Question

Use the given real number line to compute each distance. $$ \begin{array}{rrrrrrrrrrr} & A & & B & C & D & & E & & & \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline-4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \end{array} $$ $$ d(C, E) $$

Short answer

The distance d(C, E) is 4.

Step-by-step solution

Identify the positions of C and E on the number line

Locate the positions of C and E on the number line. According to the number line, point C is at -1 and point E is at 3.

Set up the distance formula

The distance between two points on a number line is calculated using the formula: \[ d(C, E) = |C - E| \].

Substitute the positions into the formula

Substitute the values of C and E into the formula. This gives us: \[ d(C, E) = |-1 - 3| \].

Simplify the expression

Simplify the absolute value expression: \[ |-1 - 3| = |-4| = 4 \].

Key concepts

real numbers

Real numbers are a broad category of numbers that include almost all the numbers you can think of. They encompass integers (like -2, 0, 3), fractions (like 1/2), and irrational numbers (like √2 and π). Real numbers can be positive, negative, or zero. When we use a number line, every point on that line represents a real number. This makes the concept of real numbers very practical and easy to visualize.

absolute value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a positive value or zero. For example:

• The absolute value of -4 is written as \(|-4|\) and equals 4.
• The absolute value of 5 is written as \(|5|\) and equals 5.

You can think of absolute value as the number of steps you take to reach zero. This makes it easier to calculate distances on the number line, as we are often interested only in the magnitude of distance, but not the direction.

distance formula

The distance between two points on a number line can be easily calculated using the distance formula. This formula uses the absolute values to provide the numerical distance without considering direction. The formula is given as:

\[ d(A, B) = |A - B| \]

Here, \(d(A, B)\) represents the distance between points A and B. By taking the absolute value of the difference between the two points, you ensure that the answer is always positive. Let's revisit the previous exercise using this formula:

number line

A number line is a straight line that represents numbers at equally spaced intervals. It usually extends indefinitely in both directions, with zero at the center. Positive numbers are placed to the right of zero and negative numbers to the left.
You can use the number line to:

• Visualize addition and subtraction.
• Understand absolute values by finding the distance of a number from zero.
• Compute the distance between any two points, which is what we did in the given exercise.

For example, on our number line, C is at -1 and E is at 3. Using the number line, we can visually see the steps needed to go from C to E, allowing us to understand and apply the distance formula effectively.