Problem 74 Explain why $$\left|a^{2}\righ... [FREE SOLUTION]

Chapter 0: Problem 74

Explain why $$\left|a^{2}\right|=a^{2}$$ for every real number $a$.

Short Answer

Expert verified

The absolute value of a squared real number equals the square itself, $\left|a^{2}\right|=a^{2}$, because squaring any real number always results in a non-negative value, and according to the definition of absolute value, if $a^2 \geq 0$, then $\left|a^{2}\right|=a^2$.

Step by step solution

Define absolute value

Absolute value is defined as the distance of a number from zero on the number line. For any real number a, the absolute value is written as |a|. The mathematical definition of an absolute value is given as follows: $|a| = \begin{cases} a &\text{if } a \geq 0 \\ -a &\text{if } a < 0 \end{cases}$

Consider that a squared real number is always non-negative

It's important to observe that the square of any real number, whether positive or negative, is always non-negative. In other words: $a^2 \geq 0$ for every real number a To see this, consider a positive real number, a positive number times itself is always positive. If a is negative, a negative number times a negative number results in a positive number. Finally, if a is zero, then the square is zero.

Explain why the absolute value of a squared real number equals the square itself

Now let's use the definition of absolute value and the fact that the square of a real number is always non-negative to explain why $\left|a^{2}\right|=a^{2}$ for every real number a. Since $a^2 \geq 0$ for every real number a, according to the definition of the absolute value, we have: $\left|a^{2}\right| = \begin{cases} a^2 & \text{ if } a^2 \geq 0 \end{cases}$ Since squaring any real number always results in a non-negative number, there is no need to consider the other case, when $a^2 < 0$, as this will never occur. Thus, we can say: $\left|a^{2}\right|=a^{2}$ for every real number a.

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Key Concepts

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

Squared Real Number

A squared real number is simply the result you get when you multiply a real number by itself. For example, if you take a real number, say 3, and square it, you obtain 9. It's essential to recognize that squaring a number involves a multiplication operation. Key points about squared real numbers:

They are always the product of a real number with itself.
Squaring is denoted by the exponent 2, such as in $a^2$.
They are crucial for understanding concepts like area and energy in physics and geometry.

This squaring operation is foundational in math and provides us insights into how numbers behave, especially in regard to their size and sign.

Non-Negative Number

Non-negative numbers are those numbers that are either positive or zero. This means they lie on or to the right of zero on the number line. It is important to note that negative numbers, which lie to the left of zero, are not part of this group. Here's why non-negative numbers matter:

They include all the whole numbers from zero onward (0, 1, 2, 3, etc.).
Negative numbers, such as -1, -2, -3, are not considered non-negative.
The square of any real number is always non-negative, meaning negative values are never produced.

Understanding non-negative numbers is vital, especially when learning about squares, as it reassures us that squaring any real number results in a non-negative outcome.

Real Numbers

Real numbers are all the numbers you can find on the number line. They consist of rational numbers (such as fractions and whole numbers) and irrational numbers (like the square root of 2 or pi). Real numbers are the foundation of most of the math you encounter in everyday life.Some characteristics of real numbers include:

They can be positive, negative, or zero.
Rational numbers can be expressed as a fraction of two integers, while irrational numbers cannot.
Real numbers do not include imaginary or complex numbers, such as $\sqrt{-1}$.

The set of real numbers is continuous, meaning there are no gaps on the number line. This continuity makes them very useful in calculus, algebra, and many real-world applications.

Mathematical Definition

Mathematical definitions are precise explanations of mathematical concepts and operations. They are crucial for understanding how different elements in math relate to each other clearly and unambiguously. Why definitions matter:

They help you understand complex mathematical operations by breaking them down into simpler, defined parts.
For instance, the absolute value definition simplifies ideas that might seem abstract by equating them to something measurable鈥攄istance from zero.
Definitions ensure consistency in solving mathematical problems.

In our context, understanding the rigorous definition of absolute value provides the foundation for knowing why $|a^2|=a^2$ for any real number $a$. It highlights how knowledge and definitions can lead to predictability and certainty in math.

91影视

Explain why $$\left|a^{2}\right|=a^{2}$$ for every real number \(a\).

Short Answer

Step by step solution

Define absolute value

Consider that a squared real number is always non-negative

Explain why the absolute value of a squared real number equals the square itself

Key Concepts

Squared Real Number

Non-Negative Number

Real Numbers

Mathematical Definition

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