- Convert the given numbers \(-1.3, 1 \frac{2}{5}, 0.9, -\frac{7}{10}\) to decimal form: \(-1.3, 1.4, 0.9, -0.7\).
- Calculate the opposites of these numbers: \(1.3, -1.4, -0.9, 0.7\).
- The values for part (a) are \(\{-1.3, -0.7, 0.9, 1.4\}\) and for part (b) are \(\{-1.4, -0.9, 0.7, 1.3\}\).
- Plotting these on a number line provides a visual representation of their positions relative to zero.
Explanation
- Data Analysis and Conversion
We are given the set of rational numbers: \(-1.3\), \(1 \frac{2}{5}\), \(0.9\), and \(-\frac{7}{10}\). To plot these on a number line, we first convert all values to their decimal representations:
- \(1 \frac{2}{5} = 1 + \frac{2}{5} = 1 + 0.4 = 1.4\)
- \(-\frac{7}{10} = -0.7\)
So the set of numbers is \(\{-1.3, 1.4, 0.9, -0.7\}\).
- Calculating Opposites
To find the opposites, we negate each value:
- The opposite of \(-1.3\) is \(-(-1.3) = 1.3\).
- The opposite of \(1.4\) is \(-(1.4) = -1.4\).
- The opposite of \(0.9\) is \(-(0.9) = -0.9\).
- The opposite of \(-0.7\) is \(-(-0.7) = 0.7\).
The set of opposites is \(\{1.3, -1.4, -0.9, 0.7\}\).
- Summary of Values for Plotting
For part (a), the points on the number line are \(-1.3, -0.7, 0.9, 1.4\).
For part (b), the points representing the opposites on the number line are \(-1.4, -0.9, 0.7, 1.3\).
These values all fall within the range \([-1.5, 1.5]\), which is appropriate for visualization.
Examples
Plotting numbers and their opposites on a number line is a fundamental skill in understanding the concept of signed numbers and distances from zero. For example, if you track the temperature changes over a day, where the morning temperature is \(-0.7^\circ\text{C}\) and it later reaches \(0.9^\circ\text{C}\), plotting these values helps visualize the progression and the magnitude of temperature shifts relative to the freezing point of \(0^\circ\text{C}\). This skill is essential for navigation, finance, and various scientific measurements where identifying positive and negative values is necessary.