Question

How many significant figures are in each of the following measured quantities? a. \(20.60 \mathrm{~mL}\) b. \(1036.48 \mathrm{~kg}\) c. \(4.00 \mathrm{~m}\) d. \(20.8^{\circ} \mathrm{C}\) e. \(60800000 \mathrm{~g}\) f. \(5.0 \times 10^{-3} \mathrm{~L}\)

Short answer

a. 4, b. 6, c. 3, d. 3, e. 3, f. 2

Step-by-step solution

- Understanding Significant Figures

Significant figures (also known as significant digits) are the digits in a measurement that carry meaningful information about its precision. This includes all the nonzero digits, any zeros between them, and trailing zeros in the decimal part.

- Count Significant Figures for 20.60 mL

Count all the digits: 2, 0, 6, 0. All these digits are significant because trailing zeros after a decimal point are significant. Hence, 20.60 has 4 significant figures.

- Count Significant Figures for 1036.48 kg

Count all the digits: 1, 0, 3, 6, 4, 8. All these digits are significant. Therefore, 1036.48 has 6 significant figures.

- Count Significant Figures for 4.00 m

Count all the digits: 4, 0, 0. The trailing zeros after a decimal point are significant. Thus, 4.00 has 3 significant figures.

- Count Significant Figures for 20.8°C

Count all the digits: 2, 0, 8. All these digits are significant. Hence, 20.8 has 3 significant figures.

- Count Significant Figures for 60800000 g

Count the significant digits: 6, 0, 8. The trailing zeros without a decimal point are not significant. Therefore, 60800000 has 3 significant figures.

- Count Significant Figures for 5.0 × 10^-3 L

Count all the digits: 5, 0. The trailing zero after a decimal point is significant. Hence, 5.0 has 2 significant figures.

Key concepts

Measurement Precision

Measurement precision is crucial in scientific experiments and calculations. It refers to how close repeated measurements are to each other. The more precise a measurement, the more significant figures it has. Precise measurements help scientists and engineers make better decisions.
When you measure something, the precision is shown by the number of digits you record. For example, a measurement written as 20.60 mL is more precise than just 20 mL. This is because 20.60 mL shows that the measurement was taken to the nearest hundredth of a milliliter, while 20 mL can imply a broader range.
To improve precision in your measurements:

• Use better equipment like digital scales or high-quality measuring cylinders.
• Take multiple measurements and average them out.
• Make measurements under stable conditions to minimize variability.

Remember, more digits mean more precision, but only if those digits are reliable and represent true measurement accuracy.

Nonzero Digits

Nonzero digits are easy to understand because they are always significant. In any given number, all nonzero digits count toward significant figures.
For example, in the number 1036.48 kg, every digit—1, 0, 3, 6, 4, and 8—is significant. That makes a total of six significant figures. Nonzero digits are straightforward and very important in counting significant figures.
Let's look at another example: 20.8°C. The digits 2, 0, and 8 are all nonzero and therefore count. This gives us three significant figures.
Whenever you see a number, just count all the nonzero digits to find out a portion of the significant figures. This rule is especially useful when the number has a mix of zero and nonzero digits.
Here’s a quick way to remember:

• If it's not a zero, it's significant!
• Every single nonzero digit counts.

Trailing Zeros

Trailing zeros can be a bit tricky but are very important when dealing with significant figures. Trailing zeros are zeros at the end of a number. Whether or not these zeros are significant depends on where the decimal point is.
For instance, in the number 20.60 mL, the trailing zero after the 6 is significant because it comes after a decimal point. Therefore, 20.60 has four significant figures.
However, in the number 60800000 g, the trailing zeros (the zeros after 8) are not significant because there is no decimal point. This number, in this form, has only three significant figures.
Another example is 4.00 m. Here, the zeros after 4 are significant because they are at the end of a number with a decimal point. Therefore, 4.00 has three significant figures.
Easy tips to remember trailing zeros:

• If there's a decimal point and the zero is at the end, it matters.
• If there's no decimal point, the zero doesn't count.
• Trailing zeros can show precision when used correctly.

Understanding trailing zeros helps you determine the accuracy of measurements and interpret numbers correctly.