next

Check Your Understanding

Summary