Problem 55 - Calculus Single Variable

Question

Prove that there is no smallest positive real number.

Step-by-Step Solution

Answer

There is no smallest positive real number because for any positive number \( x \), \( \frac{x}{2} \) is a smaller positive number.

Step 1: Understanding the Problem

We need to show that for any positive real number \( x \), there exists another positive real number \( y \) such that \( 0 < y < x \). This will prove that there is no smallest positive real number.

Step 2: Choose a Number Less than a Given Positive Number

Given any positive real number \( x \), consider the number \( y = \frac{x}{2} \). This number is clearly positive because \( x > 0 \), and dividing a positive number by a positive constant (2 in this case) results in another positive number.

Step 3: Verify the Chosen Number is Smaller

Now, we compare \( y \) and \( x \). Since \( y = \frac{x}{2} \), clearly \( y < x \) because dividing \( x \) by 2 reduces its magnitude by half. Therefore, \( 0 < y < x \).

Step 4: Conclude There is no Smallest Positive Real Number

By demonstrating that we can always find a number \( y \) such that \( 0 < y < x \) for any chosen positive real number \( x \), we conclude that there is no smallest positive real number, as any candidate can be reduced further by selecting \( y \).

Key Concepts

Positive Real Numbers

Positive real numbers are those numbers that are greater than zero and do not include any imaginary components. Think of numbers like 0.5, 5, or 500. These numbers exist on the right side of the number line.
Understanding positive real numbers is crucial because they form the basis of many mathematical concepts, including measurements in real life such as length, area, and even time.
In our exercise, we're interested in demonstrating whether there is a smallest positive real number, which requires an understanding of how these numbers work and their properties.

No Smallest Number

One of the fascinating things about positive real numbers is the absence of a smallest member in their set. In mathematics, we state that there is no smallest positive real number.
This is because, for any positive real number you pick, another positive number exists that is smaller.
To illustrate, consider a number such as 0.1. You might think it is tiny, but you can easily find something smaller, like 0.01 or even 0.001.
Mathematically, this concept is important as it allows us to understand that numbers are infinite and there's always a smaller number, reinforcing that there is no absolute minimum.

Proof Techniques

Proof techniques are vital tools in mathematics. They allow us to establish the truth or validity of mathematical statements.
In the given exercise, to prove that there is no smallest positive real number, we used a proof by contradiction. Here, we start with the assumption that there is a smallest positive number, and then show that this assumption leads to a contradiction.
By dividing this supposed smallest number by 2, we find a smaller number, contradicting our assumption.
Furthermore, understanding proof techniques helps in solving more complex mathematical problems. It involves logical reasoning, which is valuable in many areas beyond mathematics.

Division by Constants

Division by constants is a simple yet powerful mathematical operation. It means dividing one number by a fixed, non-zero number known as a constant.
This concept is used in our proof to find a smaller positive number. By dividing a given positive real number by a constant like 2, we end up with another positive real number that is smaller than the original.
This technique is useful because it maintains the property of positivity in the number we're working with. Always remember that the divisor must be positive and non-zero to ensure the outcome of division remains valid.
In the context of the exercise, division by a constant helps illustrate the concept of finding numbers indefinitely smaller than any chosen positive number, hence proving there is no smallest positive real number.