Question

Prove that \(|x|<|y| \Leftrightarrow x^{2}

Short answer

Yes, the proof shows that \\(|x|<|y|\\) if and only if \\(x^{2}<y^{2}\\).

Step-by-step solution

Establish Direction for |x|

Start with the assumption \(|x| < |y|\). By definition of absolute value, this implies \(|x|\cdot|x| \leq |x|\cdot|y|\). This inequality is strict since multiplying a smaller positive number by a larger positive number keeps the inequality's direction the same.

Apply Squaring to Both Sides

From \(|x|\cdot|x| < |y|\cdot|y|\), we have \(|x|^2 < |y|^2\). Since \(x^2=|x|^2\) and \(y^2=|y|^2\) for any real numbers \(x\) and \(y\), it follows directly that \(x^2 < y^2\).

Establish Direction for x²

Starting with \(x^2 < y^2\), it follows that \(|x|^2 < |y|^2\). Next, since the absolute value squared gives us positive real numbers, we can write \(|x|^2 - |y|^2 < 0\).

Factor the Difference of Squares

The inequality \(|x|^2 - |y|^2 < 0\) can be rewritten using the difference of squares formula: \((|x|-|y|)(|x|+|y|) < 0\).

Analyze the Product

For a product of two terms to be negative, it requires either \(|x| - |y| < 0\) while \(|x| + |y| > 0\) or vice versa. Since both absolute values are non-negative, \(|x| + |y| > 0\) is clearly satisfied. Therefore, the negative product implies \(|x| - |y| < 0\), which means \(|x| < |y|\).

Key concepts

Absolute Value

The absolute value of a number is a way of describing the distance of that number from zero on the number line without considering direction. When we talk about \(|x|\), we're simply indicating how far x is from zero. This means \(|x|\) is always a non-negative number.
Applying absolute value in inequalities provides a unique property: \(|x| < |y|\) implies that the magnitude or size of x is smaller than that of y, irrespective of whether x and y are positive or negative.

For the exercise, starting with \(|x| < |y|\), we squared both sides to show that \(|x|^2 < |y|^2\). Because square of a number is the same as the square of its absolute value, this translates directly to \(x^2 < y^2\). Thus, understanding absolute value is crucial in proving inequalities involving real numbers.

Square of a Number

Squaring a number means multiplying the number by itself, resulting in a non-negative product, irrespective of the sign of the number. For any real number x, the square is denoted as \(x^2\).

This concept helps in providing structure to many mathematical proofs as it forces all real numbers into the non-negative subset. In the given problem, squaring was a pivotal step because it allowed us to transition expressions with absolute values like \(|x|\) to more manipulable algebraic standards like \(x^2\). This step is valid because of the identity \(x^2 = |x|^2\), letting us transform the given conditions into a form we can easily analyze and prove.

Difference of Squares

The difference of squares is a crucial algebraic identity in mathematics, represented as \((a^2 - b^2) = (a-b)(a+b)\). This formula is extremely useful for factoring expressions where two square terms are involved.

In the exercise, this concept was applied to express \(|x|^2 - |y|^2 < 0\) as \((|x|-|y|)(|x|+|y|) < 0\). This transformation clarified the reasoning about the inequality of squares by breaking it down into a product involving sub-expressions related to their absolute values, making the inequality more straightforward to handle and analyze.

Proof Techniques

Proof techniques are methods used to establish the validity of a statement. In this exercise, we use direct proof, a common method in mathematics that involves a straightforward argument from assumptions to conclusion.

Start by assuming the inequality \(|x| < |y|\) or \(x^2 < y^2\) depending on the direction of the proof. Through logical reasoning like squaring, factoring, and invoking algebraic identities like the difference of squares, the proof progresses to show that the assumption leads to the conclusion robustly.

With this technique, it's essential to maintain logical coherence and consider all possible implications of the assumptions until the final conclusion is reached, ensuring clarity and precision in proving statements about inequalities among real numbers.