You can download all the important posts as PDF book "Unlocking the Secrets of Quantitative Thinking".
When I was an engineering student, I was wondering where all those number series are coming from. You know, those for which you have to prove the convergence, for instance.
I always thought there's some special reason, something behind the scenes that mathematicians used to construct those series. But, it turns out it may not be so. Or, whatever reason they had, it may not be a mathematical reason. The answer is, anyone can declare a series! By ZFC axioms, you are ALLOWED to form a series without any proof, almost directly from fundamental axioms of mathematics! You don't need to wait for high school education to do that. Once you are familiar with fractions, go ahead, create, construct, your own number series. And right there ask your teacher to introduce you to that delta-espilon definition of limit! No proof needed for constructing a series! How so? Because, essentially, fundamental axioms in math tell you that you can create numbers at free will, you can define sequences as you wish. So, you CAN define, say number 2, then multiply by itself, 2, 3, 4, 5,.. times and each time you decide to divide 1 by that and then sum them. Like here, 1/2 + 1/(2^2) + 1/(2^3) + 1/(2^4) + ..... Notice how arbitrary you came up with choosing the number and defining the sequence of operations. And you can define these sequences as much as you want. There is no other way! Nothing else behind the scene.
Now, proving whether it converges or not is a different story. However, look that YOU can define WHAT needs to be proven! Why would you define any series? Mathematician will say because you can! Do you really want to? You may not want to! That's the state of affairs too. Now, once you define it, math does not care whether you liked it or not, were you motivated or not, or, whether the series came form some physical or other process measurements, or from you just playing with numbers,. The series, as it is, can come into existence just because you can create them. You are allowed, again, by fundamental axioms in math to do that. And there can be many, many series, but again, they are all arbitrarily created. Try it by yourself! Just create a series, a sequence and try to find the sum. That's all what's to it.
You can read more:
When I was an engineering student, I was wondering where all those number series are coming from. You know, those for which you have to prove the convergence, for instance.
I always thought there's some special reason, something behind the scenes that mathematicians used to construct those series. But, it turns out it may not be so. Or, whatever reason they had, it may not be a mathematical reason. The answer is, anyone can declare a series! By ZFC axioms, you are ALLOWED to form a series without any proof, almost directly from fundamental axioms of mathematics! You don't need to wait for high school education to do that. Once you are familiar with fractions, go ahead, create, construct, your own number series. And right there ask your teacher to introduce you to that delta-espilon definition of limit! No proof needed for constructing a series! How so? Because, essentially, fundamental axioms in math tell you that you can create numbers at free will, you can define sequences as you wish. So, you CAN define, say number 2, then multiply by itself, 2, 3, 4, 5,.. times and each time you decide to divide 1 by that and then sum them. Like here, 1/2 + 1/(2^2) + 1/(2^3) + 1/(2^4) + ..... Notice how arbitrary you came up with choosing the number and defining the sequence of operations. And you can define these sequences as much as you want. There is no other way! Nothing else behind the scene.
Now, proving whether it converges or not is a different story. However, look that YOU can define WHAT needs to be proven! Why would you define any series? Mathematician will say because you can! Do you really want to? You may not want to! That's the state of affairs too. Now, once you define it, math does not care whether you liked it or not, were you motivated or not, or, whether the series came form some physical or other process measurements, or from you just playing with numbers,. The series, as it is, can come into existence just because you can create them. You are allowed, again, by fundamental axioms in math to do that. And there can be many, many series, but again, they are all arbitrarily created. Try it by yourself! Just create a series, a sequence and try to find the sum. That's all what's to it.
You can read more:
- Real world examples for rational numbers, for kids
- Math and its relationship with real world
- How math can be applied to so many different fields?
- Where the graphs in mathematics and physics come from?
- Tweets about math, physics, and how to approach math calculations
- One insight about mathematical axioms, logic and their relation to other disciplines
- How the ideas are born and notes on creative thinking
- Mathematics Axiomatic Frontier
- Why math can be an independent discipline?
- More on creative thinking, math, innovations, physics, emotions
- Comparison between making movies and math and physics
- More tweets about math
- Domains of math applications and math development
- Where all those number series in math are coming from?
- How to understand the role of math in economics, physics, engineering, and in other fields
No comments:
Post a Comment