Problem 62 - Algebra and Trigonometry

Question

Find the distance between the given numbers. $$ \begin{array}{llll}{\text { (a) } \frac{7}{15} \text { and }-\frac{1}{21}} & {\text { (b) }-38 \text { and }-57} & {\text { (c) }-2.6 \text { and }-1.8}\end{array} $$

Step-by-Step Solution

Answer

(a) \(\frac{18}{35}\), (b) 19, (c) 0.8.

Step 1: Understanding Distance in Mathematics

In mathematics, the distance between two numbers on the number line is the absolute value of their difference. This means we subtract the smaller number from the larger and remove any negative signs using absolute value.

Step 2: Solve Part (a)

The numbers given are \(\frac{7}{15}\) and \(-\frac{1}{21}\). First, compute the difference: \[\frac{7}{15} - \left(-\frac{1}{21}\right) = \frac{7}{15} + \frac{1}{21}\]To add these fractions, find a common denominator. The least common multiple of 15 and 21 is 105. Convert each fraction:\[\frac{7}{15} = \frac{49}{105}\quad \text{and}\quad \frac{1}{21} = \frac{5}{105}\]Now, add them:\[\frac{49}{105} + \frac{5}{105} = \frac{54}{105}\]The fraction \(\frac{54}{105}\) can be simplified to \(\frac{18}{35}\). Hence, the distance is \(\frac{18}{35}\).

Step 3: Solve Part (b)

The numbers given are \(-38\) and \(-57\). First, compute the difference:\[|-38 - (-57)| = |-38 + 57|\] Calculate the result:\[|19| = 19\]Thus, the distance is 19.

Step 4: Solve Part (c)

The numbers given are \(-2.6\) and \(-1.8\). First, compute the difference:\[|-2.6 - (-1.8)| = |-2.6 + 1.8|\]Calculate the result:\[|-0.8| = 0.8\]Thus, the distance is 0.8.

Key Concepts

Absolute Value

The concept of absolute value plays a crucial role in finding the distance between numbers on a number line. Simply put, the absolute value of a number is its distance from zero, without regard to which side of zero it lies on. This is why the absolute value is always a non-negative number.
For example, the absolute value of

• -3 is 3, written as \( |-3| = 3 \)
• and the absolute value of 4 is 4, noted as \( |4| = 4 \).

When calculating the distance between two numbers, we find the absolute value of their difference. This ensures the distance is always positive, as distance cannot be negative. For instance, for two numbers, \( a \) and \( b \), the distance between them is \( |a - b| \). This formula is handy because it disregards which number is larger and focuses on how far apart they are on a number line.

Fractions

In order to find the distance between two fractions, you essentially perform subtraction and find the absolute value of the result.
Let's say we have two fractions, \( \frac{7}{15} \text{ and }-\frac{1}{21} \). Converting these into a common denominator is essential before you can efficiently subtract them. To find their difference, you must:

• Identify the least common denominator, which is the smallest number that both denominators divide into evenly. Here, it would be 105.
• Convert each fraction to have this common denominator: \[\frac{7}{15} = \frac{49}{105}, \quad -\frac{1}{21} = -\frac{5}{105}. \]
• Add or subtract the equivalent numerators: \( \frac{49}{105} + \frac{5}{105} = \frac{54}{105} \).
• Simplify the result to find the minimal representation, \( \frac{54}{105} = \frac{18}{35} \).

Mastering how to convert and manipulate fractions is valuable, not only for solving exercises like this but also for everyday math tasks.

Number Line

To fully understand the concept of distance between numbers, it helps to envision a number line. A number line is a visual representation of numbers laid out in increasing order from left to right. They are useful for understanding numerical concepts intuitively, such as adding or subtracting numbers, comparing size, and measuring distance.
When we talk about the distance between two points on this line, we're really talking about how far those points are from each other.

• Any positive number you'd consider is placed to the right of zero, and negative numbers appear to the left of zero.
• The distance between two numbers essentially is how many 'steps' you would take along this line to move from one number to the other.

For instance, for numbers \( -2.6 \text{ and } -1.8 \), they both lie to the left of zero. The solution becomes apparent by simply calculating \( |-2.6 - (-1.8)| \), which rearranges to \( |-2.6 + 1.8| \), then provides a distance of \( 0.8 \).Understanding the number line is integral to visualizing not just distance, but also arithmetic operations and relationships between numbers.