Problem 136 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Multiplying a negative number by a nonnegative number will always give a negative number.

Step-by-Step Solution

Answer

The statement is false. The corrected statement is: 'Multiplying a negative number by a positive number always gives a negative number, but multiplying a negative number by zero always gives zero.'

Step 1: Understand the rule of multiplication involving negative and nonnegative numbers

The basic rule of multiplication involving negative and nonnegative numbers is that if one number is negative and the other is positive, the result is a negative number. If one number is zero (nonnegative but not positive) and the other number is negative, the result is zero. The result is never positive.

Step 2: Apply the rule to the statement

Apply the rule to the provided statement. 'Multiplying a negative number by a nonnegative number will always give a negative number' is only partially true given that if the nonnegative number is positive, the result is indeed negative. However, if the nonnegative number is zero, the result is zero not negative.

Step 3: Validate or correct the statement

Given the explanation in step 2, it can be concluded that the statement is false. Correcting the statement would give 'Multiplying a negative number by a positive number always gives a negative number, but multiplying a negative number by zero always gives zero.'

Key Concepts

Negative and Nonnegative Numbers

Understanding the nature of negative and nonnegative numbers is crucial in arithmetic and algebra. Negative numbers are those less than zero, represented with a minus sign \textbf{(-)}, such as \textbf{-2} or \textbf{-7}. They can represent deficits, such as debt or temperatures below zero. \(
\)Nonnegative numbers, on the other hand, include all positive numbers and zero. These numbers do not have a minus sign before them and are always greater than or equal to zero. For example, \textbf{3} and \textbf{0} are both nonnegative numbers. \(
\)It's important to note that while zero is neither positive nor negative, it is considered nonnegative because it is not less than zero. This subtle differentiation is key when discussing multiplication rules involving negative and nonnegative numbers.

Multiplying Negative Numbers

Multiplication involving negative numbers follows specific rules that can sometimes be counterintuitive. To multiply a negative number by another number, we look at the sign of the second number. If it is positive (greater than zero), the product will always be negative. This reflects the idea that if we have a negative quantity and we increase it, we're moving further away from zero in the negative direction. \(
\)For instance, multiplying \textbf{-4} by \textbf{2} results in \textbf{-8}. But, what happens when the second number is zero? The rule here is straightforward: any number multiplied by zero equals zero. Therefore, a negative multiplied by zero would simply result in zero, not a negative number. \(
\)Hence, the rule to remember is that a negative number multiplied by a positive number yields a negative result, while a negative number multiplied by zero yields precisely zero. This distinction is essential for the correct application of multiplication in various contexts.

True or False Mathematical Statements

Handling true or false mathematical statements involves assessing whether a given statement accurately reflects mathematical principles. A statement can claim, for example, that a certain result will 'always' occur, but such absolute terms require careful consideration. \(
\)In the exercise in question, the claim that 'Multiplying a negative number by a nonnegative number will always give a negative number' requires a small but significant clarification. While it is true for all positive nonnegative numbers, it does not hold for the special nonnegative number zero. \(
\)By applying the rules for multiplying negative numbers, as previously discussed, we correct the statement to reflect that multiplication of a negative number with zero results in zero. This practice not only helps us understand the specific rules but also nurtures a critical approach to the language used in mathematics - where precision is paramount. Recognizing the nuances in such statements enables students to deepen their understanding and avoid misconceptions as they progress in their studies of mathematics.