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Chapter 1: Problem 26

The quadratic mean of two real numbers \(x\) and \(y\) equals \(\sqrt{\left(x^{2}+y^{2}\right) / 2}\) . By computing the arithmetic and quadratic means of different pairs of positive real numbers, formulate a conjecture about their relative sizes and prove your conjecture.

Short Answer

Expert verified
The quadratic mean is always greater than or equal to the arithmetic mean.

Step by step solution

01

- Compute Arithmetic Mean

The arithmetic mean (AM) of two real numbers, x and y, is given by: \[ \text{AM} = \frac{x + y}{2} \]
02

- Compute Quadratic Mean

The quadratic mean (QM) of the two numbers is given by: \[ \text{QM} = \sqrt{\frac{x^2 + y^2}{2}} \]
03

- Compare Arithmetic and Quadratic Means

To formulate a conjecture, compare the computed AM and QM for several pairs of positive real numbers. For example:- For \(x = 1\) and \(y = 3\):\[ \text{AM} = \frac{1 + 3}{2} = 2 \]\[ \text{QM} = \sqrt{\frac{1^2 + 3^2}{2}} = \sqrt{5} \approx 2.24 \]- For \(x = 2\) and \(y = 4\):\[ \text{AM} = \frac{2 + 4}{2} = 3 \]\[ \text{QM} = \sqrt{\frac{2^2 + 4^2}{2}} = \sqrt{10} \approx 3.16 \]We can see from these examples that the quadratic mean is always greater than or equal to the arithmetic mean.
04

- Prove the Conjecture

To prove this conjecture mathematically, we need to show that for any positive real numbers \(x\) and \(y\), \( \text{QM} \geq \text{AM} \).Start with the inequality:\[ \left( \frac{x + y}{2} \right)^2 \leq \frac{x^2 + y^2}{2} \]Simplify and expand:\[ \frac{(x + y)^2}{4} \leq \frac{x^2 + y^2}{2} \]\[ \frac{x^2 + 2xy + y^2}{4} \leq \frac{x^2 + y^2}{2} \]Multiply both sides by 4:\[ x^2 + 2xy + y^2 \leq 2x^2 + 2y^2 \]Rearrange terms:\[ 0 \leq x^2 - 2xy + y^2 \]\[ 0 \leq (x - y)^2 \]Since \( (x - y)^2 \geq 0 \) for all real numbers \(x\) and \(y\), the inequality holds, thus proving that the quadratic mean is always greater than or equal to the arithmetic mean.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic Mean
The arithmetic mean (AM) is one of the most commonly used measures of central tendency. It is the average of a set of values, calculated by summing all the values and then dividing by the number of values. For two real numbers, \( x \) and \( y \), the arithmetic mean is computed as follows:

\[ \text{AM} = \frac{x + y}{2} \]
For example, if \( x = 1 \) and \( y = 3 \), the arithmetic mean would be
\( \text{AM} = \frac{1 + 3}{2} = 2 \).

This measure gives us a simple representation of the central value of \( x \) and \( y \). Understanding AM is important as it is widely used in various fields such as statistics, economics, and everyday problem solving. It allows us to compare different sets of numbers in an effective way.
Quadratic Mean
The quadratic mean (QM), also known as the root mean square (RMS), is another way to find the average of a set of values. Unlike the arithmetic mean, the quadratic mean gives more weight to larger values. For two real numbers, \( x \) and \( y \), the quadratic mean is given by the formula:

\[ \text{QM} = \frac{x^2 + y^2}{2} \]
Taking the square root of this result gives:
\[ \text{QM} = \frac{\text{sqrt}(x^2 + y^2)}{2} \]
For instance, if \( x = 2 \) and \( y = 4 \), the quadratic mean would be:
\( \text{QM} = \frac{2^2 + 4^2}{2} = \text{sqrt}( 10 \,) \approx 3.16 \).

The quadratic mean is particularly useful in scenarios where we want to measure the magnitude of a set of values, such as in engineering or physics. Because it squares the values before averaging and then takes the square root, it emphasizes larger deviations more than the arithmetic mean does.
Inequalities
Inequalities are mathematical expressions involving the symbols \(<\), \(\leq\), \(>\), and \(\geq\). These symbols indicate that one quantity is less than, less than or equal to, greater than, or greater than or equal to another quantity, respectively. In the context of comparing quadratic mean and arithmetic mean, the key inequality we focus on is:

\[ \text{QM}\text{ (Quadratic Mean)} \geq \text{AM} \text{(Arithmetic Mean)} \]

To understand and prove this inequality, consider the steps:
1. Start with the general form of the arithmetic and quadratic means.
\( \text{AM} = \frac{x + y}{2} \)
\( \text{QM} = \frac{\text{sqrt}(x^2 + y^2)}{2} \)
2. Express the inequality to be proven:
\( \frac{x + y}{2} \leq \frac{x^2 + y^2}{2} \).
3. By simplifying and rearranging both sides, you can show that the inequality \(0 \text{<=}\ (x-y)^2\). Since the square of a real number is always non-negative, it follows that the quadratic mean is always greater than or equal to the arithmetic mean.

Understanding inequalities is crucial in various mathematical contexts, as they help establish relationships and compare different quantities.

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Most popular questions from this chapter

Express each of these system specifications using predicates, quantifiers, and logical connectives, if necessary. a) At least one console must be accessible during every fault condition. b) The e-mail address of every user can be retrieved whenever the archive contains at least one message sent by every user on the system. c) For every security breach there is at least one mechanism that can detect that breach if and only if there is a process that has not been compromised. d) There are at least two paths connecting every two distinct endpoints on the network. e) No one knows the password of every user on the system except for the system administrator, who knows all passwords.

Express the negations of these propositions using quantifiers, and in English. a) Every student in this class likes mathematics. b) There is a student in this class who has never seen a computer. c) There is a student in this class who has taken every mathematics course offered at this school. d) There is a student in this class who has been in at least one room of every building on campus.

Exercises \(61-64\) are based on questions found in the book Symbolic Logic by Lewis Carroll. Let P(x), Q(x), and R(x) be the statements 鈥渪 is a clear explanation,鈥� 鈥渪 is satisfactory,鈥� and 鈥渪 is an excuse,鈥� respectively. Suppose that the domain for x consists of all English text. Express each of these statements using quantifiers, logical connectives, and P(x), Q(x), and R(x). a) All clear explanations are satisfactory. b) Some excuses are unsatisfactory. c) Some excuses are not clear explanations. d) Does (c) follow from (a) and (b)?

Prove that if \(m\) and \(n\) are integers and \(m n\) is even, then \(m\) is even or \(n\) is even.

Prove that \(m^{2}=n^{2}\) if and only if \(m=n\) or \(m=-n\)
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