Question

State the number of significant digits in each of the following: (a) \(0.5 \mathrm{~mL}\) (b) \(0.50 \mathrm{~mL}\) (c) \(5.00 \mathrm{~mL}\) (d) \(500 \mathrm{~mL}\)

Short answer

(a) 1, (b) 2, (c) 3, (d) 1

Step-by-step solution

Understand the Basics of Significant Digits

Significant digits in a number are the digits that carry meaning contributing to its precision. This includes all non-zero digits, zeroes between significant digits, and trailing zeroes (in the decimal part).

Evaluate the Number 0.5 mL

In the number 0.5, the 0 is not significant because it's a placeholder. The 5 is the only significant digit here. Thus, there is 1 significant digit in 0.5 mL.

Evaluate the Number 0.50 mL

In 0.50, both zeros after the decimal point are significant because they indicate the precision of the measurement. Therefore, the number 0.50 has 2 significant digits.

Evaluate the Number 5.00 mL

In the number 5.00, all digits are significant. The trailing zeros after the decimal point show precision. So, the number 5.00 has 3 significant digits.

Evaluate the Number 500 mL

In the number 500, there are no decimal points, and it isn't specified whether the zeros are significant. Typically, it's assumed the zeros are not significant unless stated otherwise, leaving only the 5 as significant. Hence, 500 mL has 1 significant digit.

Key concepts

Precision

Precision in the context of measurements and numbers refers to how detailed or exact a number is. When a number has more digits, specifically more significant digits, it conveys more precision. For example:

• A measurement of 0.5 mL is less precise than 0.50 mL.
• When the measurement is 5.00 mL, it gives even more precision compared to 5 mL without any decimals.

Understanding precision means knowing how much detail your measurement provides. More digits after the decimal can mean a higher level of specificity for the measurement. It's like saying, "I'm getting more precise in stating my answer." This precision is often crucial in scientific experiments where knowing the exact amount or size down to the smallest unit can make a big difference in results and interpretations.

Placeholder Zeros

Placeholder zeros are zeros used to indicate the position of digits in a number, especially in decimal points, without being significant themselves. They help in representing the scale of the number:

• For example, in 0.5 mL, the zero is a placeholder. It shows that the number 5 is in the tenths place.
• These zeros do not affect the significant digits. For instance, removing these zeros does not change the value significantly.

Understanding placeholder zeros is key to identifying which zeros in a number are necessary simply to establish the size or position of other digits versus those which may carry meaning in precise measurements.

Trailing Zeros

Trailing zeros in a number can denote important significance when placed after a decimal point. They often indicate that the measurement or calculation is meant to be exact to a particular degree, showcasing precision:

• In the number 5.00, the two zeros following the decimal point are important. They suggest that the measurement is precise to the hundredth place.
• These zeros are only considered significant if they come after a decimal point, such as in 5.00 or 0.50.
• Without a decimal point, such as in 500, they might not be significant unless specified.

Trailing zeros can often be confused with placeholder zeros, but their significance is made clear when they are part of decimal precision.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. These include all non-zero digits, any zeros between significant digits, and trailing zeros if they are after a decimal point:

• In 0.5 mL, there is 1 significant figure, as only the 5 is considered significant.
• For 0.50 mL, the digits 5 and 0 are significant, thus giving 2 significant figures.
• In the number 5.00 mL, all three digits are significant, providing 3 significant figures.
• For 500 mL, only the 5 is typically considered significant, giving it 1 significant figure unless specified otherwise.

The concept of significant figures is essential in clearly communicating the precision and reliability of numerical data and measurements.