Question

The number \(\sqrt{a b}\) is called the geometric mean of two positive numbers \(a\) and \(b\). Prove that $$ 0

Short answer

\( a < \sqrt{ab} < b \) is true if \( 0 < a < b \).

Step-by-step solution

Understand the Problem

We are given two positive numbers \( a \) and \( b \) such that \( 0 < a < b \). We need to prove that \( \sqrt{ab} \), the geometric mean of \( a \) and \( b \), lies between \( a \) and \( b \).

Prove \( a < \sqrt{ab} \)

Start by squaring both sides of the inequality \( a < \sqrt{ab} \). This yields \( a^2 < ab \). Since \( b > a \), divide both sides by \( a \) (which is positive) to get \( a < b \), which is true as given, so \( a < \sqrt{ab} \) is proven.

Prove \( \sqrt{ab} < b \)

Now, square both sides of the inequality \( \sqrt{ab} < b \). This gives \( ab < b^2 \). Divide both sides by \( b \) (which is positive) to have \( a < b \), which is true as given, thus \( \sqrt{ab} < b \) is proven.

Conclude the Proof

Having demonstrated both \( a < \sqrt{ab} \) and \( \sqrt{ab} < b \), we conclude that \( a < \sqrt{ab} < b \) holds true under the condition \( 0 < a < b \). The statement is thus proven.

Key concepts

Inequalities

Inequalities involve expressions that indicate one quantity is less than or greater than another. In the context of this problem, we are given the inequality \( 0 < a < b \) along with the task of proving \( a < \sqrt{ab} < b \). Understanding inequalities is essential for solving such problems, as it involves logical reasoning to demonstrate that certain conditions are met.
Inequalities are used to describe the range of values, and they have specific rules:

• If a number \( x \) is less than a number \( y \), this is written as \( x < y \).
• If a number is less than or equal to another, it is denoted by \( \leq \).
• When dealing with inequalities, multiplying or dividing by a negative number reverses the inequality sign.
• The transitive property is highly useful: if \( a < b \) and \( b < c \), then \( a < c \).

In our exercise, these inequality rules help us identify the correct positions of \( a \), \( \sqrt{ab} \), and \( b \) on the number line, ensuring that \( \sqrt{ab} \) is indeed between \( a \) and \( b \). This is achieved by transforming and solving the inequalities step by step, as evidenced through squaring and simplifying mathematical expressions.

Proofs in Mathematics

Mathematical proofs are logical arguments that establish the truth of a mathematical statement, using a series of logical deductions. To prove \( a < \sqrt{ab} < b \), each part of inequality has been proven separately, resulting in a thorough conclusion.
Proofs typically involve:

• Assumptions: Start by understanding given conditions and assumptions. Here, we assumed \( 0 < a < b \).
• Logical reasoning: Every step must follow logically without any leaps. We logically deduced \( a < \sqrt{ab} \) and \( \sqrt{ab} < b \), separately.
• Concluding: After proving both, putting them together concludes the proof \( a < \sqrt{ab} < b \).

The value of mathematical proof lies in its power to validate hypotheses and theorems with certainty, abolishing doubts. By demonstrating each required truth individually and combining them logically, a complete and reliable demonstration of the statement is accomplished.

Square Root

The square root is a fundamental concept in mathematics, describing the value that, when multiplied by itself, gives the original number. In this problem, the square root is applied to the product of two numbers, \( a \) and \( b \), defining their geometric mean as \( \sqrt{ab} \).
Here are key aspects of the square root you need to know:

• Denoted as \( \sqrt{x} \) for a number \( x \), it implies the number \( y \) such that \( y^2 = x \).
• Square roots only apply inherently to nonnegative numbers for real numbers. Hence, \( a \) and \( b \) must be positive in this scenario.
• Knowing properties of square roots, such as \( \sqrt{xy} = \sqrt{x} \cdot \sqrt{y} \) and \( \sqrt{x^2} = |x| \), helps discern how roots behave in equations.
• A square root operation relates to many real-world contexts, like geometry, where it references dimensions of squares.

In showing that \( \sqrt{ab} \) falls between \( a \) and \( b \), the square root serves not merely as a calculation, but also as a conceptual bridge linking arithmetic to geometry, enhancing comprehension of numerical relationships.