Question

The number \(\frac{1}{2}(a+b)\) is called the average, or arithmetic mean, of \(a\) and \(b\). Show that the arithmetic mean of two numbers is between the two numbers; that is, prove that $$ a

Short answer

The arithmetic mean \(\frac{a+b}{2}\) is between \(a\) and \(b\) if \(a<b\).

Step-by-step solution

Understand the inequality to prove

We need to show that if \(a < b\), then \(a < \frac{a+b}{2} < b\). This means the average of \(a\) and \(b\) is greater than \(a\) and less than \(b\).

Prove that \(a < \frac{a+b}{2}\)

Start by expanding the inequality: \\[ a < \frac{a+b}{2} \\] By multiplying both sides by 2 to eliminate the fraction, we get: \\[ 2a < a + b \\] Which simplifies to: \\[ a < b \\] Since \(a < b\) is given, this part of the inequality holds true.

Prove that \(\frac{a+b}{2} < b\)

Now consider the inequality: \\[ \frac{a+b}{2} < b \\] Multiply both sides by 2: \\[ a + b < 2b \\] Simplifying gives: \\[ a < b \\] Again, this is given, so this part of the inequality is also true.

Conclusion: Combine the results

Since both parts of the original inequality \(a < \frac{a+b}{2}\) and \(\frac{a+b}{2} < b\) are proven under the assumption \(a < b\), it follows that \(a < \frac{a+b}{2} < b\). Thus, the arithmetic mean of two numbers is indeed between those two numbers.

Key concepts

Inequality Proof

An inequality proof is a way of demonstrating that one quantity is consistently greater than or less than another. In this exercise, we aim to prove that the arithmetic mean, or average, of two numbers, lies between them. This requires a clear step-by-step approach.

Let's break down the proof for the statement: \(a < b \Rightarrow a < \frac{a+b}{2} < b\). This tells us that when one number \(a\) is less than another number \(b\), the midpoint, or average \(\frac{a+b}{2}\), should be greater than \(a\) and less than \(b\).

• The first step involves understanding the expression \(a < \frac{a+b}{2}\); this means confirming that the average is greater than \(a\).
• The second step is validating \(\frac{a+b}{2} < b\), showing that the average is less than \(b\).

By proving both conditions under the assumption that \(a < b\), we can confidently conclude that the arithmetic mean is sandwiched between \(a\) and \(b\). This type of proof is fundamental in mathematics, aiding us in understanding and demonstrating the behavior of numbers consistently.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thought to understand and explain mathematical concepts. In this context, it helps us systematically prove inequalities by working through logical steps.

For this exercise, we start with a given, \(a < b\), which is our starting point for reasoning. We then apply logical steps to transform the inequality into a form that is easier to analyze.

• First, to show \(a < \frac{a+b}{2}\), we simplify the inequality by eliminating the fraction using basic algebra.
• This involves multiplying both sides by 2, resulting in \(2a < a + b\), simplifying to \(a < b\), which matches our given condition.
• We apply similar reasoning for the statement \(\frac{a+b}{2} < b\), again eliminating the fraction to reach \(a< b\).

By logically discussing and breaking down these inequalities, mathematical reasoning helps us navigate from the assumptions to the conclusion effectively.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. In this exercise, it is primarily used to handle the inequalities involving fractions.

Simplifying Inequalities: Eliminate Fractions
Both parts of our inequality involve dividing by 2, so we multiply every term by 2 to get rid of the fractions. This step is crucial as it simplifies the problem to its core components:

• For \(a < \frac{a+b}{2}\), multiplying by 2 gives \(2a < a + b\).
• Simplification then results in \(a < b\), which confirms part of the proof.

Working with the Other Side
Similarly, the part \(\frac{a+b}{2} < b\) is simplified through multiplication by 2, leading to \(a + b < 2b\). Again, after simplifying, we arrive at \(a < b\).

Algebraic manipulation is a fundamental skill that enables mathematicians to transform complex expressions into simpler forms, revealing the information needed to solve problems or prove statements.