Question

State the number of significant figures in each of the following numbers: (a) \(40.0\) (b) \(0.081\) (c) 129,042 (d) \(4.090 \times 10^{-3}\)

Short answer

3, 2, 6, 4

Step-by-step solution

Identifying significant figures in 40.0

The number 40.0 has three digits. To determine the number of significant figures, note that trailing zeros to the right of the decimal point are significant. Thus, 40.0 has three significant figures.

Identifying significant figures in 0.081

The number 0.081 has three digits. Leading zeros are not significant, but the digits after the leading zeros are significant. Thus, 0.081 has two significant figures.

Identifying significant figures in 129,042

The number 129,042 has six digits and contains no decimal point. All the digits are significant. Therefore, 129,042 has six significant figures.

Identifying significant figures in 4.090 x 10^{-3}

For the number 4.090 x 10^{-3}, only the digits in 4.090 are considered for significant figures. The scientific notation part (10^{-3}) does not affect the count of significant figures. Therefore, 4.090 has four significant figures.

Key concepts

significant figures

In chemistry, significant figures reflect the precision of a number. They include all the non-zero digits, any zeros between these digits, and any trailing zeros when they are to the right of the decimal point.
Let's break it down:

• Non-zero digits are always significant. For instance, in 129,042, each digit is counted.
• Zeros between non-zero digits are significant. Take 4.090, where all zeros between 4 and 9 count.
• Leading zeros before non-zero digits are not significant. For example, in 0.081, the zeros before 8 are not counted.
• Trailing zeros in a number with a decimal point are significant. In 40.0, the zeros after the decimal count.

Understanding these rules helps you interpret measurements accurately.

leading zeros

Leading zeros appear before the first non-zero digit. They don't contribute to the number of significant figures. They only serve as place holders.
For example:

• In the number 0.081, the zeros before 8 are leading zeros, making only 8 and 1 significant.

These rules ensure precision is maintained and numbers are interpreted correctly. Even when written in scientific notation, such as 4.090 × 10^{-3}, leading zeros are ignored.

trailing zeros

Trailing zeros are zeros that follow a non-zero digit. Their significance depends on the presence of a decimal point.
If a decimal is present:

• Trailing zeros are significant. For example, in 40.0, all digits are significant, making it three significant figures.

If a decimal is absent:

• Trailing zeros are not significant. For example, 4000 would only have one significant figure unless further specified.

This rule helps scientists communicate precise measurements and avoid ambiguity.

scientific notation

Scientific notation is a way to express very large or very small numbers in a more concise form. Significant figures in numbers written this way are determined by the digits before the multiplication sign.
For instance:

• In 4.090 × 10^{-3}, focus on 4.090. It has four significant figures.
• The exponent (10^{-3}) does not impact the significant figures count.

Scientific notation helps in maintaining precision and readability of measurements, which is fundamental in chemistry.