Problem 158 - Introductory Chemistry Atoms First

Question

Round each number to three significant digits and express the answer in scientific notation: (a) \(0.592861\) (b) 438932 (c) \(0.000073978\) (d) \(0.235469\) (e) \(82.550\) (f) \(529.8\)

Step-by-Step Solution

Answer

The rounded numbers in scientific notation are: (a) \(5.93 \times 10^{-1}\) (b) \(4.39 \times 10^5\) (c) \(7.40 \times 10^{-5}\) (d) \(2.35 \times 10^{-1}\) (e) \(8.25 \times 10^1\) (f) \(5.30 \times 10^2\)

Step 1: (a) Identify the three most significant digits in \(0.592861\)

The three most significant digits in \(0.592861\) are \(5, 9,\) and \(2\). Since the next digit after \(2\), which is \(8\), is greater than or equal to \(5\), we should round up to get \(0.593\).

Step 2: (a) Express \(0.593\) in scientific notation

Put \(0.593\) in the form \(a \times 10^b\), where \(1 \leq a < 10\) and \(b\) is an integer. Here, we have \(5.93 \times 10^{-1}\).

Step 3: (b) Identify the three most significant digits in \(438932\)

The three most significant digits in \(438932\) are \(4, 3,\) and \(8\). Since the next digit after \(8\), which is \(9\), is greater than or equal to \(5\), we should round up to get \(439000\).

Step 4: (b) Express \(439000\) in scientific notation

Put \(439000\) in the form \(a \times 10^b\), where \(1 \leq a < 10\) and \(b\) is an integer. Here, we have \(4.39 \times 10^5\).

Step 5: (c) Identify the three most significant digits in \(0.000073978\)

The three most significant digits in \(0.000073978\) are \(7, 3,\) and \(9\). Since the next digit after \(9\), which is \(7\), is greater than or equal to \(5\), we should round up to get \(0.0000740\).

Step 6: (c) Express \(0.0000740\) in scientific notation

Put \(0.0000740\) in the form \(a \times 10^b\), where \(1 \leq a < 10\) and \(b\) is an integer. Here, we have \(7.40 \times 10^{-5}\).

Step 7: (d) Identify the three most significant digits in \(0.235469\)

The three most significant digits in \(0.235469\) are \(2, 3,\) and \(5\). Since the next digit after \(5\), which is \(4\), is less than \(5\), we do not need to round. The number remains \(0.235\).

Step 8: (d) Express \(0.235\) in scientific notation

Put \(0.235\) in the form \(a \times 10^b\), where \(1 \leq a < 10\) and \(b\) is an integer. Here, we have \(2.35 \times 10^{-1}\).

Step 9: (e) Identify the three most significant digits in \(82.550\)

The three most significant digits in \(82.550\) are \(8, 2,\) and \(5\). Since the next digit after \(5\), which is \(5\), is equal to \(5\), we need to look at the following digit, which is \(0\) and less than \(5\). Thus, we don't need to round. The number remains \(82.5\).

Step 10: (e) Express \(82.5\) in scientific notation

Put \(82.5\) in the form \(a \times 10^b\), where \(1 \leq a < 10\) and \(b\) is an integer. Here, we have \(8.25 \times 10^1\).

Step 11: (f) Identify the three most significant digits in \(529.8\)

The three most significant digits in \(529.8\) are \(5, 2,\) and \(9\). Since the next digit after \(9\), which is \(8\), is greater than or equal to \(5\), we should round up to get \(530\).

Step 12: (f) Express \(530\) in scientific notation

Put \(530\) in the form \(a \times 10^b\), where \(1 \leq a < 10\) and \(b\) is an integer. Here, we have \(5.30 \times 10^2\).

Key Concepts

Scientific Notation

Scientific notation is a way to express numbers that are too big or too small in a more manageable form. It allows us to write numbers as a product of a number between 1 and 10, and a power of ten. This is particularly useful in science and engineering where you might encounter very large measurements or very tiny particles. Scientific notation provides a concise and standardized way to represent such numbers. For example, the number 438,932 can be written as 4.38932 × 10⁵ in scientific notation. The steps to convert a number into scientific notation are simple:

• Move the decimal point in the number until you have a coefficient between 1 and 10.
• The number of places you moved the decimal point becomes the exponent of 10.
• If you move the decimal to the left, the exponent is positive; if you move it to the right, the exponent is negative.

Scientific notation is beneficial for handling extremely large or small numbers and simplifies comparative calculations.

Rounding Numbers

Rounding numbers is all about reducing the digits in a number while keeping its value close to the original. This is especially useful when dealing with measurements or calculations where a couple of digits past the decimal may not considerably affect precision. The basic rule of rounding is to look at the digit immediately after the last significant digit you want to keep.

• If it's 5 or above, round up the last significant digit.
• If it's below 5, keep the last significant digit the same.

For example, in rounding the number 82.550 to three significant digits, we look at the digit after the last desired digit, which is 0. Since 0 is less than 5, the number remains 82.5. Rounding is crucial when you need to simplify a number while maintaining a reasonable level of accuracy.

Significant Digits

Significant digits, or significant figures, are the numbers in a measurement that provide meaningful information about its precision. They include all non-zero digits, zeros between significant digits, and trailing zeros in the decimal portion. Understanding significant digits is important when recording scientific data and for precise calculations. Here are some tips for identifying significant digits:

• Non-zero numbers are always significant. (e.g., in 45.6, all digits are significant)
• Any zeros between significant digits are significant. (e.g., in 405, the zero is significant)
• Leading zeros are never significant. (e.g., in 0.0067, the zeros merely indicate the position of the decimal point)
• Trailing zeros in a number with a decimal point are significant. (e.g., in 85.00, all digits are significant)

Applying these simple rules helps ensure that the data you work with reflects the precision of your measurements or calculations accurately.