Problem 72
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
Step-by-Step Solution
Verified Answer
The arithmetic mean \(\frac{a+b}{2}\) is between \(a\) and \(b\) if \(a
1Step 1: 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 \(b\).
2Step 2: 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.
3Step 3: 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.
4Step 4: 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 ProofMathematical ReasoningAlgebraic Manipulation
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 \(b\).
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 \(b\).
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.
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 \(a< b\).
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:
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.
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.
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.
Other exercises in this chapter
Problem 71
Which of the following are true? Assume that \(x\) and \(y\) are real numbers. (a) For every \(x, x>0 \Rightarrow x^{2}>0\). (b) For every \(x, x>0 \Leftrightar
View solution Problem 72
Suppose that \((a, b)\) is on the circle \(x^{2}+y^{2}=r^{2}\). Show that the line \(a x+b y=r^{2}\) is tangent to the circle at \((a, b)\).
View solution Problem 72
Which of the following are true? Unless it is stated otherwise, assume that \(x, y\), and \(\varepsilon\) are real numbers. (a) For every \(x, x0\), there exist
View solution Problem 73
The number \(\sqrt{a b}\) is called the geometric mean of two positive numbers \(a\) and \(b\). Prove that $$ 0
View solution