Question

If \(x<0\), which of the following is true? \(O x^2\) is positive. \(x^2\) is negative. \(x^2\) could be either positive or negative. I don't know.

Short answer

The statement ' \(x^2\) is positive' is correct when \(x<0\).

Step-by-step solution

Understand the property of squaring

Consider the key property of squaring: Any real number, when squared, becomes a positive number. This is because a negative number times a negative number equals a positive number.

Apply the property to our case

Given that \(x\) is less than zero, it means x is a negative number. When we square \(x\) (or \(x^2\)), we multiply \(x\) by itself. Since a negative number times a negative number yields a positive number, \(x^2\) will always be positive.

Analyze the options

Now, we examine the provided statements. The first one \(x^2\) is positive, aligns with our solution. The remaining options suggest that \(x^2\) could be negative or has an uncertain value. From our understanding of the squaring operation, we know that these options are not valid.

Key concepts

Understanding Negative Numbers

Negative numbers might seem intimidating at first, but they simply represent values less than zero. These numbers are positioned on a number line to the left of zero. Negative numbers are crucial in mathematics because they allow us to express values below zero, such as temperature, depth, or financial loss. In calculations, handling negative numbers follows specific rules:

• Adding a negative number is the same as subtracting its positive counterpart.
• Subtracting a negative number is equivalent to adding its positive counterpart.
• Multiplying or dividing two negative numbers results in a positive number.
• Multiplying or dividing a negative and a positive number results in a negative number.

These rules ensure that we can work with negative numbers as effectively as positive ones.

Principles of Squaring Numbers

Squaring a number means multiplying that number by itself. The result, or square, of any real number is always non-negative. This includes both positive numbers and negative numbers. Let's dive into what happens:

• For a positive number, squaring it yields another positive number.
• A negative number squared will also be positive because two negative factors result in a positive product.

For example, if we take a negative number like a hypothetical digit \(-2\) and square it, which means doing \(-2 \times -2\) the result is \(4\). Thus, when you square any number, you can be confident of achieving a non-negative outcome.

What Are Real Numbers?

Real numbers make up a comprehensive and fundamental part of mathematics. They include all the numbers you're probably familiar with: positive integers, negatives, fractions, and irrational numbers (like √2 or π). Think of the number line as endlessly stretching in both precise directions, capturing every conceivable number between. Real numbers can be:

• Rational numbers: Numbers that can be expressed as a fraction of two integers, like 1/2 or -3.
• Irrational numbers: Numbers that cannot be neatly written as a fraction, such as √2 or π.

Understanding real numbers is crucial because nearly all real-world quantities, from measurements to calculations, rely on them. They provide the scaffold for our entire number system, capturing everything from the smallest fraction to the largest whole number.

Embracing Mathematical Reasoning

Mathematical reasoning is the process of carefully thinking through a mathematical problem to arrive at a logical conclusion. This skill is important because it enables you to tackle complex problems by breaking them down into understandable parts. It involves:

• Identifying patterns: Recognizing sequences or relationships between numbers.
• Understanding properties: Such as knowing squaring a number always results in a non-negative value.
• Making logical deductions: Using all the information you know to make informed decisions.
• Checking consistency: Ensuring that all parts of your solution fit together logically.

Mathematical reasoning allows you not only to solve the problem at hand but also primes you for approaching new challenges confidently. It’s about cultivating a mindset that questions, analyzes, and reaches conclusions based on solid evidence and logic.