Q.State the number of significant figures in the following:

Concept: Significant Figures Calculation – the number of reliably known digits in a value, excluding leading zeros but including trailing zeros after a decimal.

Reasoning:

• (a) $0.007\ \text{m}^2$: Leading zeros are not significant. Only the digit 7 counts → 1.
• (b) $2.64 \times 10^{24}\ \text{kg}$: The coefficient $2.64$ has three digits → 3.
• (c) $0.2370\ \text{g cm}^{-3}$: Leading zero ignored; trailing zero after decimal is significant → 4.
• (d) $6.320\ \text{J}$: All digits, including the trailing zero after decimal, are significant → 4. …

The rule for counting significant figures is simple once you see the pattern: zeros are the only troublemakers. Non-zero digits are always significant. Zeros can be either placeholders (not significant) or measured digits (significant), depending on where they sit.

The key distinction: leading zeros — zeros that come before the first non-zero digit — are never significant. They only tell you the decimal place. Trailing zeros after a decimal point are always significant because they indicate the measurement was precise enough to record that zero. Zeros between non-zero digits are always significant.

Let's apply this to each case.

1. (a) $0.007\ \text{m}^2$

   The digits are $0.0\ 0\ 7$. The first two zeros are leading zeros — they just position the decimal. Only the $7$ is a measured digit.

   Watch out

   A common mistake is to count the zeros before the 7. They are not significant — they only tell you the order of magnitude.

   Significant figures: 1

2. (b) $2.64 \times 10^{24}\ \text{kg}$

   Scientific notation is a gift: the coefficient $2.64$ contains all the significant figures. The $10^{24}$ part only tells you the scale. All three digits in $2.64$ are non-zero, so all are significant.

   Significant figures: 3

3. (c) $0.2370\ \text{g cm}^{-3}$

   The digits are $0.\ 2\ 3\ 7\ 0$. The leading zero before the decimal is not significant. The $2$, $3$, and $7$ are non-zero, so they count. The trailing zero after the decimal — the last $0$ — is significant because it was actually recorded. It tells us the measurement was precise to four decimal places.

   Significant figures: 4

4. (d) $6.320\ \text{J}$

   All digits are after a decimal point: $6$, $3$, $2$, $0$. The trailing zero is significant — it was written, so it was measured. …

Method: The "Non‑Zero / Zero Rule" Method for Counting Significant Figures

This method uses three simple rules to decide whether a digit is significant.

Steps

1. All non‑zero digits are always significant.

   (1–9 count, no exceptions.)

2. Zeros between non‑zero digits are always significant.

   (Captive zeros count.)

3. Leading zeros (zeros to the left of the first non‑zero digit) are never significant.

   (They only fix the decimal place.)

4. Trailing zeros are significant only if the number contains a decimal point.

   (If no decimal point is shown, trailing zeros are ambiguous and are not counted as significant in standard convention.)

Applying the Steps

(a) $0.007\ \text{m}^2$

• Leading zeros: not significant.
• Only the 7 is non‑zero.
• Answer: 1 significant figure

(b) $2.64 \times 10^{24}\ \text{kg}$

• The coefficient $2.64$ contains 2, 6, 4 — all non‑zero.
• The power of 10 does not affect the count.
• Answer: 3 significant figures

(c) $0.2370\ \text{g cm}^{-3}$

• Leading zero: not significant.
• Digits: 2, 3, 7 are non‑zero → significant.
• Trailing zero after the decimal point: significant (rule 4).
• Answer: 4 significant figures

(d) $6.320\ \text{J}$

• All digits are non‑zero or trailing after a decimal.
• 6, 3, 2, 0 — the zero is trailing and after the decimal.
• Answer: 4 significant figures

(e) $6.032\ \text{N m}^{-2}$ …

Here is a breakdown of the common mistakes students make when determining significant figures, specifically for the given examples, and how to avoid them.

Mistake 1: Misinterpreting Leading Zeros (The "Decimal Point Trap")

The Mistake: Students often count all zeros, including those at the beginning of a number. For example, in (a) $0.007\ \text{m}^2$, a common error is to count the three zeros (0.007) and say the answer is 3 significant figures.

Why it happens: Students see zeros and assume they are "significant" without checking their position relative to the decimal point and the first non-zero digit.

How to Avoid (The Rule):

• Leading zeros are NEVER significant. They are only placeholders to locate the decimal point.
• The Rule: Start counting from the first non-zero digit from the left.
• For (a) $0.007\ \text{m}^2$: The first non-zero digit is 7. The zeros before it are not significant.
  • Correct Answer: 1 significant figure.

Mistake 2: Forgetting the "Trailing Zero" Rule for Numbers Without a Decimal

The Mistake: This is less common in the given examples (since they all have decimals), but it's a classic exam trap. For a number like $7000$, students often say it has 4 significant figures.

Why it happens: They assume all zeros are significant, ignoring the ambiguity of trailing zeros in whole numbers.

How to Avoid (The Rule):

• Trailing zeros are significant ONLY if the number contains a decimal point.
• Example: $7000$ has 1 significant figure (the 7). $7000.$ (with a decimal) has 4 significant figures. $7000.0$ has 5 significant figures.
• For the given examples: All numbers have a decimal point, so this specific trap is avoided, but it's crucial to remember for other questions.

Mistake 3: Ignoring the "Trailing Zero" Rule for Numbers WITH a Decimal

The Mistake: Students stop counting at the last non-zero digit, ignoring zeros at the end of a number that has a decimal point. For example, in (c) $0.2370\ \text{g cm}^{-3}$, they might say the answer is 3 significant figures (2, 3, 7), ignoring the final zero.

Why it happens: They think the zero at the end is "unnecessary" or just a placeholder.

How to Avoid (The Rule):

• Trailing zeros in a number with a decimal point ARE significant. They indicate the precision of the measurement.
• For (c) $0.2370\ \text{g cm}^{-3}$: The zeros before the 2 are leading (not significant). The digits 2, 3, 7, and the final 0 are all significant.
  • Correct Answer: 4 significant figures.
• For (d) $6.320\ \text{J}$: The trailing zero is significant.
  • Correct Answer: 4 significant figures.

Mistake 4: Misinterpreting Zeros Between Non-Zero Digits (The "Sandwich" Rule)

The Mistake: Students sometimes think zeros between non-zero digits are not significant, or they get confused about how to count them. For example, in (e) $6.032\ \text{N m}^{-2}$, they might say 3 significant figures (6, 0, 3, 2? or 6, 3, 2?).

Why it happens: They don't have a clear rule for "captive" zeros.

How to Avoid (The Rule):

• Zeros between non-zero digits are ALWAYS significant. They are "captive" zeros.
• For (e) $6.032\ \text{N m}^{-2}$: The zero is between 6 and 3, so it is significant. All digits (6, 0, 3, 2) are significant.
  • Correct Answer: 4 significant figures.
• For (f) $0.0006032\ \text{m}^2$: The zeros before the 6 are leading (not significant). The zero between 6 and 3 is captive (significant). The digits 6, 0, 3, 2 are significant.
  • Correct Answer: 4 significant figures.

Mistake 5: Confusing Scientific Notation with the Number of Significant Figures …