Problem 43 - 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. Multiply the number by 4. Add 8 to the product. Divide this sum by 2 . Subtract 4 from the quotient.

Step-by-Step Solution

Answer

The conjecture that the sequence of operations on any number would result in double of the original number is proved.

Step 1: Apply the procedure on chosen numbers

Select four numbers, let's say we choose the numbers 1, 2, 3 and 4. Apply the given procedure to these numbers. For 1: multiply by 4 gives 4. Add 8 to get 12. Divide by 2 to get 6. Subtract 4 to get 2.For 2: multiply by 4 gives 8. Add 8 to get 16. Divide by 2 to get 8. Subtract 4 to get 4.For 3: multiply by 4 gives 12. Add 8 to get 20. Divide by 2 to get 10. Subtract 4 to get 6.For 4: multiply by 4 gives 16. Add 8 to get 24. Divide by 2 to get 12. Subtract 4 to get 8.

Step 2: Conjecture a relationship

Based on the results, a relationship can be noticed. The final result always appears to be double of the original number.

Step 3: Prove the conjecture using a variable

Let the original number be \( n \). After applying the procedure, we have:Step 1: Multiply the number by 4, results in \( 4n \),Step 2: Add 8 to the product, results in \( 4n+8 \),Step 3: Divide this sum by 2, results in \( \frac{(4n+8)}{2}=2n+4 \),Step 4: Subtract 4 from the quotient, results in \( (2n+4)-4=2n \).This simplified result \( 2n \) matches the conjecture, as it is double of the original number \( n \).

Step 4: Conclusion

Thus, the conjecture that the result of these operations on any number \( n \) would be \( 2n \) is proved.

Key Concepts

Deductive Reasoning

Deductive reasoning involves reaching a logically certain conclusion based on the given premises. In our exercise, we began by performing a series of mathematical operations on several numbers to observe a pattern. This systematic process allows us to form a conjecture, which we can then set out to prove.

• Firstly, we applied the steps to each number such as doubling and subtracting results, ensuring consistent results to work with.
• Observational findings led to hypothesizing that the output was double the input number.
• This led us into deductive reasoning, where understanding each algebraic transformation verifies our hypothesis step by step.

By following these logical steps, we verify that our conclusion must be true where our initial assumptions are accurate. This application of deductive reasoning is central in mathematics to establish truths beyond assumptions and observations.

Number Manipulation

Number manipulation refers to altering numbers through mathematical operations like addition, subtraction, multiplication, or division. This exercise demonstrates how complex results can be achieved through simple sequential operations.

• When you multiply a number by 4, you are scaling the number to a higher value, intensifying its original magnitude.
• Adding 8 could be seen as increasing the total by a fixed increment, shifting the scaled number upward.
• By then dividing by 2, we are normalizing the result slightly, reducing it to regain balance.
• Finally, subtracting 4 achieves a consistent standardized result, essential in verifying our conjecture.

By methodically applying these steps, meticulous predictions can be made about the behavior and final outcomes of numbers, facilitating a structured approach to find number patterns and outcomes.

Algebraic Proofs

Algebraic proofs are systematic steps used to verify a conjecture or mathematical statement. Here, algebra helps to prove the observation that one result is double the original input.

• By using the variable \( n \) to represent any number, we generalized our operations.
• Subsequential operations were expressed algebraically such as multiplying and adding with \( 4n + 8 \).
• Transformation through division prompts \( \frac{(4n+8)}{2} \), which simplifies to \( 2n + 4 \).
• The final step, subtraction of 4, precisely derives \( 2n \), validating our initial claim algebraically.

These algebraic transformations confirm our understanding of the exercise, translating observed patterns into a formal proof, shifting conjectures into verifiable truths. Each algebraic expression matters in ensuring the proof's conclusiveness and universal applicability.