Problem 45 - Thinking Mathematically

Question

Exercises 43-46 describe procedures that are to be applied to numbers. In each exercise, a. Repeat the procedure for four numbers of your choice. Write a conjecture that relates the result of the process to the original number selected. b. Use the variable n to represent the original number and use deductive reasoning to prove the conjecture in part (a). Select a number. Add 5. Double the result. Subtract 4 . Divide by 2 . Subtract the original selected number.

Step-by-Step Solution

Answer

The result of the given procedure when applied to any number is always 1.

Step 1: Apply the procedure to four numbers

Let's choose four numbers - 2, 3, 5 and 10. We have to add 5 to each number, then double the result, after that subtract 4, divide by 2 and finally subtract the original selected number. Here are the results: For 2, the result is 1. For 3, the result is 1. For 5, the result is 1. For 10, the result is 1. It seems like the result is always 1 regardless of the original number.

Step 2: Make a conjecture

Considering the results from Step 1, a possible conjecture could be: For any number, when we add 5, double the result, subtract 4, divide by 2 and then subtract the original number, the result is always 1.

Step 3: Prove the conjecture

Now, let's test the conjecture on a variable \( n \). We add 5 to \( n \), double the result to get \( 2(n+5) \), subtract 4 to get \( 2(n+5) - 4 \), divide by two to get \( \frac{2(n+5) - 4}{2} \) and finally subtract \( n \) to get \( \frac{2(n+5) - 4}{2} - n \). When reduced, we find that the result is \( 1 \), proving the conjecture. The correct use of mathematical deductive reasoning confirms the proposed conjecture.

Key Concepts

Deductive Reasoning

Deductive reasoning is a logical process where conclusions are drawn from a set of premises or facts. In the exercise, we use deductive reasoning to confirm our conjecture. Here’s how it works:

• We start with a general principle or procedure – for instance, the series of mathematical operations applied to any number.
• We perform the operations systematically on several specific cases. This moves us from a particular example to a general rule.
• Finally, we validate or prove the general rule using algebra, ensuring it holds for any number.

Deductive reasoning ensures that if our premises are true, the conclusion must also be true. This kind of reasoning is essential in mathematics, where accuracy and proof are crucial.

Number Theory

Number theory is the branch of mathematics dealing with the properties and relationships of numbers. This exercise is embedded in number theory since we work with operations on numbers and seek to find patterns and general rules.

• By applying operations to various numbers, we search for a consistent result.
• Our goal is to identify a conjecture for all natural numbers.

Tools from number theory help us understand why operations behave a certain way. They provide insights into factors, divisibility, and many other properties that allow us to make accurate conjectures about numbers in general. While our exercise is simple, it applies core number theory principles, like forming and testing conjectures about numerical properties.

Mathematical Proof

A mathematical proof is a logical argument that verifies the truth of a mathematical statement. It provides a full, step-by-step deduction from known facts. In our exercise, proof is crucial to verify the conjecture.

• We start with the operations applied to any number represented by a variable, say, \( n \).
• The proof involves working through each operation systematically to confirm a consistent result.
• We compute each step algebraically, ensuring all steps are clear and logically follow.
• When simplified, the expression becomes \( 1 \), confirming that our conjecture holds true for any initial number.

Mathematical proofs are necessary for solidifying ideas in mathematics. Without them, what we believe to be true may just be unverified guesses. Proofs help build a strong foundation of knowledge.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In our exercise, we represent the process using algebraic expressions. This is important for simplifying and solving problems.

• We use variables such as \( n \) to stand in for any number, allowing us to generalize results for all numbers.
• Operations on \( n \) create new expressions: adding, doubling, subtracting, and dividing all form our work process.
• The steps of simplifying the expression reflect each part of the operation pieced together: \( 2(n+5) - 4 \) becomes \( \frac{2(n+5) - 4}{2} - n \).
• Simplifying algebraic expressions allows us to see the fundamental relationship being expressed, in this case \( 1 \), for any number \( n \).

Algebraic expressions are powerful tools in mathematics. They enable us to simplify complex operations, proving or disproving conjectures with clear reasoning.