Problem 48 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true for all real numbers \(a\) and \(b\). $$ \left|b^{2}\right|=b^{2} $$

Step-by-Step Solution

Answer

The statement \(\left|b^{2}\right|=b^{2}\) is true for all real numbers \(a\) and \(b\) because the square of any real number, \(b^2\), is always non-negative, and the absolute value of any real number is also non-negative.

Step 1: An absolute value of a real number is denoted by \(|x|\) and is defined as the distance of the number from zero on the real number line. It is non-negative for all real numbers. Formally, the absolute value function is defined as \( |x| = \begin{cases} x & \text{if } x \geq0 \\ -x & \text{if } x < 0 \end{cases} \) #Step 2: Analyze properties of squaring a real number#

Squaring a real number means multiplying the number by itself. Given a real number \(x\), its square is denoted by \(x^2\) and is the result of \(x * x\). It is important to note that, for all real numbers, their square is always non-negative. #Step 3: Compare the statement to the properties#

Step 2: Now, let's look at the given statement and compare it to the properties we've just established: \(\left|b^{2}\right|=b^{2}\) Given that the square of any real number is non-negative, this means \(b^2\) will always be non-negative as well. Since the absolute value of any real number is also non-negative, the left side of the statement, \(\left|b^{2}\right|\), will be non-negative. #Step 4: Determine if the statement is true#

Since both the left side, \(\left|b^{2}\right|\), and the right side, \(b^{2}\), of the statement are non-negative, we can deduce that this statement is true for all real numbers \(a\) and \(b\). In conclusion, the statement \(\left|b^{2}\right|=b^{2}\) is true for all real numbers \(a\) and \(b\).

Key Concepts

Real Numbers

Real numbers are the set of numbers that include all the rational and irrational numbers. These numbers can be found anywhere on the number line. Real numbers encompass:

• Rational numbers: These are numbers that can be expressed as fractions, such as 1/2 or 3/4, which include integers and fractions.
• Irrational numbers: These numbers cannot be expressed as simple fractions. Examples include the square root of 2 and pi (\(\pi\)).

A few key attributes of real numbers include:

• They can be positive, negative, or zero.
• They are located on the continuous number line, making it possible to precisely measure the distance between any two numbers.

Understanding real numbers is crucial since almost every concept in mathematics, such as absolute values and squaring numbers, applies to them. This broad category enables us to engage with a diverse range of arithmetic and algebraic functions!

Squaring Real Numbers

Squaring is a simple yet powerful mathematical operation, particularly with real numbers. When you square a real number, you multiply it by itself. If you have a real number denoted as \( b \), squaring it means calculating \( b^2 \), that is \( b \times b \).

One important property of squaring is that the result is always non-negative. This means:

• If \( b \) is positive or zero, then \( b^2 \) is positive or zero.
• If \( b \) is negative, then \( b^2 \) is still positive because multiplying two negative numbers results in a positive number.

This property is fundamental to the exercise we reviewed. It aligns with the absolute value concept because both squared values and absolute values yield non-negative results.

Understanding that squaring a number always leads to a non-negative result helps while evaluating expressions and solving equations. It assures that no matter the sign of the original real number, squaring will lead to a consistent and predictable outcome.

Non-negative Integers

Non-negative integers are a subset of real numbers and include whole numbers that are greater than or equal to zero. Simply put, they consist of all positive integers and zero. Examples of non-negative integers are:

• 0
• 1
• 2
• 15, 234, etc.

These integers are the building blocks of many mathematical concepts and are crucial in discussions involving measures, counting, and quantity.

When handling non-negative integers, their key characteristic is that they never fall below zero. This is similar to both the absolute value and squared value, which you probably noticed share the trait of being non-negative.

When dealing with exercises like the one analyzed, understanding non-negative integers allows you to quickly deduce that operations yielding non-negative results verify the mathematical statements associated with them. Thus, reinforcing why the absolute value of a squared real number comfortably equals the original squared value without altering its outcome.