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Magazine Chapter 2: Problem 39
Challenge Round each number to four significant figures, and write the answer in scientific notation. a. 0.00054818 \(\mathrm{g} \quad\) c. 308,659,000 \(\mathrm{mm}\) b. 136,758 \(\mathrm{kg} \quad\) d. 2.0145 \(\mathrm{mL}\)
Short Answer
Expert verified
a. \(5.482 \times 10^{-4}\) g, b. \(1.370 \times 10^{5}\) kg, c. \(3.090 \times 10^{8}\) mm, d. \(2.015 \times 10^{0}\) mL.
Step by step solution
01
Identifying Significant Figures for 0.00054818 g
The number 0.00054818 has 6 significant figures: 5, 4, 8, 1, and 8. We need to round it to 4 significant figures.
02
Rounding 0.00054818 g to Four Significant Figures
To round 0.00054818 to four significant figures, we consider the fifth figure, which is 1. Since 1 is less than 5, we do not need to round up. So, the number becomes 0.0005482, which contains the significant figures: 5, 4, 8, and 2.
03
Converting 0.0005482 g to Scientific Notation
To convert 0.0005482 into scientific notation, we move the decimal point 4 places to the right which gives us 5.482. Therefore, in scientific notation, it is written as \(5.482 \times 10^{-4}\) g.
04
Identifying Significant Figures for 136,758 kg
The number 136,758 has 6 significant figures: 1, 3, 6, 7, 5, and 8. We need to round it to 4 significant figures.
05
Rounding 136,758 kg to Four Significant Figures
To round 136,758 to four significant figures, we consider the fifth figure, which is 7. Since 7 is 5 or greater, we round the fourth figure up, changing 6 to 7. Thus, it becomes 137,000.
06
Converting 137,000 kg to Scientific Notation
To convert 137,000 into scientific notation, we move the decimal point 5 places to the left which gives us 1.3700. Therefore, in scientific notation, it is written as \(1.370 \times 10^{5}\) kg.
07
Identifying Significant Figures for 308,659,000 mm
The number 308,659,000 has 9 significant figures: 3, 0, 8, 6, 5, 9, 0, 0, 0. We need to round it to 4 significant figures.
08
Rounding 308,659,000 mm to Four Significant Figures
To round 308,659,000 to four significant figures, we consider the fifth figure, which is 6. Since 6 is 5 or greater, we round the fourth figure up, changing 6 to 9 in the 8, resulting in 309,000,000.
09
Converting 309,000,000 mm to Scientific Notation
To convert 309,000,000 into scientific notation, we move the decimal point 8 places to the left which gives us 3.090. Therefore, in scientific notation, it is written as \(3.090 \times 10^{8}\) mm.
10
Identifying Significant Figures for 2.0145 mL
The number 2.0145 has 5 significant figures: 2, 0, 1, 4, and 5. We need to round it to 4 significant figures.
11
Rounding 2.0145 mL to Four Significant Figures
To round 2.0145 to four significant figures, we consider the fifth figure, which is 5. Since 5 is equal to 5, we round the fourth figure up, changing 4 to 5, resulting in 2.015.
12
Converting 2.015 mL to Scientific Notation
To convert 2.015 into scientific notation, we move the decimal point 0 places, keeping it as 2.015. Therefore, in scientific notation, it is written as \(2.015 \times 10^{0}\) mL.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rounding Numbers
Rounding numbers is a fundamental skill in mathematics that helps simplify numbers to make them easier to manage or understand. When rounding numbers to a certain number of significant figures, you focus on the digits of the number that have meaningful contributions to its measurement. Here鈥檚 a simplified approach:
- Identify all the digits in the number. For instance, in the number 308,659,000, all digits are significant.
- Decide how many significant figures you need. Suppose you need to round 308,659,000 to 4 significant figures.
- Look at the fifth digit. If it is 5 or greater, round up the fourth digit. In our example, the fifth digit is 6, so we increase the fourth digit by one, resulting in 309,000,000.
- If the fifth digit is less than 5, keep the fourth digit the same as in the case of 0.00054818 rounded to 0.0005482.
Scientific Notation
Scientific notation is a widely-used method to write very large or small numbers conveniently. It expresses numbers as a product of a coefficient and a power of ten, simplifying calculations and comparisons. Here's how you convert a number into scientific notation:
- Identify the significant digits and ignore zeros that are not significant. For 309,000,000, the significant digits are 3, 0, and 9.
- Move the decimal point right after the first non-zero digit. In 309,000,000, moving it eight places to the left gives us 3.09.
- Count how many places the decimal point was moved. This number becomes the exponent of 10. In this case, it's 8, so we write it as \(3.09 \times 10^8\).
- For numbers less than one, the exponent will be negative. For example, 0.0005482 becomes \(5.482 \times 10^{-4}\) after moving the decimal 4 places to the right.
Decimal Point Movement
The movement of the decimal point plays a vital role in both rounding numbers and converting them into scientific notation. Here鈥檚 how you handle it:
- Rounding Numbers: When rounding, the position of the decimal point doesn't change; you adjust the digits based on rounding rules, such as increasing or maintaining digits based on the number following your rounding cut-off.
- Scientific Notation: Here, moving the decimal point is key to getting the exponent of 10.
- For large numbers: Move the decimal point to the left until only one non-zero digit remains to the left of the decimal point, e.g., 137,000 becomes 1.37 after moving the decimal 5 places to the left.
- For small numbers: Move it to the right until one non-zero remains on the left side, like turning 0.0005482 into 5.482 by shifting 4 places.
