Question

What is the number of significant figures in each of these measured quantities? (a) 4867 miles, (b) \(56 \mathrm{~mL}\) (c) 60,104 tons, (d) \(2900 \mathrm{~g}\).

Short answer

(a) 4867 miles has 4 significant figures. (b) 56 mL has 2 significant figures. (c) 60,104 tons has 5 significant figures. (d) 2900 g has 2 significant figures.

Step-by-step solution

Analyzing significant figures

The first step is to understand the rules for significant figures: Any non-zero digit is a significant figure. Any zero between two significant figures becomes significant. The left most zeroes appearing in front of non-zero digits are not significant. Zeroes at the end or right of a number are only significant if the number contains a decimal point.

Apply rule to each measured quantity

(a) For 4867 miles, all are non-zero digits, so all are significant. It has 4 significant figures. (b) For 56 mL, both are non-zero and hence significant. It has 2 significant figures. (c) For 60,104 tons, zeros between two non-zero digits are significant, so it has 5 significant figures. (d) For 2900 g, zeros to the right of the number without a decimal point are not significant, so it has 2 significant figures.

Key concepts

Measurement Accuracy

Measurement accuracy refers to how close a measured value is to the true value or accepted standard. In scientific terminology, it involves precision and reliability of measurement tools and methods. Understanding measurement accuracy is vital because it affects how we interpret data in experiments.

• High measurement accuracy means that repeated measurements yield values very close to the true value.
• Inaccurate measurements can lead to misleading conclusions.
• Calibration and maintenance of instruments play a crucial role in ensuring accuracy.

Considering significant figures is an essential aspect of improving measurement accuracy. They help us understand the precision of a measurement. Each significant figure represents the certainty of a digit in the number. Thus, when recording or rounding measurements, keeping precision in mind avoids diminishing accuracy.

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a concise and standardized form. It helps simplify calculations by expressing numbers as a product of a number between 1 and 10 and a power of ten. For example, the number 5,600 can be written as \(5.6 \times 10^3\).
Because scientific notation uses powers of ten, it's particularly useful in science and engineering where extreme values are common.

• Using scientific notation reduces the number of significant figures to only those that are relevant.
• It helps avoid errors in calculations by maintaining precision across computations.
• By expressing a number in scientific notation, we gain a clearer sense of its scale.

For example, instead of writing 0.00052, you can write \(5.2 \times 10^{-4}\), making the number easier to read and compare. This notation is essential for maintaining significant figures in scientific work.

Zero Significance Rules

Understanding zero significance rules is key to correctly determining how many significant figures are in a number. Significant figures are the digits in a measurement that contribute to its precision.
The rules for zero significance help identify whether zeros in a measurement are significant:

• Zeros between non-zero digits are always significant.
• Leading zeros (zeros in front of non-zero digits) are never significant; e.g., in 0.0035, only '3' and '5' count as significant figures.
• Trailing zeros (zeros at the end of a number) are significant only if there is a decimal point present; for example, 50.0 has three significant figures, whereas 50 has just one.

Grasping these rules helps when assessing measurements and ensuring recorded figures accurately reflect the precision of the instrument used. For instance, in the measurement "2900 g," treating the zeros as non-significant without a decimal point signifies that the measurement is rounded to the nearest hundred, thus having only two significant figures.