Question

State the number of significant digits in each of the following: (a) \(5.0 \mathrm{~g}\) (b) \(5.05 \mathrm{~g}\) (c) \(0.5 \mathrm{~g}\) (d) \(1055 \mathrm{~g}\)

Short answer

(a) 2, (b) 3, (c) 1, (d) 4 significant digits.

Step-by-step solution

Understanding Significant Digits

Significant digits are the numbers in a measurement that contribute to its precision. These include all certain numbers plus one uncertain or estimated digit. Zeroes can be significant if they are between other numbers or after a decimal point.

Analyze (a) - 5.0 g

In '5.0 g', the number '5' and the '0' after the decimal point are significant because the zero indicates precision. Hence, there are 2 significant digits.

Analyze (b) - 5.05 g

For '5.05 g', each '5', and the zero in the middle, are considered significant. The middle zero is significant because it is between two other numbers. Thus, there are 3 significant digits.

Analyze (c) - 0.5 g

In '0.5 g', the '5' is significant while the leading zero is not. Leading zeroes are not counted as significant digits. Therefore, there is only 1 significant digit.

Analyze (d) - 1055 g

For '1055 g', each digit ('1', '0', '5', '5') is counted as significant, as all numbers including zeroes between significant digits are significant. So, there are 4 significant digits.

Key concepts

Measurement Precision

Measurement precision is like the level of detail in a measurement. It tells us how finely tuned an instrument is to detect different measurements. When you're measuring something, say the weight of an apple, the numbers you obtain are only as precise as the measuring tool allows.
For example, if a scale reads to the nearest gram, it can detect differences as small as one gram. However, if it reads to the nearest 0.1 gram, it can detect much smaller differences. The more precise the measurement, the more confidence you have that you're capturing the true value of what's being measured.
More precision means more significant figures, which are the digits that matter in your measurement. Just make sure to understand that more precise instruments usually come with a cost, in terms of both money and complexity.

Significant Figures Rules

Significant figures help us understand how precise a measurement is. There are a few simple rules you should know:

• All non-zero numbers are significant. For example, in '127', all the three numbers are significant because they are not zero.
• Any zero between other significant figures is also significant, like in '202' which has three significant digits.
• In decimal numbers, zeros to the right of a non-zero number are significant. Therefore, '5.00' has three significant digits.
• Leading zeros, or zeros to the left of the first non-zero number, are not significant. Thus, '0.05' has only one significant digit.

These rules help ensure that your calculations are as accurate as possible while communicating just how precise your measurements really are.

Zero Significance in Measurements

Zeros in measurements can be a bit tricky when it comes to determining whether they are significant. Here's how you can tell which zeros count:

**Between Other Numbers:** If a zero is sandwiched between two non-zero digits, it is always significant. For instance, in '5.05', the zero is significant because it falls between two '5s'.

**After a Decimal:** Zeros that appear after a decimal point and after a non-zero number are significant because they show precision. For instance, '2.30' shows a level of precision that distinguishes it from just '2.3'.

**Leading Zeros:** These are the zeros that precede all non-zero digits and are never significant. They're important as placeholders but don't indicate precision. In the number '0.003', only the '3' is significant.

By understanding how zeros function in measurements, you can better evaluate the precision of your data.