Problem 122 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The difference between 0 and a negative number is always a positive number.

Step-by-Step Solution

Answer

The statement is True

Step 1: Understanding the Statement

The statement to verify is: 'The difference between 0 and a negative number is always a positive number'. To understand this, one needs to grasp that the difference between two numbers is calculated by subtracting one from the other. In this case, the exercise specifies that 0 is the number to subtract from, while the other number is a negative number.

Step 2: Subtraction Operation

When subtracting a negative number from 0, the operation becomes a bit counter-intuitive. In general, the subtraction of a negative number is the same as the addition of a positive number. Therefore, the equation can be rewritten as 0 plus the absolute value of the negative number. That is, 0 + |-n|, where n is any negative number.

Step 3: Solution and Verification

Any positive number added to zero will preserve its sign, meaning it will remain positive. Hence, the initial statement: 'The difference between 0 and a negative number is always a positive number.' is True.

Key Concepts

Subtraction

Subtraction is a fundamental arithmetic operation that involves taking away one quantity from another. When expressed in a formula, subtraction looks like this: \( a - b \), where \( a \) is the minuend and \( b \) is the subtrahend. The result of this operation is called the difference.

In our exercise, we focus on subtracting a negative number from 0, which might seem tricky. However, it's essential to remember: when you "subtract a negative," you are adding the positive equivalent of that number. For example, \( 0 - (-3)\) translates to \( 0 + 3 \). This switch from subtraction to addition occurs because of the properties of negative numbers, which we will discuss in more detail later. Always ensure that you understand subtraction well, as it forms the basis of many more complex mathematical concepts.

Negative Numbers

Negative numbers are numbers less than zero. They are often used to represent loss, debt, or decrease, like temperatures below freezing or money owed. Representing them with a minus sign (-), they have unique rules when used in arithmetic operations.

When subtracting a negative number, as seen in our exercise, it is crucial to recognize that two negatives make a positive. This rule can be visualized as moving to the right on a number line when "subtracting a negative." So, subtracting \(-2\) is equivalent to adding \(2\). This operation changes direction—adding rather than subtracting—and results in a positive outcome, which confirms the truth of our original statement.

True or False Statements

True or false statements are logical expressions that declare facts which can be verified as either correct (true) or incorrect (false). In mathematics, these statements often involve numerical relationships and operations that can be tested for validity.

To verify the truth of a statement, like our example about differences between numbers, we must check the norms and mathematical rules involved. By understanding the properties of subtraction and negative numbers, we can establish if a given statement is true or false. Additionally, when a statement is false, adjustments are necessary to transform it into a truthful claim. Rigorous analysis allows us to confirm that the difference between 0 and a negative number results indeed in a positive value, validating the statement as true.