Q.What do you mean by significant figures?

Concept: Significant Figures – the digits in a measured value that carry meaning contributing to its precision. They include all certain digits plus the first uncertain (estimated) digit.

Essential reasoning:

1. Non-zero digits are always significant (e.g., 123 has 3 sig figs).
2. Zeros between non-zero digits are significant (e.g., 1002 has 4 sig figs).
3. Leading zeros are never significant (e.g., 0.005 has 1 sig fig). …

Why Significant Figures Matter

When you measure something — say, the length of a table with a ruler — you can never be absolutely exact. The ruler might show millimetre marks, so you can say "it's definitely between 152.3 cm and 152.4 cm", and you estimate the last digit as 152.35 cm. That last digit (the 5) is uncertain, but it still carries information. Significant figures are the way we communicate how much of that measurement is reliable.

Without significant figures, you could write "152.35 cm" and someone might think you measured to the nearest hundredth of a centimetre — which you didn't. The concept prevents us from pretending our measurements are more precise than they actually are.

The Rules at a Glance

Step-by-Step Examples

1. Non-zero digits are always significant.

The number $123.45$ has 5 significant figures — every digit is non-zero.

2. Zeros between non-zero digits count.

In $1002$, the two zeros are sandwiched between 1 and 2, so they are significant. This number has 4 significant figures.

3. Leading zeros never count.

Consider $0.0034$. The zeros before the 3 are just placeholders — they tell us the magnitude (thousandths), not the precision. Only the 3 and 4 are significant, so this has 2 significant figures.

4. Trailing zeros after a decimal point are significant.

$45.00$ has 4 significant figures. The two zeros after the decimal tell us the measurement was made to the nearest hundredth — they are not just placeholders.

5. Trailing zeros in a whole number without a decimal are tricky.

The number $1500$ could have 2, 3, or 4 significant figures depending on context. Was it measured to the nearest hundred? Nearest ten? To avoid ambiguity, scientists use scientific notation: $1.5 \times 10^3$ (2 sig figs), $1.50 \times 10^3$ (3 sig figs), or $1.500 \times 10^3$ (4 sig figs). …

Concept: Significant Figures (Precision in Measurement)

Significant figures are the digits in a measured value that are known reliably plus one digit that is uncertain. They indicate the precision of a measurement.

Key Rules (for exam accuracy):

1. All non-zero digits are significant (e.g., 123 has 3 sig figs).
2. Zeros between non-zero digits are significant (e.g., 1005 has 4 sig figs).
3. Leading zeros are not significant (e.g., 0.002 has 1 sig fig).
4. Trailing zeros are significant only if a decimal point is present (e.g., 100 has 1 sig fig; 100. has 3 sig figs).

Example: …

Common Mistakes: SI Unit of Mass — and How to Avoid Them

The Correct Answer First

• SI unit of mass: kilogram (kg)
• Definition: The kilogram is defined by taking the fixed numerical value of the Planck constant $h$ to be $6.62607015 \times 10^{-34}$ when expressed in the unit $\text{J·s}$, which is equal to $\text{kg·m}^2\text{·s}^{-1}$, where the metre and the second are defined in terms of $c$ and $\Delta \nu_{\text{Cs}}$.

(This is the current definition, revised in 2019. The old definition — the mass of the international prototype kilogram — is no longer valid.)

Mistake #1: Writing "gram" instead of "kilogram"

What students do wrong:

They write "gram (g)" as the SI unit of mass.

Why it's wrong:

The SI base unit of mass is the kilogram, not the gram. The gram is a submultiple (1 kg = 1000 g), but it is not the base unit.

How to avoid:

• Memorise: kg is the only SI base unit with a prefix (kilo-) built in.
• Remember the mnemonic: "The kilogram is the king of mass units."

Mistake #2: Giving the old definition (the "Paris prototype")

What students do wrong:

They say: "One kilogram is the mass of the international prototype kilogram kept at Sèvres, France."

Why it's wrong:

This definition was replaced in 2019. The prototype is no longer the standard — it was retired because its mass changed slightly over time.

How to avoid:

• Know the 2019 revision — the kilogram is now defined using the Planck constant $h$.
• Use the phrase: "Defined in terms of fundamental constants, not a physical object."

Mistake #3: Confusing mass with weight

What students do wrong:

They say: "The SI unit of mass is the newton (N)" or "Mass is measured in newtons."

Why it's wrong:

The newton is the SI unit of force (weight), not mass. Mass is a scalar quantity; weight is a force that depends on gravity.

How to avoid:

• Remember: kg for mass, N for weight.
• Think: "Mass is what you are; weight is what the scale shows under gravity."

Mistake #4: Omitting the Planck constant in the definition

What students do wrong:

They give a vague definition like: "It is defined using the Planck constant." — without the exact value or the equation.

Why it's wrong:

Exams expect precision. The definition must include:

• The fixed value: $h = 6.62607015 \times 10^{-34} \ \text{J·s}$
• The relationship: $\text{kg} = \frac{h}{6.62607015 \times 10^{-34} \ \text{m}^2\text{s}^{-1}}$

How to avoid:

• Write the definition in two parts:
  1. "The kilogram is defined by fixing $h = 6.62607015 \times 10^{-34} \ \text{J·s}$."
  2. "This implies $1 \ \text{kg} = \frac{h}{6.62607015 \times 10^{-34}} \ \text{m}^{-2}\text{s}$."
• Practise writing it exactly as in the official SI brochure.

Mistake #5: Forgetting the role of the metre and second

What students do wrong:

They define the kilogram in isolation, without mentioning that the definition depends on the definitions of the metre and second.

Why it's wrong: …