Question

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). There exists a positive real number \(x\) for which \(x^{2}<\sqrt{x}\).

Short answer

The statement is true. For example, for \(x = 0.5\), \(x^{2} = 0.25\) is less than \(\sqrt{x} = 0.7071\). It is important to note that this is a specific instance where the statement holds true. However, it would not hold true for all positive real numbers.

Step-by-step solution

Understand the Problem

The problem is stating that there is a positive real number that when squared is less than its square root. This means that if we denote this positive real number by \(x\), then \(x^{2} < \sqrt{x}\). Our task is to prove this statement.

Assume a positive real number

Consider a positive real number, \(x\). For small enough values of \(x\) (typically between 0 and 1), squaring the number actually makes it smaller, while the square root makes it larger. This is because the square of a number less than one is less than the number itself, while the square root of the same number is larger.

Test our Assumption

We can test our hypothesis by plugging values. Let's say we designate \(x = 0.5 \rightarrow x^{2} = (0.5)^{2} = 0.25, \sqrt{x} = \sqrt{0.5} \approx 0.7071\). Here, \(x^{2} = 0.25 < \sqrt{x} = 0.7071 \). Thus, for \(x = 0.5\) indeed \(x^{2} < \sqrt{x}\). This effectively proves the original proposition, demonstrating that there exists a positive real number \(x\) for which \(x^{2}<\sqrt{x}\). However remember, this does not constitute a proof for all possible positive real numbers \(x\), it only shows a particular example where it is true.

Key concepts

Proof Techniques

When faced with mathematical assertions like the one in our exercise, it's essential to employ the correct proof techniques to establish their validity. In mathematics, a proof is a logical argument that demonstrates the truth of a statement. There are various methods through which a statement can be proven, including direct proof, proof by contradiction, and proof by counterexample, among others.

For the given exercise, a technique that comes to our aid is testing specific instances or examples, which, while not a proof method in itself, can often lead to insights on how to approach a full proof. In more complex cases, one might need to construct an argument using algebraic manipulations, consider limiting behavior, or apply theorems that have already been established. The crucial aspect is understanding the nature of the problem to choose an appropriate technique to tackle the proof.

Real Numbers

The term real numbers refers to both rational and irrational numbers, which can be represented on the number line. They are the building blocks of calculus and many other mathematical concepts. In the context of the exercise, when we discuss a positive real number, we are specifically talking about real numbers greater than zero.

Characteristics of Positive Real Numbers

• A positive real number multiplied by itself (squared) will always yield a positive result.
• The square of a number less than one is smaller than the original number.
• The square root of a positive number is the positive number that, when squared, will give the original number as the result.

These properties of positive real numbers are crucial for understanding inequalities involving squares and square roots, as we see in our exercise.

Inequalities

An inequality tells us how one value is lesser or greater than another value. Inequalities are fundamental in mathematical discussions about ranges, estimates, and when exact values are not required. The exercise poses an inequality involving squares and square roots.

To understand inequalities, it's important to note that they maintain their truth under certain operations, such as adding the same number to both sides or multiplying by a positive number. However, the inequality reverses when we multiply or divide by a negative number. Here's a critical insight: for positive numbers less than one, squaring them makes them smaller, which is the basis for our inequality statement that a positive real number can be found such that squaring it results in a number smaller than its square root.