Question

How many significant figures are understood for the numbers in the following definition: \(1 \mathrm{mi}=5280 \mathrm{ft} ?\)

Short answer

The numbers involved in the definition are 1 mile and 5280 feet. 1 mile has 1 significant figure, while 5280 feet has 4 significant figures, making a total of 5 significant figures in the given definition.

Step-by-step solution

Identify the numbers involved in the definition

In the given definition, 1 mile is equivalent to 5280 feet. The numbers involved are 1 and 5280.

Determine the significant figures for the number 1 mile

The number 1 has only one digit, and it is non-zero. Therefore, it has 1 significant figure.

Determine the significant figures for the number 5280 feet

The number 5280 has all non-zero digits, which are all significant. However, the trailing zero (0) may or may not be significant, depending on how the number was measured or documented. It is unclear whether it is significant or not, but we can consider all four digits to be significant figures for the sake of this exercise, giving us a total of 4 significant figures.

Combine the number of significant figures for both numbers

We found that 1 mile has 1 significant figure and 5280 feet have 4 significant figures. Therefore, the given definition has 1 + 4 = 5 significant figures in total.

Key concepts

Measurement Precision

Precision in measurement refers to how detailed and exact a measurement is generally. It's all about how repeatable and consistent the results are when measuring is done multiple times under the same conditions.

In the context of significant figures, measurement precision gives us an idea of how much confidence we can have in a number's accuracy.
Here are some key points:

• More significant figures indicate a higher level of precision because more digits represent a closer approximation to the true value.
• The least precise measurement limits the precision of a calculated result because uncertainties can add up.
• Measurement tools have a certain precision cap, indicated by how many significant digits they can reliably produce.

Being aware of the precision of the numbers utilized is key to understanding the reliability and accuracy of the measurement.

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a concise form, which makes them easier to read and more manageable.

This is especially useful in scientific calculations where numbers can vary widely in magnitude. Scientific notation expresses numbers as the product of a number between 1 and 10 and a power of ten.
For example, the number 5280 can be expressed in scientific notation as:

• 5280 = 5.280 × 10^3

This format helps to clearly define the number of significant figures; in this case, four significant figures are present.
Some benefits of using scientific notation include:

• It makes it easier to compare magnitudes of numbers without getting bogged down by zeros.
• It clarifies the precision of the number, as leading and trailing zeros can be ambiguous in traditional notation.
• It facilitates straightforward arithmetic operations, such as multiplication or division, by simply adjusting exponents.

Numerical Accuracy

Numerical accuracy refers to how close a measured or calculated value is to its true value.

It's important to distinguish numerical accuracy from precision, although they are closely related.
Accuracy is about the closeness of the finish line, while precision is about the consistency of every step taken towards it.

• A reading with high numerical accuracy is likely to reflect the true value more closely.
• Errors, such as systematic and random errors, can affect accuracy.
• Accurate measurements often require carefully calibrated tools and methods.

Understanding numerical accuracy will help ensure that calculations or measurements provide a meaningful and correct representation of the quantity or concept being evaluated.