Problem 8 - Precalculus: Mathematics for Calculus

Question

Yes or No? If No , give a reason. Assume that \(a\) and \(b\) are nonzero real numbers. (a) Is the distance between any two different real numbers always positive? (b) Is the distance between \(a\) and \(b\) the same as the distance between \(b\) and \(a\) ?

Step-by-Step Solution

Answer

(a) Yes. (b) Yes.

Step 1: Understanding the Concept of Distance

The distance between two real numbers, say \(a\) and \(b\), is defined as \(|a-b|\). This is the absolute value of the difference between \(a\) and \(b\), which always yields a non-negative result. Let's analyze the given parts of the exercise based on this information.

Step 2: Analyzing Part (a)

Part (a) asks whether the distance between any two different real numbers is always positive. Since \(a\) and \(b\) are nonzero and different, \(|a-b|\) will not be zero. Indeed, the absolute difference of two different real numbers is always positive.

Step 3: Conclusion for Part (a)

For part (a), we conclude that the statement is correct. Thus, the answer for part (a) is 'Yes', the distance between any two different real numbers is always positive.

Step 4: Analyzing Part (b)

Part (b) asks if the distance between \(a\) and \(b\) is the same as the distance between \(b\) and \(a\). Using the formula for distance, \(|a-b| = |b-a|\), because both expressions represent the absolute difference between \(a\) and \(b\). Absolute values negate the direction, so they are equal.

Step 5: Conclusion for Part (b)

For part (b), we conclude that the statement is correct as well. Therefore, the answer for part (b) is 'Yes', the distance between \(a\) and \(b\) is the same as the distance between \(b\) and \(a\).

Key Concepts

Absolute Value

Absolute value is a crucial concept in understanding the distance between real numbers. It represents the non-negative magnitude of any real number, without considering its sign. This is formatted as

• \(|x|\) if the number \(x\) is positive or zero, is equal to \(x\)
• \(|x|\) if the number \(x\) is negative, is equal to \(-x\)

Thus, absolute value ensures that all distances are non-negative.
The absolute value function transforms any number into its distance from zero on the number line.
For example, the absolute value of

• -3 is 3
• 5 is 5

Absolute value is used to define the distance between two real numbers, such as

• \(|a-b|\)

This guarantees the distance is always non-negative, a key property in real analysis.

Nonzero Real Numbers

Real numbers represent all possible numbers on the number line, including both positive and negative numbers, and zero. When we talk about nonzero real numbers, we refer to all real numbers except zero.
In definitions of distance like

• \(|a-b|\)

using nonzero real numbers ensures that

• \(a eq b\)

When

• \(a eq b\)

\(|a-b|\) will never be zero.
Since the set excludes the only number that doesn't have an inverse (zero), it helps when calculating values like distances. Nonzero real numbers are important in mathematics because they avoid cases where divisions or operations might be undefined if zero were involved.
This property guarantees that different numbers genuinely have a measurable, positive distance between them, confirming that distances are positive.

Properties of Distance

The concept of distance relies on several fundamental properties that help in the uniform measurement on a number line.
When measuring the distance between two points \(a\) and \(b\), we use the absolute value

• \(|a-b|\)

Some essential properties of distance between real numbers include:

• Non-negative Property: Distance is never negative; it's either positive or zero (only if the points coincide).
• Symmetric Property: Distance between \(a\) and \(b\) is the same as the distance between \(b\) and \(a\), mathematically \(|a-b| = |b-a|\).
• Identity of Indiscernibles: Distance between two numbers is zero only if they are identical, i.e., \(a = b\).

These properties ensure a consistent and logical framework for measuring how far apart two numbers are.
The symmetric property is particularly interesting in showing why distance is the same regardless of direction, highlighting that what matters is the magnitude, not the order of subtraction.