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 digits in a number that carry meaning and contribute to its measurement precision. To represent one thousand with varying significant figures, we have: \(1\) significant figure: \(1 \times 10^3\); \(2\) significant figures: \(1.0 \times 10^3\); \(3\) significant figures: \(1.00 \times 10^3\); and \(4\) significant figures: \(1.000 \times 10^3\). For the given calculation, \(\frac{1.5-1.0}{0.50}\), after performing the subtraction and division with proper consideration of significant figures, the result is \(1.0\). Contrary to the question, the correct answer is \(1.0\). This might be due to a mistake in the text or an attempt to demonstrate the importance of using significant figures correctly, leading to the right answer, which is \(1.0\).

Step-by-step solution

Understand significant figures

Significant figures are the digits in a measurement or number that carry meaning and contribute to its precision. A number with more significant figures is more precise than a number with fewer significant figures. When performing calculations, it's essential to consider significant figures to avoid losing precision or introducing false precision.

Indicate one thousand with varying significant figures

Let's show the number one thousand to 1, 2, 3, and 4 significant figures. \(1\) significant figure: \(1 \times 10^3\) \(2\) significant figures: \(1.0 \times 10^3\) \(3\) significant figures: \(1.00 \times 10^3\) \(4\) significant figures: \(1.000 \times 10^3\)

Analyze the given calculation

The given calculation is: \[ \frac{1.5 - 1.0}{0.50} \] First, we'll calculate the numerator and denominator individually, taking care of significant figures.

Calculate the numerator

In the numerator, we have: \[ 1.5 - 1.0 \] Both numbers have one decimal place (precision to the tenths), so the answer of the subtraction should also have one decimal place. We get: \[ 1.5 - 1.0 = 0.5 \]

Calculate the denominator

The denominator is already given as 0.50, and there is no calculation required.

Perform the given calculation

Divide the numerator by the denominator, considering the significant figures: \[ \frac{0.5}{0.50} \] Both the numerator and denominator have one decimal place, so our division result must have at least one decimal place as well. The result of the division is: \[ \frac{0.5}{0.50} = 1.0 \]

Understand the "not 1.0" question

The question asks why the answer is not \(1.0\) for the calculation. However, the answer to the given calculation \emph{is} \(1.0\). This might be a mistake in the text of the exercise or an attempt to show that the correct use of significant figures leads to the right answer, which is \(1.0\) indeed.