Problem 3 - Algebra 1

Question

Use a counterexample to show that the following statement is false. The opposite of a number is never positive.

Step-by-Step Solution

Answer

The statement 'The opposite of a number is never positive' is false. A counterexample is -5, because its opposite is 5, which is positive.

Step 1 Identify the Problem

The problem statement asserts that 'the opposite of a number is never positive.' To disprove this we need a counter example, which is an example that violates the conditions of the statement.

Step 2 Find a Counterexample

To find a counterexample of a number whose opposite is positive, we take a negative number. For instance, let's take the number -5. The opposite of -5 is 5, which is a positive number.

Step 3 Conclude the reasoning

Since we have found a number (-5) whose opposite is positive (5), this is a counterexample that proves the statement 'the opposite of a number is never positive' is false.

Key Concepts

Understanding Opposite Numbers

Opposite numbers are pairs of numbers that are the same distance away from zero on the number line but on opposite sides. Every number has an opposite, and together they form a kind of symmetry around zero.
When you take a number and find its opposite, you change its sign. Here’s how it works:

• The opposite of a positive number is negative. For instance, the opposite of 3 is -3.
• The opposite of a negative number is positive. For example, the opposite of -7 is 7.
• Zero is its own opposite since it is neither negative nor positive, and lies at the center of the number line.

Understanding the concept of opposite numbers helps in many mathematical operations, including solving equations and understanding numerical patterns.

Delving into Negative Numbers

Negative numbers extend our number system beyond zero into the realm below zero. They are critical in representing values that are less than nothing, which can occur in many real-world and mathematical situations.
Here's what to know about them:

• Negative numbers are represented with a "-" sign before the number, such as -1, -5, or -100.
• They are crucial in various applications, such as representing debts, temperatures below zero, and descents below sea level.
• On a number line, negative numbers are positioned to the left of zero, with values decreasing as you move further left.

When finding opposites of negative numbers, noticing their transformation to positive numbers is key. This transformation aids in solving the original exercise by directly providing counterexamples.

Exploring Proof by Contradiction

Proof by contradiction is a logical approach used to demonstrate that a statement is false by showing that it leads to an inconsistency or absurdity if assumed true.
This type of proof involves several key steps:

• Start by assuming the statement in question is true.
• Using this assumption, work through logical steps to identify a contradiction or illogical conclusion.
• Once a contradiction is found, the original assumption that the statement is true is determined to be false.

In the provided exercise, the assertion ‘the opposite of a number is never positive’ was assumed true. By finding a counterexample, such as the number -5 whose opposite is +5, we showed a contradiction to this statement. Thus, the conclusion is that the statement is false.