Problem 34 - Fundamentals of Mathematics

Question

For the following 10 problems, on the number line, how many units are there between the given pair of numbers? -6 and 2

Step-by-Step Solution

Answer

There are 8 units between -6 and 2.

Step 1: Identify the Given Numbers

The problem provides two numbers: -6 and 2. We need to find the distance between these two numbers on a number line.

Step 2: Calculate the Distance

On a number line, the distance between two points is the absolute value of the difference between the numbers. To find this difference, subtract the smaller number from the larger number. Calculate: \[ 2 - (-6) = 2 + 6 \] The result of this calculation is 8.

Step 3: Apply Absolute Value

Since distance cannot be negative, we take the absolute value of the result. However, in this case, the calculated difference is already positive: 8.

Key Concepts

Absolute Value

Absolute value is a crucial concept in mathematics that helps us find the distance between numbers, regardless of their direction. Imagine standing on a number line at any point, whether it's a positive position or a negative one. Absolute value transforms your number into a positive distance from zero. It is symbolized by vertical bars, like this: \(|x|\). So, for any number \(x\), the absolute value \(|x|\) is simply the non-negative amount that number is away from zero.
For example:

• The absolute value of 5 is \(|5| = 5\)
• The absolute value of -5 is \(|-5| = 5\)

When calculating distance on the number line, absolute value ensures that our result is always positive. No matter the order of subtraction, taking the absolute value of the difference gives us the correct distance. This property of always being positive makes it very useful when discussing distances.

Subtracting Integers

Subtracting integers is all about understanding how the number line works moving in different directions. To subtract one integer from another, you take the second number and reverse its sign before adding. This process can be remembered by the phrase "add the opposite." For instance, if you need to solve \(a - b\), it’s the same as \(a + (-b)\).
Consider this example:

• If we have to subtract -6 from 2, we can write it as: \(2 - (-6)\). This becomes \(2 + 6\).
• Another example, subtracting 3 from 5: \(5 - 3\) or \(5 + (-3)\).

This method simplifies particle interactions on the number line, allowing clear visualization of how differences between points are represented. Always remember that subtracting a negative is just like adding a positive, a perfect tool that helps when determining distances.

Number Line

A number line is a visual representation that assists in understanding numbers and their relationships. It features a horizontal line with numbers placed at intervals, ranging from negatives on the left to positives on the right. Zero is commonly placed in the center.
Using a number line is highly effective for identifying the distance between two points. If you want to determine how far -6 is from 2, locate each point on the line. The distance is simply how many units you move from one point to the other without considering direction, which is where absolute value plays a role.
Steps on a number line can be seen as counting units from one point to another. For instance:

• To find how far -6 is from 2, start at -6 and count units moving towards 2, across the zero marker, resulting in a total of 8.

This straightforward model of addition and subtraction results is especially useful for students to grasp concepts easily using actual visuals.