Problem 18 - Fundamentals of Mathematics

Question

How many units are there between the given pair of numbers? -7 and -4

Step-by-Step Solution

Answer

The distance between -7 and -4 is 3 units.

Step 1: Understand the Concept of Distance Between Numbers

To find the number of units between two numbers on a number line, think about the distance between them without considering the direction. You can find the distance by subtracting one number from the other and taking the absolute value of the result.

Step 2: Subtract the Smaller Number from the Larger Number

Identify which of the two numbers is larger. In this case, -4 is larger than -7. Compute the difference by subtracting -7 from -4: \[-4 - (-7) = -4 + 7 = 3\].

Step 3: Take the Absolute Value of the Difference

After calculating the difference, you take the absolute value to ensure the distance is non-negative: Since the result from Step 2 is already positive, the absolute value of 3 is simply 3.

Key Concepts

Absolute Value

When we talk about absolute value, we're referring to the distance a number is from zero on the number line. It's always a non-negative number because distance can't be negative.
The absolute value is denoted by two vertical lines around the number, like this: \(|x|\).
For example, the absolute value of both 4 and -4 is 4, written as \(|4| = 4\) and \(|-4| = 4\).
- The absolute value of zero is zero: \(|0| = 0\). The purpose of using absolute value in our problem is to ensure the distance between two points is always positive.
So when we calculate the distance between -7 and -4, it doesn't matter which number we subtract from which, since taking the absolute value makes the result positive.

Number Line

The number line is a visual representation of numbers laid out in a straight line. It helps us see the relative positions of numbers and understand their distances from each other.
When you look at a number line, you can see that numbers grow larger as you move to the right and smaller as you move to the left.
Here's how to use the number line in our exercise:

• Locate both -7 and -4 on the number line.
• You can count the units between -7 and -4 moving right, which naturally shows a distance of 3 units.

Using the number line is a great way to intuitively find distances between numbers, reinforcing what we find using calculations.

Subtraction

Subtraction is the process of finding the difference between two numbers. It's like asking, "How much more do I need to go from one number to another?"
In terms of our exercise, subtraction helps us calculate the distance between two numbers on a number line.
When subtracting:

• Identify the larger number and subtract the smaller one from it to find the difference.
• Iin the specific example, you subtract -7 from -4: \[-4 - (-7) = -4 + 7 = 3\]
• This result shows the distance between the two numbers before applying the absolute value.

Remember, subtraction often works hand-in-hand with absolute value to help us ensure the result (the distance) is always non-negative.
Using subtraction efficiently makes math problems like this one even more approachable!