Question

Number Theory Jacqui made the following conjecture about the information in the table. If the first number is negative and the second number is positive, the sum is always negative. Find a counterexample for her conjecture.

Short answer

A counterexample is -3 and 5, as their sum is 2 (positive).

Step-by-step solution

Understanding the Conjecture

The conjecture states that if you have two numbers, where the first number is negative and the second is positive, their sum will always be negative. We want to find a situation where this is not true.

Choosing Numbers

To find a counterexample, let's choose specific numbers. Select a negative number, for example, \[ a = -3 \]and a positive number, larger in magnitude, for example,\[ b = 5 \]

Calculating the Sum

Now calculate the sum of these two numbers:\[ -3 + 5 = 2 \]

Analyzing the Result

The result is 2, which is a positive number. This sum contradicts Jacqui’s conjecture because the sum is positive, proving that the conjecture is not always true.

Key concepts

Conjecture

A conjecture in mathematics is a statement or proposition that is believed to be true based on observations, but which has not yet been proven or disproven. In the context of number theory, conjectures often involve patterns or properties of numbers. For example, Jacqui's conjecture suggests a specific outcome when summing numbers of different signs. Conjectures serve as important steps in mathematical reasoning, guiding investigations and leading to either proofs, confirmations, or refutations. While they may seem obvious or likely based on limited observations, it's important to test a conjecture with various examples to ensure its validity before accepting it as a mathematical fact.

Negative Numbers

Negative numbers are numbers less than zero and are typically found on the left side of a number line. They serve various purposes, such as representing losses, debts, or temperatures below zero. In mathematical operations, negative numbers have specific rules:

• When added to a smaller positive number, the result can still be negative.
• Multiplying two negative numbers results in a positive number.
• Dividing a negative number by a positive number results in a negative number.

Understanding how negative numbers interact with positive numbers, as in Jacqui's conjecture, is crucial in predicting the outcome of operations like addition or subtraction.

Counterexample

A counterexample in mathematics is an example that disproves a conjecture or theorem. It's a critical component in mathematical reasoning because finding even a single counterexample is enough to show that a statement is not universally true.
To find a counterexample for Jacqui's conjecture, we need a situation where the first number is negative, and the second is positive, yet their sum is positive. For instance, using -3 and 5, the sum is 2, proving the conjecture false. In research and learning, actively seeking counterexamples helps refine or abandon hypotheses, ensuring they withstand rigorous scrutiny before being widely accepted.

Positive Numbers

Positive numbers are numbers greater than zero and are located to the right of zero on the number line. These numbers are used to describe quantities, such as distances, temperatures above zero, or profits. In arithmetic,

• Adding two positive numbers results in a positive number.
• Both multiplying and dividing two positive numbers yield positive results.
• However, adding a positive number to a sufficiently larger negative number results in a negative sum.

This interaction between positive and negative numbers is central to understanding when exceptions to mathematical conjectures might occur, such as in the case of Jacqui's conjecture.