Problem 20 - Fundamentals of Mathematics

Question

How many units are there between the given pair of numbers? -1 and 1

Step-by-Step Solution

Answer

There are 2 units between -1 and 1.

Step 1: Identify the Numbers

First, recognize the two numbers involved. In this case, they are -1 and 1.

Step 2: Apply the Distance Formula

Calculate the distance between two numbers on a number line using the formula: distance = |a - b| or |b - a|. Here, choose a = -1 and b = 1, thus the formula becomes |1 - (-1)|.

Step 3: Calculate the Absolute Value

Compute the value inside the absolute value: 1 - (-1) = 1 + 1 = 2. The distance, or the absolute value of 2, is still 2.

Key Concepts

Number Line

When we talk about a number line, we mean a simple straight line where numbers are placed at equal intervals. The center of this line is usually zero, and numbers extend infinitely in both positive and negative directions. Each point on this line corresponds to a number, which makes it an excellent tool for visually understanding concepts like the distance between numbers.

• Positive numbers are found to the right of zero.
• Negative numbers are located to the left of zero.

To find the distance between any two numbers on a number line, imagine hopping from one number to the other. The number of hops you take corresponds to the distance between those numbers. In the problem with -1 and 1, if you start at -1 and hop three times to the right, you first reach 0 and then end at 1, making the total distance 2 units.

Absolute Value

Absolute value is a concept that refers to the distance of a number from zero on the number line, without considering direction. It is always a non-negative value, since distance can't be negative. We denote absolute value using two vertical lines, for example, the absolute value of a number \( a \) is written as \(|a|\).

So, when we calculate the distance between two numbers like -1 and 1, we use their absolute values to determine how far apart they are without worrying about their sign.

• For example, the absolute value of -1 is \(|-1| = 1\).
• The absolute value of 1 is \(|1| = 1\).

If you want to find the distance between -1 and 1, you apply the concept of absolute value to the operation. It involves finding the absolute value of their difference, \(|1 - (-1)|\). In this case, it simplifies to \(2\), which tells you that these two numbers are 2 units apart on the number line.

Distance Formula

The distance formula is a mathematical method used to find the space between two points, which in the context of a number line, can easily be applied to calculate how far apart two numbers are. This formula is: \[ \text{distance} = |a - b| \] Where \(a\) and \(b\) are the two numbers you're considering. This formula essentially gives you the absolute value of the difference between those two numbers, effectively removing any negative sign while providing the distance.

Applying the formula: \( |1 - (-1)| = |2| = 2\).

• You subtract the smaller number from the larger one, or vice versa, as the order doesn't matter due to the absolute value.
• Use the absolute value to ensure the answer is positive, representing the physical distance between the points on a number line.

This straightforward formula is a key tool in various mathematical applications, helping turn complex distance calculations into simple arithmetic tasks.