Question

Determine the number of significant figures in each measurement \(\begin{array}{ll}{\text { a. } 508.0 L} & {\text { c. } 1.0200 \times 10^{5} \mathrm{kg}} \\ {\text { b. } 820,400.0 \mathrm{L}} & {\text { d. } 807,000 \mathrm{kg}}\end{array}\)

Short answer

The number of significant figures in each measurement is as follows: a. 508.0 L has 4 significant figures, b. 820,400.0 L has 7 significant figures, c. \(1.0200 \times 10^{5}\) kg has 5 significant figures, and d. 807,000 kg has 3 significant figures.

Step-by-step solution

a. 508.0 L

To find the significant figures in 508.0, we apply the rules mentioned above. All non-zero digits are significant, and the trailing zero after the decimal is also significant. Therefore, there are 4 significant figures in 508.0 L.

b. 820,400.0 L

For 820,400.0, all non-zero digits are significant, and any zeros between significant digits are significant as well. The trailing zero after the decimal is also significant. So, there are 7 significant figures in 820,400.0 L.

c. 1.0200 × 10^5 kg

In the scientific notation, the significant figures are present in the number before the multiplication symbol (×). Here, 1.0200 has 5 significant figures as non-zero digits are significant and trailing zeros after the decimal point are significant as well. The exponent 10^5 does not affect the significant figures count.

d. 807,000 kg

For 807,000, we can see that all non-zero digits are significant, but the trailing zeros in a whole number are not significant. So, there are 3 significant figures in 807,000 kg.

Key concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers more conveniently. It involves representing numbers as a product of a decimal number and a power of 10. For example, instead of writing 0.0000123, we can express it as \(1.23 \times 10^{-5}\).
In scientific notation, only the digits before the multiplication sign (typically \(\times\)) count toward the significant figures. This makes it easy to identify how precise a measurement is without being distracted by zeros that simply set the scale of size through the power of 10. In the example \(1.0200 \times 10^5\), all digits in 1.0200 are considered significant.

• The exponent (\(10^5\) in this case), simply indicates the scale of the number and does not affect the count of significant figures.
• Scientific notation provides clarity when identifying significant figures in measurements.

Non-Zero Digits

Non-zero digits are any digits from 1 to 9. They are always considered significant figures whenever they appear in a number. This is because non-zero digits contribute to the actual value of the number.
For instance, in the number 508.0, both the '5' and '8' are non-zero digits. They help determine the magnitude and precision of the number. Non-zero digits are unambiguous in conveying a measurement's precision.
Whenever counting significant figures:

• Every non-zero digit counts as a significant figure.
• They are visible within measurements no matter where they appear in the number, whether at the beginning, middle, or end.

By understanding the role of non-zero digits, you can always start by counting them first when determining significant figures.

Trailing Zeros

We often encounter trailing zeros, which are zeros at the end of a number. Whether these zeros count as significant figures depends on the presence of a decimal point.
For example, in the number 820,400.0, the zeroes are trailing but the decimal indicator at the end signals that these zeros are significant. This implies a precise and reliable measurement to seven significant figures.
Here are important rules for trailing zeros:

• If there is a decimal point present, trailing zeros are significant.
• In numbers without a decimal point, trailing zeros are not significant as seen in 807,000 kg which suggests only three significant figures (807).

Recognizing the significance of trailing zeros can greatly affect reported measurements and their reliability.

Decimal Point Significance

Decimal points play a crucial role in understanding significant figures. They help indicate whether zeros are simply placeholders or part of the measurement precision.
A clear example can be seen in 508.0 L, where the decimal point makes the zero significant, implying increased precision and thus four significant figures.
With decimal point significance, remember:

• A decimal point following a zero makes that zero significant.
• Numbers with a decimal point convey more precision through their zeros, highlighting accurate measurement.

By understanding the importance of decimal points, you can better interpret and record precise measurements with confidence.