Conjecture in Math: Definition & Example

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  • 0:01 What Is a Conjecture?
  • 1:11 Writing a Conjecture
  • 1:30 Uses of Conjecture
  • 2:24 Counterexample
  • 3:48 Lesson Summary
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Lesson Transcript
Instructor: Beverly Maitland-Frett

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

This lesson will define the term conjecture, provide examples, and discuss conditions for writing them. We'll also highlight the use of conjectures by professionals and other people in their daily routines.

What is a Conjecture?

Parents make conjectures all the time; without even realizing that they do, they form conclusions about their children. Susie notices that when she buys strawberry ice cream, her 3-year-old son Johnny always ask for seconds, but when she buys vanilla, he leaves some in the bowl. What conclusions do you think Susie would make? Of course, she would think that Johnny likes strawberry more than vanilla.

Informally, we can say a conjecture is just using what you know and observe to form conclusions about something. Formally, a conjecture is a statement believed to be true based on observations. In general, a conjecture is like your opinion about something that you notice or even an educated guess. Looking at the following numbers: 2, 4, 6, 8, 10, 12. What would be the next number? Most likely, you are thinking 14. Why did you make that conclusion? You perhaps looked at the pattern and noticed that the list is counting by 2s.

Your conjecture would be: The next number is 14 because the list is counting by 2s. You didn't prove anything; you just noticed the pattern and formed a conclusion.

Writing a Conjecture

Therefore, when you are writing a conjecture two things happen:

  • You must notice some kind of pattern or make some kind of observation. For example, you noticed that the list is counting up by 2s.
  • You form a conclusion based on the pattern that you observed, just like you concluded that 14 would be the next number.

Uses of Conjecture

A conjecture is like a hypothesis to a scientist. Scientists write hypotheses and test them to see if they are true. A conjecture is just an initial conclusion that you formed based on what you see and already know. Making a conjecture doesn't mean that you are correct or incorrect. All mathematical theorems began with a conjecture. Mathematicians notice a pattern in numbers or shapes, then they perform a number of operations and solve numerous equations to prove their conjecture.

As we mentioned, parents also make conjectures about their child's health and well-being. If they notice something, they will make a few more observations and form some conclusions. Please note that forming a conjecture is only the first step, doing something about the conjecture to prove or disprove it is another step and has other names.

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