Question

What are significant figures? Show how to indicate the number one thousand to 1 significant figure, 2 significant figures, 3 significant figures, and 4 significant figures. Why is the answer, to the correct number of significant figures, not \(1.0\) for the following calculation? $$ \frac{1.5-1.0}{0.50}= $$

Short answer

Significant figures are the digits in a number that carry meaning and contribute to the precision of the measurement. To represent 1,000 to different significant figures, we have: 1. One significant figure: \(1 \times 10^3\) 2. Two significant figures: \(1.0 \times 10^3\) 3. Three significant figures: \(1.00 \times 10^3\) 4. Four significant figures: \(1.000 \times 10^3\) For the given calculation, \(\frac{1.5-1.0}{0.50} = \frac{0.5}{0.50} = 1\). The answer should be rounded to one significant figure (1), not 1.0, based on the rules of significant figures for subtraction and division operations.

Step-by-step solution

Representing one thousand in different significant figures

Let's express the number 1,000 in: 1. One significant figure 2. Two significant figures 3. Three significant figures 4. Four significant figures

One significant figure

To represent the number 1,000 to one significant figure, simply round the number to the nearest ten-thousandth. The most significant digit is 1, so the number becomes: $$ 1 \times 10^3 $$

Two significant figures

To represent the number 1,000 to two significant figures, round the number to the nearest thousandth. Since all digits after the most significant digit (1) are zeros, the number remains the same: $$ 1.0 \times 10^3 $$

Three significant figures

To express the number 1,000 in three significant figures, we add a decimal after the most significant digit, followed by two zeros after (no rounding needed): $$ 1.00 \times 10^3 $$

Four significant figures

To express the number 1,000 in four significant figures, we add a decimal after the most significant digit, followed by three zeros after (no rounding needed): $$ 1.000 \times 10^3 $$

Analyzing the given calculation

Now let's analyze the given calculation: $$ \frac{1.5-1.0}{0.50} $$

Performing the calculation

Perform the subtraction in the numerator and division: $$ \frac{0.5}{0.50} = 1 $$

Rounding based on significant figures

The subtraction operation has two significant figures (1.5 and 1.0), and the division operation has two significant figures (0.5 and 0.50). Additionally, the rules of significant figures indicate that when performing division, the result should be rounded to the least number of significant figures present in either the numerator or denominator. In this case, 1.5 - 1.0 has one significant figure (0.5 has only one non-zero digit). Therefore, the answer should be rounded to one significant figure, which is 1, not 1.0.

Key concepts

Scientific Notation

Scientific notation is a method of writing numbers that are too large or too small to be conveniently written in decimal form. It's commonly used in science and engineering to express measurements with a high degree of precision. This notation represents numbers as the product of a single digit or a combination of digits ranging from 1 to 9, followed by a decimal point, and the appropriate number of significant figures, multiplied by 10 raised to an exponent.

For example, the number one thousand can be written in scientific notation as follows:

• One significant figure: \( 1 \times 10^3 \)
• Two significant figures: \( 1.0 \times 10^3 \)
• Three significant figures: \( 1.00 \times 10^3 \)
• Four significant figures: \( 1.000 \times 10^3 \)

These variations show how altering the number of significant figures modifies the precision of the numerical representation.

Rounding Numbers

Rounding numbers is an essential skill in chemistry and other sciences that deals with precision and accuracy of measurements. It involves adjusting a number to a specific degree of significance, which is often determined by the context in which the measurement is used. The main rule for rounding is that if the digit to the right of the target significant figure is 5 or greater, you round up. If it's less than 5, you round down.

For instance, if we are to round the number 1.5 to one significant figure, we simply round to the nearest whole number resulting in 2. In the context of significant figures, zeros may or may not be considered significant, depending on their position and whether a decimal point is present. For example, when rounding the number 1.5 to one significant figure, it becomes 2, whereas rounding the number 0.50 to one significant figure results in 0.5, maintaining the non-zero digit as the significant figure.

Arithmetic Operations in Chemistry

Arithmetic operations in chemistry, such as addition, subtraction, multiplication, and division, must be performed with an understanding of significant figures. Each operation has its own rules for determining the number of significant figures in the final answer.

When adding or subtracting numbers, the result should have the same number of decimal places as the number with the fewest decimal places in the calculation. For multiplication and division, the result should have as many significant figures as the number with the least amount of significant figures used in the calculation. In our example exercise, \(\frac{1.5 - 1.0}{0.50} = 1\), the answer is 1 because division (and subtraction) requires us to limit the result to the least number of significant figures present in either number, which is one significant figure in this case. Whereas, if one mistakenly considered 1.0 to be two significant figures, the result would inaccurately be recorded as 1.0, which does not follow the rule for significant figures in division.