How Accurate Are You? Determining The Significant Figures

Summary

Measurements have limited precision because different instruments may produce slightly different readings. This variation is called measurement uncertainty. When uncertainty is not stated explicitly, significant figures indicate how precise a measured value is: they include all digits considered certain, plus one estimated digit.

For example, in 59.4 cm, the 5 and 9 are treated as certain, while the 4 is the estimated digit, giving three significant figures.

Rules for Identifying Significant Figures

• Nonzero digits are always significant. For example, 12 has 2 significant figures, and 1467 has 4.
• In scientific notation, count the digits in the coefficient, not the power of ten. For example, (2.326 \times 10^{12}) has 4 significant figures.
• Zeros between nonzero digits are significant. For example, 2304 has 4 significant figures.
• Leading zeros are not significant. For example, 0.00038 has 2 significant figures.
• Trailing zeros in a number with a decimal point shown are significant. For example, 84.00 has 4 significant figures, and 8400.00 has 6.
• Trailing zeros in a whole number without a decimal point shown are treated as not significant in the video. For example, 840 is said to have 2 significant figures.

Rounding and Calculations

To round to a specified number of significant figures, identify the last significant digit needed and inspect the digit immediately after it:

• If that digit is greater than 5, increase the last retained digit by one.
• If it is less than 5, drop the following digits.
• If it is exactly 5, the video recommends rounding to the nearest even number.

For multiplication and division, the video says to round the result to the same number of significant figures as the input with the fewest. For example, (0.0281 \times 1.3605 = 0.03823005), which rounds to 0.0382 (3 significant figures).

Accuracy Notes

The video’s addition example applies the fewest-significant-figures rule to (1.312 + 0.45), giving 1.8 kg. Under the usual scientific convention, addition and subtraction are rounded according to the fewest decimal places, so this example would give 1.76 kg. The fewest-significant-figures rule is generally used for multiplication and division.

Rounding to an even digit applies to an exact tie—when the discarded part is precisely halfway between two values. The video’s examples, 1.05598 and 1.02598 rounded to 3 significant figures, are not exact ties. Using ordinary rounding, they become 1.06 and 1.03, respectively.

The video concludes that significant figures matter in fields that rely on measurements, including engineering, astronomy, and medicine. Understanding them helps students report results appropriately.

Speakers and Sources Featured

• Narrator: An unidentified voiceover presenting the lesson.
• Other speakers or external sources: None identified in the subtitles.