Problem 112 - Chemistry Matter and Change

Question

Round 20.56120 \(\mathrm{g}\) to three significant figures. (Chapter 2\()\)

Step-by-Step Solution

Answer

20.56120 g rounds to 20.6 g to three significant figures.

Step 1: Identify the Significant Figures

Determine which are the significant figures in the given number. The number is 20.56120. The significant figures start from the first non-zero digit on the left. In this case, it is '2'. We will count three significant figures.

Step 2: Count and Identify the Third Significant Figure

Count three significant figures. Start counting from '2'. The number looks like this: 2, 0, 5. So, the third significant figure is '5'.

Step 3: Check the Fourth Figure for Rounding

Look at the digit immediately after the third significant figure. The fourth digit is '6'. This will determine whether you round up or leave the third figure as is.

Step 4: Apply Rounding Rule

Since the digit '6' is greater than 5, you round up the third significant figure, which is '5', to '6'. So, the rounded figure is 20.6.

Key Concepts

Rounding Numbers

Rounding numbers is a fundamental concept in mathematics and science that simplifies numbers to make them easier to work with. This process is especially important when dealing with significant figures, as it affects the precision of measurement results. For example, when rounding 20.56120 grams to three significant figures, you need to determine which numbers are significant.

• Start with the leftmost non-zero number.
• Count the desired number of significant figures.
• Use the number immediately after your last significant figure to decide whether to round up or stay the same.

In our case, you begin counting significant figures with '2', then '0', and the third one is '5'. The next number is '6', which tells us the '5' needs to be rounded up, giving a result of 20.6 grams.

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a simplified manner. This notation is common in science and engineering to make data easier to read and compare. Instead of writing out long strings of zeros, scientific notation uses powers of ten.

• A number in scientific notation is typically shown as: \[a \times 10^n\]
• The "a" is a number, known as the significand, usually a value between 1.0 and 10.0.
• The "n" is an integer representing the power of ten necessary to get the actual number.

For instance, 3500 can be written as \(3.5 \times 10^3\) when expressed in scientific notation. This format makes it easier to grasp large numerical values and compare them without confusion.

Measurement Precision

Measurement precision is crucial for determining the reliability and consistency of measurements. It refers to how close repeated measurements are to each other under unchanged conditions. The precision of a measurement can affect the number of significant figures you include when rounding.

• Higher precision implies more significant figures and a potentially smaller margin of error.
• Measurements with low precision may require rounding, which can help streamline data without significantly affecting accuracy.

For example, if your scale can measure to the thousandths place, rounding 20.56120 to three significant figures (20.6) remains relatively accurate and reflects the precision of the measurement instrument. Understanding measurement precision helps ensure that data integrity is maintained across scientific studies and reports.