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Chapter 1: Problem 27

Determine the precision of each measurement. \(3.00 \times 10^{-4} \mathrm{~kg}\)

Short Answer

Expert verified
3 significant figures.

Step by step solution

01

Identify the Given Measurement

The measurement given is written in scientific notation as \(3.00 \times 10^{-4} \mathrm{~kg}\). This represents a weight or mass.
02

Analyze the Precision Indicator

Precision in a measurement refers to the number of significant figures shown. The number "3.00" in \(3.00 \times 10^{-4} \mathrm{~kg}\) has three significant figures. This is determined by counting all non-zero digits and any zeros that are between them or trailing in the decimal portion.
03

State the Precision of the Measurement

With three significant figures in "3.00", the precision of the measurement \(3.00 \times 10^{-4} \mathrm{~kg}\) is accurately represented to three significant figures.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Significant Figures
Significant figures are a fundamental part of accurately reporting measurements. They represent meaningful digits in a number that contribute to its precision. When you examine a number like 3.00, you count all the non-zero digits and any zeros that follow a decimal point. This means you have three significant figures. Why are significant figures important? They help convey how precise a measurement is. If a number includes more significant figures, it suggests a higher level of precision. This is critical in scientific calculations where accurate data can significantly influence experimental outcomes. To summarize:
  • Count all non-zero digits as significant.
  • Include zeros if they are between non-zero digits or after a decimal point.
  • Significant figures indicate the certainty of a measurement.
The Role of Scientific Notation
Scientific notation simplifies how we express large or small numbers, making them easier to read and compare. For instance, 3.00 x 10^{-4} helps you clearly see both the significant figures and the magnitude of the number.Numbers are written in the form \(a \times 10^n\), where \(a\) is a number between 1 and 10, and \(n\) is an integer. This format highlights the significant figures, making it easier to distinguish them from other digits.Benefits of scientific notation:
  • Simplifies comparisons between numbers of different sizes.
  • Makes calculations more straightforward, especially in scientific work.
  • Highlights the precision of measurements through clarity in significant figures.
Ensuring Measurement Accuracy
Measurement accuracy involves obtaining results that are as close as possible to the true value. While precision and accuracy are related, they are not the same. Precision concerns repeatability and the number of significant digits. Accuracy, on the other hand, evaluates how close a measurement is to the actual or true value. Achieving high accuracy requires reliable tools and methods. Systematic errors need to be minimized to ensure results are not just precise, but also accurate. Key points about measurement accuracy:
  • It鈥檚 about proximity to the true value, not just repeatability.
  • Both precision and the right methods contribute to accuracy.
  • Regular calibration of equipment can help enhance accuracy.
Understanding the balance between precision and accuracy is vital in any scientific experiment or measurement task.

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