Significant Figures: Class 11 Physics Notes, Definition, Working Principle, Formula & Real-Life Applications

As per NCERT:

“As discussed above, every measurement involves errors. Thus, the result of measurement should be reported in a way that indicates the precision of measurement. Normally, the reported result of measurement is a number that includes all digits in the number that are known reliably plus the first digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures.“

It means that when you measure anything like length or weight, there is always some tiny uncertainty. So when you write down a measurement, you include all the digits you know for sure, plus one extra digit that might be a little off. Those certain digits, plus that one “maybe” digit at the end, are called significant figures. They tell anyone reading your result how precise your measurement really is.

Those who are going to appear in the CBSE board exam must practice the NCERT solutions of the Units and Measurement chapter.

The NCERT textbook (Class 11, Physics, Chapter 2) provides clear rules for determining significant figures:

1. Every nonzero digit counts

Why it’s true: If a digit isn’t zero, it came directly from your measuring device (say, a ruler or a stopwatch). There’s no doubt it’s “real” data.

Example: “123.4”

Here, 1, 2, 3, and 4 each represent something your instrument measured directly. So you have 4 significant figures.
Tip: Whenever you see a digit from 1 to 9, just count it—no questions asked.

2. Zeros between nonzero digits count

Why it’s true: A zero that sits between two nonzero digits wasn’t just a placeholder—it’s part of the actual value.
Example: “5003”
The two middle zeros tell you that the measurement really was “five thousand and three,” not “five thousand.” In other words, your instrument resolved those zeros as real information. So all four digits (5, 0, 0, 3) are significant.
Common mistake: Thinking that “5003” is really just “five” because zeros look empty. In reality, if those zeros weren’t measured, the number would have been written “5×10³” or something similar, not “5003.”

3. Leading zeros (zeros before the first nonzero digit) do not count

Why it’s true: Leading zeros simply place the decimal point—they don’t tell you anything about the precision of your measurement. They’re just placeholders to show “where” the real digits start.
Example: “0.045”
The zeros before the 4 (“0.0”) are not significant. The only digits you’re sure about are 4 and 5, so there are 2 significant figures.
Tip: Anything written purely to “push” the decimal point leftward does not count. You only start counting once you hit a nonzero digit.

4. Zeros at the end of a decimal number always count

Why it’s true: If you write a number like “4.330,” that last zero tells the reader that your instrument can measure to the thousandths place. It’s not just a placeholder—it’s evidence of precision.
Example: “4.330”
All four digits (4, 3, 3, and 0) are significant because the meter (or caliper) showed you that 4.330 was the reading, not simply 4.33 or 4.3. You believed that final zero.
Common pitfall: Dropping the final zero (writing 4.33) might suggest your tool only measured to the hundredths place. But if your device really did measure to the thousandths, you have to show that by keeping the zero.

5. Zeros at the end of a whole number (without a decimal) can be ambiguous

Why it’s tricky: In a number like “5400,” you don’t know if those two zeros were actually read by the instrument, or if they’re just placeholders. Did the scale only measure to the hundreds place (giving 5.4×10³), or did it measure to the tens (5.40×10²), or even to the ones (5.400×10¹)?
Example: “5400”
If someone writes it as 5.40 × 10³, you know for sure it’s three significant figures (5, 4, and 0). If it is written just as “5400,” it might be only two significant figures (“5.4 × 10³”), or it might be four (“5.400 × 10³”), depending on how precise the measuring tool was.
Solution: Whenever possible, switch to scientific notation. That way you explicitly show how many zeros (and decimal places) you trust.

In physics calculations, the number of significant figures in the result depends on the operation:

• Addition and Subtraction: The result has the same number of decimal places as the measurement with the fewest decimal places. Example: 33.3+3.11+0.313=36.723, rounded to 36.7 ( 1 decimal place, as 33.3 has one decimal place).
• Multiplication and Division: The result has the same number of significant figures as the measurement with the fewest significant figures. Example: 142.06×0.23=32.6738, rounded to 33 (2 significant figures, as 0.23 has 2).

For JEE Main, intermediate calculations should retain more significant figures to avoid rounding errors, but the final answer must match the least precise input. For example, in $v=\sqrt{2gh},\mathrm{if} g=9.8 \frac{m}{s^2}$ (2 significant figures) and h=10.25" " m (4 significant figures), the result for v is reported with 2 significant figures.

• Scientific Notation: To avoid ambiguity, use scientific notation. Example: 5400 with 3 significant figures is $5.40\times {10}^3$ .
• Instrument Precision: The least count of an instrument determines the number of significant figures. For a ruler with a 1 mm least count, a measurement of 12.3 cm has 3 significant figures.
• Consistency: In exams like JEE Main, numerical answers must match the significant figures of the least precise data provided in the problem to avoid losing marks.

• Over-reporting Precision: Reporting too many significant figures (e.g., 12.3456 instead of 12.3) suggests false precision.
• Ignoring Rules: Misapplying significant figure rules in calculations, such as not rounding correctly in addition/subtraction.
• Ambiguous Zeros: Failing to clarify the significance of trailing zeros in whole numbers (use scientific notation to resolve).