Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

State the real number property that iustifies the statement $$ 0(2 a+3 b)=0 $$

Step-by-Step Solution

Answer

The real number property that justifies the statement \(0 * (2a + 3b) = 0\) is the \(multiplication\ property\ of\ zero\).

Step 1: Identify the relevant real number property

To solve this exercise, we need to recognize the real number property that applies to the given statement. In this case, we are looking for a property that deals with the multiplication of 0 and an expression.

Step 2: Apply the multiplication property of zero

The multiplication property of zero states that the product of any real number (or expression) and 0 is 0. Mathematically, this can be expressed as: \(a * 0 = 0\) or \(0 * a = 0\) where \(a\) is any real number or expression.

Step 3: State the real number property that justifies the given statement

The given statement is: \(0 * (2a + 3b) = 0\) According to the multiplication property of zero, any real number or expression multiplied by 0 results in 0. So, the real number property that justifies the given statement is the \(multiplication\ property\ of\ zero\).

Key Concepts

Multiplication Property of Zero

The multiplication property of zero is a fundamental concept in mathematics. It states that when you multiply any real number or expression by zero, the result is always zero. This property simplifies mathematical computations, especially when dealing with complex expressions or equations. Let's break it down a bit further.

Imagine you have a basket with various fruits, no matter how many fruits you multiply by zero, you'll always end up with none—zero fruits. In algebraic terms, if you let "fruit" represent any number or expression, the result is the same: zero multiplied by any "fruit" is always zero.

• If you have an expression like \(2a + 3b\), and you multiply it by 0, it equates to 0.
• This principle holds true regardless of the complexity of the expression or the values of \(a\) or \(b\).

It's important to understand this property because it enables us to simplify calculations and recognize outcomes quickly when multiplication by zero is present.

Mathematical Justification

Mathematical justification involves providing logical reasoning for why certain properties or results hold true. In mathematics, it's not just about solving an equation or finding an answer, but understanding why certain operations lead to certain results.

When questioning why \(0(2a + 3b) = 0\), **you refer to** the multiplication property of zero. This justifies the step mathematically because:

• The rule is consistent: multiplying by zero regardless of the number or expression still results in zero. This consistency is a corner stone in arithmetic.
• This principle applies universally within the framework of real numbers, ensuring our calculations are based on logical foundations.

Having solid mathematical justification for operations like these ensures that you're not only getting the right answer but also grasping the reasoning behind it.

Real Numbers

Real numbers encompass a wide variety of numbers that permeate the field of mathematics. These include positive and negative integers, fractions, irrational numbers like \(\pi\), and decimals. Essentially, any number you can think of that doesn't involve imaginary or complex components is a real number.

In the context of the multiplication property of zero, real numbers play an essential role. When we state that any real number or expression multiplied by zero results in zero, we are referring specifically to this broad group of numbers that have wide-ranging applications in everything from simple arithmetic to advanced calculus.

• Real numbers form the basis for most algebraic expressions and equations.
• They help in real-world applications since they can represent continuous quantities.

Understanding real numbers is crucial for recognizing how they interact through properties like the multiplication property of zero, ensuring sound reasoning behind mathematical operations.