Question

Round off or add zeros to each of the following to give an answer with two significant figures: a. \(5100000 \mathrm{~L}\) b. \(26711 \mathrm{~s}\) c. \(0.003378 \mathrm{~m}\) d. \(56.982 \mathrm{~g}\)

Short answer

a. \(5.1 \times 10^6\) L, b. 27000 s, c. 0.0034 m, d. 57 g.

Step-by-step solution

Identify the significant figures

Determine which digits are the significant figures for each value given in the exercise. Significant figures include all non-zero digits, any zeros between significant digits, and trailing zeros in the decimal part.

Round or add zeros for 5100000 L

For the number 5100000 L, the significant figures are the first two digits: 5 and 1. The value rounded to two significant figures is 5.1, and we need to account for the place value by adding zeros: 5100000 L \(\rightarrow\) 5.1 x 10^6 L.

Round or add zeros for 26711 s

For the number 26711 s, the significant figures are 2 and 6. The digit after the 2 and 6 is 7, which means we round up. Therefore, 26711 s \(\rightarrow\) 27000 s.

Round or add zeros for 0.003378 m

For the number 0.003378 m, the significant figures are 3 and 3. The digit after the second 3 is 7, which means we round up. Therefore, 0.003378 m \(\rightarrow\) 0.0034 m.

Round or add zeros for 56.982 g

For the number 56.982 g, the significant figures are 5 and 6. The digit after the 6 is 9, which means we round up. Therefore, 56.982 g \(\rightarrow\) 57 g.

Key concepts

Rounding Numbers

Rounding numbers is an essential skill in mathematics and science. It simplifies figures while preserving their meaningful parts. Here are a few key points:

• Identify the digit at the position to which you are rounding.
• If the next digit is 5 or higher, you round up.
• If the next digit is less than 5, you round down.

Let's consider the number 26711. To round it to two significant figures, look at the first two digits: 2 and 6. The third digit (7) is more than 5, so we round up and get 27000. It's like a simple game of deciding whether to go up or down the hill based on what's immediately ahead!
This technique ensures that rounded values remain close to the original numbers, making them easier to work with without losing essential information.

Precision in Measurements

Precision in measurements refers to how close repeated measurements of the same quantity are to each other. Accurate measurements have high precision and are essential for reliable scientific work.

• Identify which digits in your measurement are significant.
• Significant figures include all non-zero digits, zeros between significant digits, and trailing zeros in the decimal part.
• For example, in 5100000 L, only the digits 5 and 1 are significant.

This helps you understand the exactness and reliability of your measurements. When you record measurements with appropriate precision, it reflects the capability of your measuring instrument. For instance, the value 56.982 g rounded to two significant figures is 57 g, indicating that the mass has been measured with a precision that can distinguish up to the nearest gram.

Scientific Notation

Scientific notation is a way to express very large or very small numbers compactly. It involves powers of 10.

• Identify the significant figures in your number.
• Place the decimal point after the first significant figure.
• Count the number of places the decimal has 'moved' to get to its new position.
• That count becomes the exponent of 10.

For example, the number 5100000 L can be written as 5.1 x 10^6 L. The 5.1 represents the significant figures, and 10^6 shows that the decimal point has moved 6 places to the left. This method keeps calculations manageable and easily interprets very large or small numbers.