Question

What is the number of significant figures in each of the following measured quantities? (a) \(601 \mathrm{kg},\) (b) 0.054 \(\mathrm{s}\) ,(c) \(6.3050 \mathrm{cm},(\mathbf{d}) 0.0105 \mathrm{L},(\mathbf{e}) 7.0500 \times 10^{-3} \mathrm{m}^{3},(\mathbf{f}) 400 \mathrm{g}\)

Short answer

(a) 3 significant figures (b) 2 significant figures (c) 5 significant figures (d) 3 significant figures (e) 4 significant figures (f) 1 significant figure

Step-by-step solution

(a) 601 kg

This quantity contains non-zero digits only, so all of them are significant. Hence, there are \(\boxed{3}\) significant figures in 601 kg.

(b) 0.054 s

Leading zeroes are not significant, so we will ignore 0 before the decimal point. Now, we have two non-zero digits (5 and 4), both of them are significant. The number of significant figures in 0.054 s is \(\boxed{2}\).

(c) 6.3050 cm

This quantity has one non-zero digit before the decimal point and four digits after the decimal point. The trailing zero (the last zero) is considered significant because it's after the decimal point. So, there are \(\boxed{5}\) significant figures in 6.3050 cm.

(d) 0.0105 L

In this case, leading zeros (0 before the decimal point and 0 after the decimal point) are not significant. Then, we have three digits, where one zero in between significant digits (1 and 5). So, there are \(\boxed{3}\) significant figures in 0.0105 L.

(e) 7.0500 x 10^{-3} m^3

Here we have all non-zero digits (7, 5) and two trailing zeroes after the decimal point (zeros are significant because they are after the decimal point). The digit in the exponent of 10 (-3) doesn't count towards the significant figures. So, there are \(\boxed{4}\) significant figures in 7.0500 x 10^{-3} m^3.

(f) 400 g

This quantity contains one non-zero digit (4) and two trailing zeroes. However, since there is no decimal point, the trailing zeroes are not significant. Therefore, there is only \(\boxed{1}\) significant figure in 400 g.

Key concepts

Measurement

Measurement is a key part of science and engineering, allowing us to quantify and understand the world around us. When we measure something, we're trying to determine a specific attribute—whether it be length, weight, time, etc.—as accurately and precisely as possible.
Accurate measurement involves both a correct value representation and understanding of any errors or uncertainties that might occur. The count of the significant figures (important digits) in a measurement tells us how precise it is. More significant figures typically mean a more precise measurement, which reflects lesser uncertainty.
When performing measurements, it's crucial to note:

• Each digit provides information about the precision of the measurement.
• Significant figures stem from the limitations of the measuring instrument.
• Errors or uncertainties can impact how we determine the significant figures in a measurement.

As such, we need to understand significant figures to report the results of measurements accurately.

Trailing zeros

Trailing zeros are the zeros that appear after non-zero digits in a number. Whether these trailing zeros are counted as significant figures depends on the placement of the decimal point.
For example, in the number 6.3050 cm, the trailing zero after the decimal point counts as a significant figure. This is because once a zero comes after a decimal point and a number, it helps specify the precision of the measurement.
However, in a number like 400 g, there are trailing zeros but no decimal point. In this case, these zeros aren't considered significant because they don't show any measured precision.

• Significant if a decimal point follows the zero, as in 50.0 (3 significant figures).
• Not significant when there is no decimal point, such as in 1500 (2 significant figures).
• Clear communication whether a zero is significant often requires writing in scientific notation.

The idea here is that the presence of a decimal point essentially 'anchors' the zeroes as part of the accuracy of the measurement.

Leading zeros

Leading zeros are zeros that precede the first non-zero digit in decimal numbers. They are not considered significant figures because they only position the decimal point, not indicate precision.
For instance, in the number 0.054 s, the zeros to the left of 5 do not count as significant figures. In this case, the measurement's significant figures are only the 5 and 4.
It is essential to distinguish these leading zeros because they do not add accuracy to a measurement, which affects calculation and communication clarity.

• In numbers like 0.0035, only the non-zero digits count, so it has 2 significant figures.
• Properly identifying leading zeros aids in explaining the measured quantity's precision and reliability.

Understanding leading zeros helps us precisely utilize and present data consistently in scientific notation and other forms.

Scientific notation

Scientific notation is a valuable way to represent very large or very small numbers in a concise and clear manner. It helps in keeping track of significant figures.
This notation expresses numbers as a product of a base number and ten raised to an exponent. For example, 7.0500 x 10^{-3} m^3 uses scientific notation to express a measurement. Here, 7.0500 tells us about the significant figures—emphasizing precision, while the 10^{-3} shows the scale of that number.
When using scientific notation, the digits making up the base number are usually all significant:

• The base number must reflect the number of significant figures, as it embodies the measurement accuracy.
• The exponent does not affect the significant figures; it simply places the decimal point correctly.
• It's especially useful when dealing with computations that require tonne of zeroes or very high precision.

Understanding how to use scientific notation correctly ensures that the numerical information you're presenting is precise and easily comprehensible.