Question

How many significant digits are in the following unit factors? (a) \(1 \mathrm{~m} / 10 \mathrm{dm}\) (b) \(1 \mathrm{lb} / 454 \mathrm{~g}\) (c) \(1 \mathrm{~L} / 1000 \mathrm{~mL}\) (d) \(1 \mathrm{qt} / 946 \mathrm{~mL}\)

Short answer

(a) Infinite, (b) 3, (c) Infinite, (d) 3 significant digits.

Step-by-step solution

Understand the significance of unit factors

Unit factors represent conversion factors between two units and are exact by definition. This means that the relationships they describe are considered to have infinite significant digits.

Analyze each unit factor

Examine each given unit factor and consider the quantities involved. Since they are standard conversions, the numbers used in the conversion (other than '1') are considered definitively known quantities.

Count significant digits in each unit factor

- (a) For the unit factor \(1 \mathrm{~m} / 10 \mathrm{~dm}\), the conversion is exact and has infinite significant digits.- (b) For \(1 \mathrm{lb} / 454 \mathrm{~g}\), 454 is typically precise to three significant digits.- (c) For \(1 \mathrm{~L} / 1000 \mathrm{~mL}\), the conversion involves powers of 10 and is exact, leading to infinite significant digits.- (d) For \(1 \mathrm{qt} / 946 \mathrm{~mL}\), 946 is given to three significant digits.

Conclusion

Only counts from scientific measurements (rather than exact definitions or powers of ten) limit significant digits: hence, the answers will be 3 significant digits for (b) and (d), and infinite for (a) and (c).

Key concepts

Unit Conversion

Unit conversion is a fundamental concept in mathematics and science that involves changing a measure to an equivalent one in a different unit. Unit factors, or conversion factors, are the key to this process. They are used to convert between different systems of measurement, like metric to imperial, by multiplying or dividing.
For example, if we have a unit factor like 1 meter = 100 centimeters, this implies that multiplying or dividing by this factor will not alter the quantity's value. Instead, it helps to represent the same quantity in different units. In calculations involving unit factors, it's important to remember that these are exact by definition.
The exact nature of these conversions, such as meters to centimeters or liters to milliliters, means they have infinite significant figures. This is because they are precise definitions. As such, they do not limit the number of significant digits in the results of calculations.

Exact Numbers

Exact numbers are numbers that are known with complete certainty. These often arise in situations where measurements are defined or set by definition. Examples include the number of items in a dozen (12) and 1 foot being 12 inches.
In the context of unit conversions, exact numbers often represent the conversion factor. Like in the exercise examples, where unit conversions such as 1 meter being exactly 10 decimeters fall into the category of exact numbers. These exact relationships mean that these values have infinite significant figures, as there is no uncertainty.
Understanding this concept allows us to distinguish between measured quantities, which have a degree of uncertainty, and defined quantities, which are completely certain. This understanding is critical to ensure precision and accuracy in scientific calculations.

Measurement Precision

Measurement precision refers to how detailed or refined a measurement is. It links directly to significant figures, which indicate the precision of a measured or calculated quantity.
In measurements, precision is influenced by the instrument used or the method of measurement. For unit factors involving numbers like in the example of '1 lb / 454 g', here the number '454' has three significant figures. This means this unit factor is precise to three digits.
It's essential to recognize how precision affects calculations. When combining numbers of different precisions, the result should be reported to the lowest precision number used. Thus, when solving problems, understanding the precision of each figure ensures accurate and reliable conclusions.