Notes on Inexact vs Exact Numbers, Accuracy & Precision, and Significant Figures
Inexact vs Exact Numbers
- Inexact numbers: numbers obtained from measuring devices (balance, ruler, volumetric pipette, graduated cylinder, stopwatch, etc.). Each device has a limit of precision and an inherent uncertainty. The reported value must reflect that uncertainty.
- Exact numbers: counting items or using defined quantities that do not require a measuring device. Examples:
- There are in a kilometer.
- A pair of shoes consists of shoes.
- Why uncertainty matters: you cannot report more precision than the device allows; calculators can display many digits, but measured values must include uncertainty through the last reported digit.
Accuracy vs Precision
- Accuracy: how close a measured value is to the accepted/true value (a standard or reference).
- Example: density of water at 4°C is (a true value under standard conditions).
- Precision: how close repeated measurements are to each other (reproducibility of the method).
- You can be precise without being accurate, and accurate without being highly precise. High accuracy means measurements cluster near the true value; high precision means measurements cluster tightly together.
How to determine inexact numbers with measuring devices
Step 1: identify the largest calibrated mark on the device.
Step 2: count the subdivisions between marks to determine which place value is directly known.
Step 3: determine the estimated digit (the last digit, which is uncertain) based on the zone of uncertainty between calibrated marks.
Step 4: combine known digits and the estimated digit to report the measurement with the appropriate precision.
Example: Ruler comparisons
- Ruler B (finer subdivisions between marks):
- There are 10 lines between, e.g., 2 and 3, so subdivisions correspond to tenths (the tenths place is calibrated).
- You might record a value near 2.5–2.6, with the tenth place as the estimated digit and the hundredths place as the next uncertain digit.
- In this setup, the estimated digit is associated with the tenths place.
- Ruler A (fewer subdivisions between marks):
- The observer estimates further, and the last uncertain digit can fall in the hundredths place, making this device more precise than Ruler B.
Example: Graduated cylinder (volume)
- Identify the largest marks: 40 and 50 (tens place is known).
- There are 10 subdivisions between them; thus the ones place can be resolved as 41, 42, 43, etc.
- If the bottom of the meniscus lines up with the 43 mark, you might report 43.0 when the hundredths place is the estimated digit.
- The last digit is within the zone of uncertainty (e.g., it could be 43.0 or 42.9 depending on interpretation).
Takeaway: once you know the device’s precision, you can determine the number of significant figures in a measurement.
Significant figures: rules and examples
Rule 1: Any nonzero digit is always significant.
- Example: has two significant figures.
Rule 2: Zeros can be significant or not depending on position and context.
- Internal zeros (sandwiched between nonzeros) are significant: 3.09 has four? Actually 3, 0, and 9 are significant, giving three sig figs.
- Trailing zeros after a decimal point are significant because the decimal indicates precision to that place: e.g., 200. has three significant figures (2, 0, and 0).
Rule 3: Trailing zeros without a decimal point are ambiguous (and often not considered significant).
- Example: 200 (without a decimal) is typically reported as having one significant figure.
Rule 4: Leading zeros are never significant.
- Example: 0.01 has one significant figure (the 1).
Rule 5: If a number is measured (not exact) and there is a decimal point, the trailing zeros are significant because they reflect precision.
- Example: 15.0 has three significant figures (1, 5, and 0).
Rule 6: Exact numbers (counting or defined quantities) are not limited by measurement precision.
- Examples: meters in a kilometer, and there are shoes in a pair.
Special note on decimal presence:
- When a decimal point is present, trailing zeros are generally significant because they indicate measured precision.
- When no decimal point is shown, trailing zeros may not be significant (depends on context and convention in measurement reporting).
Representing significant figures with scientific notation
- Scientific notation is often used to clearly indicate significant figures, especially for large or small numbers or when zeros could be ambiguous.
- All digits in the scientific notation coefficient are significant.
- The exact handling of significant figures in scientific notation will be covered in a later video, but the general idea is that the exponent does not affect the count of sig figs.
Practical implications and connections
- When interpreting a measurement from a device, always report the value to the precision the device can support (i.e., include the estimated digit).
- When performing calculations, carry the appropriate number of significant figures, then round only at the end to reflect the precision of the measurement data.
- The concept of significant figures connects to the idea of uncertainty propagation: the precision of inputs affects the precision of outputs.
- In scientific reporting, the distinction between exact and inexact numbers helps maintain transparency about the reliability of measurements and calculations.
Quick reference: summary rules
- Nonzero digits are always significant: e.g., → 2 sig figs.
- Internal zeros are significant: e.g., → 3 sig figs.
- Trailing zeros with a decimal point are significant: e.g., → 3 sig figs.
- Trailing zeros without a decimal point are ambiguous; often not significant: e.g., → 1 sig fig.
- Leading zeros are not significant: e.g., → 1 sig fig.
- Exact numbers do not limit sig figs: e.g., , and shoes in a pair.
- Scientific notation clarifies sig figs: all digits in the coefficient are significant.