CONJECTURE

Conjecture

Definition Of Conjecture

Conjecture is a statement that is believed to be true but not yet proved.

Examples of Conjecture

The statement "Sum of the measures of the interior angles in any triangle is 180° " is a conjecture.
Here is another such conjecture: "If two parallel lines are cut by a transversal, the corresponding angles are congruent."

Video Examples: Making and Proving Conjectures

Solved Example on Conjecture

Ques: Sam proposed that "3n + 1 yields a prime number for any even number n." and called it a conjecture. He explained his conjecture using the numbers 2, 4, and 6. He got 7, 13, and 19. Is his proposition a conjecture?

Choices:

A. Yes
B. No
Correct Answer: B

Solution:

Step 1: A conjecture is a statement that is believed to be true but not yet proved or disproved.
Step 2: Sam's proposition is not a conjecture, because for n = 8, 3n + 1 gives 25, which is not a prime number.

Quick Summary

A conjecture is a statement proposed as true.
It requires proof to be accepted as a theorem.
Conjectures can be disproven with counterexamples.

\[ N/A (Conjectures are statements, not formulas) \]

⚠️ Common Mistakes

Assuming a conjecture is true without proof.
Confusing a conjecture with a proven theorem.
Believing a conjecture is true based on limited examples.

🌍 Real-World Uses

Cryptography: Forming conjectures about prime numbers and their distribution to develop secure encryption algorithms.
Data Analysis: Making conjectures about trends and patterns in data sets before applying statistical tests to confirm them.
Engineering Design: Proposing conjectures about the optimal shape or configuration of a structure before running simulations or building prototypes.

📋 Standards Alignment

CCSS.MATH.CONTENT.HSG.CO.C.9
CCSS.MATH.PRACTICE.MP3

🍎 Teacher Insights

Emphasize the importance of rigorous proof and the role of counterexamples in disproving conjectures. Encourage students to formulate their own conjectures.

🎓 Prerequisites

Basic Logic
Mathematical Proofs

Check Your Knowledge

Q1: Which of the following best describes a conjecture?

A statement that is always true A statement that has been proven A statement believed to be true but not yet proven A statement that is always false

Q2: If a student provides several examples to support a statement, does that prove the statement is a conjecture?

Yes, examples always prove a conjecture No, a conjecture requires a formal proof Sometimes, if there are enough examples Only if the examples are complex

Frequently Asked Questions

Q: What's the difference between a conjecture and a theorem?
A: A conjecture is a statement believed to be true, whereas a theorem is a statement that has been proven to be true.

Q: How can you disprove a conjecture?
A: A conjecture can be disproven by finding a single counterexample.