Question

Indicate whether the trailing zero in each value is significant, not significant, or possibly significant: (a) \(540 \pm 0.5\) (b) \(540 \pm 5\) (c) \(0.540\) (d) \(0.000540\) (e) 540

Short answer

(a) Possibly significant, (b) Not significant, (c) Significant, (d) Significant, (e) Possibly significant

Step-by-step solution

(a) Determine the significance of trailing zero in \(540 \pm 0.5\)

In this case, the value is \(540 \pm 0.5\). The uncertainty given is \(\pm 0.5\), meaning the actual value could be between \(539.5\) and \(540.5\). Since we can't know whether the zero is required to hold its place, the trailing zero can be considered possibly significant.

(b) Determine the significance of trailing zero in \(540 \pm 5\)

For the value \(540 \pm 5\), the actual value could be between \(535\) and \(545\). In this case, the trailing zero is not significant, because it doesn't affect the range of uncertainty.

(c) Determine the significance of trailing zero in \(0.540\)

The value is given as \(0.540\). The decimal has been used to indicate precision, and thus the trailing zero is indeed significant, as it indicates there is a higher level of precision.

(d) Determine the significance of trailing zero in \(0.000540\)

For the given value of \(0.000540\), the trailing zero is also significant. Here, the zero is necessary to maintain the given level of precision.

(e) Determine the significance of trailing zero in 540

The value is given as 540, which doesn't have any decimal. In this case, the significance of the trailing zero is ambiguous without any other context. So, the trailing zero is possibly significant.

Key concepts

trailing zeros

Trailing zeros are zeros that appear at the end of a number. Whether they are significant or not depends on the context.

• If a trailing zero is present after a decimal point, like in 0.540, it is considered significant because it indicates precision.
• For whole numbers without a decimal, such as 540, the significance of trailing zeros can be ambiguous without further information like a measurement uncertainty.
• When accompanied by a margin of error, as in 540 ± 0.5, the trailing zero might be possibly significant to reflect precision within the context of the measurement uncertainty.
• If there is a larger uncertainty range, like in 540 ± 5, the trailing zero is non-significant since it doesn't affect the measurement's meaningful range.

Overall, understanding the context and placement of zeros will help determine their significance.

uncertainty in measurements

Uncertainty in measurements refers to the doubt that exists about the result of any measurement. It is important for understanding the reliability of a measured value.

• The given uncertainty, like "+/- 0.5", specifies the range within which the true value lies.
• If a smaller uncertainty is given, it typically suggests higher confidence in the precision of the measurement.
• A larger uncertainty, such as "+/- 5", implies less precision and less significance for trailing digits.

In practical terms, uncertainty communicates the limits of what was physically measured and helps in assessing how precise a given measurement is. Always consider the margin of error along with the actual value when evaluating the significance of digits.

precision in scientific measurements

Precision in scientific measurements indicates the level of detail in which a measurement is defined, and it often relates to the number of significant figures.

• More digits of precision, such as in 0.540, mean the measurement is more finely tuned.
• A measurement like 540 ± 0.5 shows precision by providing a fairly narrow range of uncertainty.
• An expression with fewer significant digits, like 540 ± 5, suggests a broader range and less precision.
• Precision is critical in scientific experiments and data collection, as it guides how results can be compared or replicated.

Maintaining and understanding precision helps communicate the reliability and exactness of conclusions based on these measurements.

significance determination

Significance determination evaluates whether the digits in a number are meaningful in contributing to the precision and reliability of the measurement.

• Significant figures include all non-zero numbers, any zeros between significant digits, and trailing zeros if they occur after a decimal point.
• They indicate the accuracy of a measured or reported value.
• In 540 ± 0.5, determining significance might involve interpreting the context: whether that zero is needed for precision or maintaining the measurement's meaning.
• Significance is often context-driven, such as in scientific documentation where each figure might reflect precision, reliability, and the accepted methods of measurement.

Understanding significance ensures the proper communication of measurement accuracy and aids in standardizing calculations across different scientific studies and applications.