Question

How many significant figures do these numbers have? a) 0.009 b) 0.0000009 c) 65,444 d) 65,040

Short answer

a) 1, b) 1, c) 5, d) 4

Step-by-step solution

Understanding Significant Figures

Significant figures in a number are the digits that carry meaningful contributions to its measurement accuracy. This includes all non-zero numbers, zeros between significant numbers, and trailing zeros in a decimal.

Analyzing Number (a) 0.009

In the number 0.009, the leading zeros are not significant. However, the '9' is a significant figure. Therefore, there is only 1 significant figure in 0.009.

Analyzing Number (b) 0.0000009

In the number 0.0000009, similar to the first number, all the leading zeros are not significant. The '9' is the only significant figure here. Thus, there is 1 significant figure in 0.0000009.

Analyzing Number (c) 65,444

In the number 65,444, all digits are non-zero and are therefore significant. This means the number has 5 significant figures.

Analyzing Number (d) 65,040

For the number 65,040, the digits '6', '5', and '4' are significant. The trailing zero (after 4) is a placeholder here and is not significant unless specified by a decimal point. Therefore, this number has 4 significant figures.

Key concepts

Measurement Accuracy

Measurement accuracy is crucial in scientific and mathematical calculations. It refers to how close a measured value is to its actual value. Significant figures play a key role in representing this accuracy.

When a number is represented with significant figures, it tells us how precise the measurement is. The more significant figures a number has, the more accurate the measurement is. For instance:

• 0.009 has only one significant figure, indicating less precision in measurement.
• 65,444 has five significant figures, showing a greater degree of accuracy.

Understanding measurement accuracy helps us to determine how reliable and exact our calculations are. It ensures that we are aware of the potential errors or uncertainties in our data, allowing for more informed decision-making and analysis.

Non-Zero Digits

Non-zero digits are always considered significant because they contribute directly to the value of a number.

These digits immediately add meaningful information about a measurement's precision. In the example of 65,444, each digit is significant, totaling five significant figures.

When we examine a number, any non-zero digit should automatically be counted as a significant figure:

• In 65,444, the non-zero digits 6, 5, 4, and the two trailing 4's carry significant information about the measurement's accuracy.

Recognizing non-zero figures as significant allows us to distinguish between essential data and non-essential placeholders in numbers.

Trailing Zeros

Trailing zeros can be tricky when it comes to identifying significant figures, as their significance depends on the presence of a decimal point.

- Trailing zeros in a number with a decimal point are significant. For instance, 3.00 has three significant figures because the trailing zeros follow a decimal point and show precision.
- However, in numbers without a decimal point, such as 65,040, the trailing zero typically acts as a placeholder and is not always significant.

Understanding the role of trailing zeros helps in correctly identifying significant figures, which in turn contributes to more accurate calculations and assessments of data precision.

Decimal Significance

Decimal significance is pivotal when counting significant figures, as the presence of a decimal point can affect the interpretation of zeros.

For example, consider the numbers:

• 0.0000009 has only one significant figure, the '9', as all preceding zeros are not significant since they fall before a non-zero digit.
• 3.00 has three significant figures because the zeros are after a decimal and after a non-zero digit, thereby indicating precision.

Decimals play a huge role in determining how we perceive zeros and interpret the exactness of a number's representation. Recognizing decimal significance thus helps in better defining the precision level of a measurement.