Question

Determine the number of significant figures in each of the following measurements: (a) \(4867 \mathrm{mi}\), (b) \(56 \mathrm{~mL}\) (c) 60,104 tons, (d) \(2900 \mathrm{~g}\), (e) \(40.2 \mathrm{~g} / \mathrm{cm}^{3}\) (f) \(0.0000003 \mathrm{~cm},(\mathrm{~g}) 0.7 \mathrm{~min}\) (h) \(4.6 \times 10^{19}\) atoms.

Short answer

(a) 4, (b) 2, (c) 5, (d) 2, (e) 3, (f) 1, (g) 1, (h) 2 significant figures.

Step-by-step solution

Understanding Significant Figures

Significant figures in a number are the digits that carry meaning contributing to its measurement resolution. This includes all non-zero numbers, any zeros between significant digits, and any trailing zeros in a decimal number.

Analyzing each measurement

We'll go through each measurement given and apply the rules of significant figures to determine the count for each one. Let's start with the measurements:

Measurement (a) 4867 miles

For the number \(4867\), all the digits are non-zero. Thus, all four digits \(4, 8, 6,\) and \(7\) are significant.**Significant Figures: 4**

Measurement (b) 56 mL

For the number \(56\), both digits \(5\) and \(6\) are non-zero and thus significant.**Significant Figures: 2**

Measurement (c) 60,104 tons

The number \(60,104\) includes as significant all non-zero digits \(6, 0, 1,\) and \(4\), and the zero is between significant figures so it is also counted.**Significant Figures: 5**

Measurement (d) 2900 grams

In \(2900\), the digits \(2\) and \(9\) are significant, but zeros may or may not count depending on if there's a decimal point. Without a decimal, they are not significant by convention, and thus only \(2\) and \(9\) are significant.**Significant Figures: 2**

Measurement (e) 40.2 g/cm³

Here \(40.2\) has the digits \(4\), \(0\), and \(2\). All are significant since the zero is between significant figures.**Significant Figures: 3**

Measurement (f) 0.0000003 cm

The leading zeros are not significant. The only significant digit is \(3\).**Significant Figures: 1**

Measurement (g) 0.7 minutes

In \(0.7\), the leading zero is not significant, but \(7\) is a significant non-zero digit.**Significant Figures: 1**

Measurement (h) \(4.6 \times 10^{19}\) atoms

Any exponent does not alter the significant figures, so we focus on \(4.6\); here both digits \(4\) and \(6\) are significant.**Significant Figures: 2**

Key concepts

Measurement Resolution

Measurement resolution refers to the smallest possible change that can be accurately measured. It's like seeing how sensitive a measuring tool is. With significant figures, we determine how precise our measurements are. For example, in the measurement of 2900 grams, the zeros here don’t add to the measurement resolution because there’s no decimal indicating their significance.

• Accuracy is reflected in the number of significant figures. More significant figures mean more precise measurements.
• Measurement resolution helps in understanding the source of error in calculations and measurements.

Breaking down measurements into significant figures helps scientists accurately report data, improving the reliability of data comparisons.

Non-Zero Digits

Non-zero digits are any numbers between 1 to 9 in a measurement. They are always considered significant because they indicate the measured value. For example, in the number 4867 miles, the digits 4, 8, 6, and 7 are significant.

• Non-zero digits always carry meaning in measurements since they contribute directly to the resolution.
• They are simple to identify as they inherently convey information about what's being measured.

In computations, neglecting any non-zero digit can lead to incorrect interpretations of precision and accuracy.

Trailing Zeros

Trailing zeros are zeros that appear after a non-zero digit in a number. They can be tricky because their significance depends on whether they are in a decimal number. For instance, in 60,104 tons, the zero is counted because it is between non-zero digits.

• Trailing zeros in a whole number without a decimal are not considered significant (e.g., in 2900).
• However, in decimal numbers, zeros following a significant digit are significant themselves (e.g., 40.200).
• Remember, a decimal point adds clarity about zero significance.

Recognizing when trailing zeros are significant can prevent mistakes in data interpretation and reporting.

Decimal Numbers

Decimal numbers are numbers that contain a decimal point. They play an important role in expressing precision in measurements. The presence of a decimal point affects the significance of zeros, commonly enhancing the clarity

• Zeros between numbers or after a non-zero digit in decimal numbers are significant.
• Example: In 40.2, all three digits, including zero, are significant because of the decimal.

Decimals allow clearer representation of measurements, making sure that zeros are not overlooked when indicating accuracy.