How are creative math ideas come up with to solve problems?

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I am not sure if this is the type of answer you are looking for, but I hope it helps!In summary, the conversation discusses different ways the mind can approach solving math problems, including giving new definitions and posing new problems. It also touches on the motivations for doing so and how it is still practiced in modern times. The conversation also mentions the importance of out of the box thinking and provides historical examples of breakthroughs in math and other fields. It concludes with a personal viewpoint on the topic.
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trees and plants
Hello.Well, this is a general question.when solving a math problem what are the ways the mind finds to prove the problem?How about giving new definitions for abstract objects or posing new problems on your own?What are the motivations to do these things?Should they be of very high value to be accepted?Should they be very difficult or be interesting to be accepted?I know that Aristotle in ancient Greece tried to solve problems other posed and he also posed his own problems and then tried to solve them.Is this done today in math or physics?Are mathematicians of two kinds, problem solvers and theory builders?What are theory builders?Thank you.
 
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What is the context for your questions? Is this for a school project? Is it personal interest? Do you have any other questions? Should I answer these? or should I wait until others decide to answer first? ...

The first person to prove something or the first person to come up with a new idea in math and other fields often uses out of the box thinking. Others may have tried before but didn't think in the same way. There is no method to this style of thinking but you must be open to breaking the rules that you learned in order to see beyond then.

Some historical examples:
- Archimedes and his mathematical solutions to the surface area and volume of a sphere.
- Gallileo's observation about falling objects
- DaVinci's art, many inventions...

Sometimes ideas are extensions of existing ideas that are powerful in themselves:
- having counting numbers and then debit counting to negative numbers
- adding zero to the counting numbers
- fractional numbers to rational numbers to real numbers

Sometimes problems like the quadratic polynomial have no solution in real numbers:
- imaginary numbers then to complex numbers
- then to quaternions
- and octonians

These are just my personal feelings on these ideas.
 

Related to How are creative math ideas come up with to solve problems?

1. How do mathematicians come up with new and creative ideas to solve problems?

Mathematicians use a variety of techniques and approaches to generate new and creative ideas for solving problems. These may include brainstorming, breaking down complex problems into simpler parts, looking for patterns and connections, and collaborating with other mathematicians.

2. Are there any specific strategies or tools that mathematicians use to come up with creative math ideas?

Yes, mathematicians often use strategies such as analogy, visualization, and abstraction to generate new ideas. They may also use tools such as computer simulations, mathematical models, and mathematical software to explore and test their ideas.

3. How important is creativity in mathematics?

Creativity is essential in mathematics, as it allows mathematicians to come up with new and innovative solutions to problems. It also helps in making connections between different mathematical concepts and theories, leading to further advancements in the field.

4. Can anyone learn to be creative in math?

Yes, like any other skill, creativity in math can be learned and developed. By practicing problem-solving and exploring different approaches, anyone can improve their creative thinking abilities in mathematics.

5. Are there any real-world applications of creative math ideas?

Absolutely. Many real-world problems, such as optimizing traffic flow, predicting weather patterns, and designing efficient algorithms, require creative math ideas to be solved effectively. Creative thinking in math also plays a crucial role in technological advancements and scientific research.

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