Question

State the number of significant digits in each of the following: (a) \(1 \times 10^{-1} \mathrm{~mL}\) (b) \(1.0 \times 10^{-2} \mathrm{~mL}\) (c) \(1.00 \times 10^{1} \mathrm{~mL}\) (d) \(1.000 \times 10^{3} \mathrm{~mL}\)

Short answer

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

Step-by-step solution

Identify Significant Digits in 1 x 10^-1 mL

The number is written as \(1 \times 10^{-1}\). Here, the number '1' is the only digit and it is significant by default when appearing in scientific notation. Therefore, there is 1 significant digit.

Identify Significant Digits in 1.0 x 10^-2 mL

The number is represented as \(1.0 \times 10^{-2}\). The presence of the decimal indicates that the trailing zero is significant. Therefore, this number has 2 significant digits.

Identify Significant Digits in 1.00 x 10^1 mL

In \(1.00 \times 10^{1}\), both the '1' and '00' are significant because the decimal point signifies that trailing zeros are significant. Hence, this number has 3 significant digits.

Identify Significant Digits in 1.000 x 10^3 mL

For the number \(1.000 \times 10^{3}\), all the digits including the trailing zeros after the decimal point are significant. Thus, this number contains 4 significant digits.

Key concepts

Scientific Notation

Scientific notation is a convenient way to express very large or very small numbers using powers of ten.
It simplifies calculations and makes numbers more manageable.

When a number is written in scientific notation, it is displayed as the product of a number between 1 and 10 and a power of ten. For example, rather than writing 0.0001, it's expressed as \(1 \times 10^{-4}\). The first part ("1" in this example) is known as the coefficient, and it plays a crucial role in determining the number of significant figures.

• The coefficient must be greater than or equal to 1 and less than 10.
• The exponent shows how many times the base of 10 is multiplied.
• Scientific notation helps in easily recognizing the precision of a measurement.

Adopting scientific notation not just streamlines the work, but also enhances the understanding of numbers' scales, offering clear indications of significant digits.

Measurement Precision

Measurement precision refers to the exactness with which a quantity is measured in terms of significant digits.
It indicates the reliability and consistency of measurement results.

Precise measurements are essential in science and engineering, as small errors in measurement can lead to significant errors in final results.

• Precision tells about the repeatability of a measurement.
• It is not just about the number of digits, but about how reliably those digits can be used in further calculations.

For instance, a measurement of 1.000 mL implies a higher precision compared to 1 mL, suggesting that the measurement has been taken with more care and advanced equipment, which can provide more reliable data in complex calculations.
Precision is integral to ensure that scientific calculations and interpretations maintain their validity and accuracy.

Significant Digits Identification

Significant digits, also known as significant figures, are vital in determining the accuracy of a measurement. These are the digits that carry meaningful contributions to a number's precision.

To identify significant digits, consider:

• All non-zero digits are always significant. For example, in 123, all three digits are significant.
• Any zeros between significant digits are also significant. Thus, 1002 has four significant figures.
• Leading zeros, which precede all non-zero digits, are not significant. These zeros merely indicate the position of the decimal point. As in 0.0025, the significant digits are 2 and 5.
• Trailing zeros in a number with a decimal point are significant. For instance, in 1.2300, all five digits are significant, as they affirm precision.

Understanding significant digits helps in correctly rounding numbers during calculations and provides insightful knowledge on the precision level of numerical expressions in scientific and practical environments.