Question

Indicate the number of significant figures in each of the following: a. This book contains over 500 pages. b. A mile is just over \(5000 \mathrm{ft}\). c. A liter is equivalent to \(1.059 \mathrm{qt}\) d. The population of the United States is approaching 250 million. e. A kilogram is \(1000 \mathrm{g}\). f. The Boeing 747 cruises at around \(600 \mathrm{mi} / \mathrm{h}\).

Short answer

a. 1 significant figure b. 2 significant figures c. 4 significant figures d. 2 significant figures e. 4 significant figures f. 1 significant figure

Step-by-step solution

a. This book contains over 500 pages.

The number given is 500, but since "over" is used, it implies that the exact number of pages is not provided and there is some uncertainty. So, there is just 1 significant figure, which is "5".

b. A mile is just over 5000 ft.

The number given is 5000, but since "just over" is used, it implies that the exact number of feet in a mile is not provided and there is some uncertainty. So, there are 2 significant figures, "5" and "0".

c. A liter is equivalent to 1.059 qt.

In this case, there is no uncertainty attached to the given number 1.059, so all the digits are considered significant. Hence, there are 4 significant figures: "1", "0", "5", and "9".

d. The population of the United States is approaching 250 million.

The given number is 250 million. The actual population is not provided, and since all the trailing zeroes don't contribute to accuracy, there are only 2 significant figures "2" and "5".

e. A kilogram is 1000 g.

As there's no leading or trailing zero that doesn't contribute to accuracy, the number given is 1000, and all the zeroes are significant. Hence, there are 4 significant figures, one each of "1" and the three "0"s.

f. The Boeing 747 cruises at around 600 mi/h.

The number given is 600, but since "around" is used, it implies that the exact speed is not provided and there is some uncertainty. So, there is just 1 significant figure, which is "6".

Key concepts

Scientific Notation

Scientific notation is a method of expressing numbers that are too big or too small to be conveniently written in decimal format. It is often used in science and engineering to simplify calculations and to reflect the precision of measurements. In scientific notation, a number is written as the product of a coefficient and a power of 10. The coefficient must be a number greater than or equal to 1 and less than 10, and the power of 10 indicates the number of decimal places the decimal point was moved.

For example, the distance between the Earth and the Sun is approximately 149,600,000 kilometers. In scientific notation, this can be expressed as 1.496 x 10⁸ kilometers. This represents a significant way to reflect the precision and convey a clear idea about the quantity being measured.

Measurement Uncertainty

Measurement uncertainty is an essential concept in science and engineering, signifying the doubt that exists about the result of any measurement. You can never be certain in the exactness of a measured value, and uncertainty arises due to various factors like limitations of the measuring instrument, the skill of the operator, or the conditions where measurements are taken. It's important to report this uncertainty to give a complete picture of the result.

For instance, when we say 'a mile is just over 5000 feet,' the 'just over' part indicates there is some uncertainty regarding the exact length. It could be 5001 feet, 5002 feet, and so on. As such, when we account for significant figures in measurements with uncertainty, we are recognizing the limits of what we can confidently claim to know about the measured value.

Significant Figures Rules

The rules for significant figures are conventions that help in determining the number of meaningful digits in a number. These rules play a crucial role when performing operations like multiplication, division, addition, and subtraction, where the precision of the result is based on the precision of the numbers being used.

Here are some key rules for counting significant figures:

• All non-zero digits are considered significant. For example, 1234 has four significant figures.
• Zeros between non-zero digits are significant. E.g., 1002 has four significant figures.
• Leading zeros are not significant as they only indicate the position of the decimal point. For example, 0.0005 has only one significant figure.
• Trailing zeros in a decimal number are significant. Thus, 2.00 has three significant figures.
• In a whole number without a decimal point, trailing zeros may or may not be significant based on the context or if a decimal point is explicitly included. E.g., 200 could have one, two, or three significant figures, but 200. has three significant figures.

Understanding these rules is essential for accurately reporting numerical data, ensuring consistency and precision in scientific communication.