The world of mathematics through the eyes of AI


Prompt: Robot teaches math

Prompt: Robot teaches math

Hi all! In this article, I want to see how AI sees the world of mathematics, or rather, which sections it knows, as far as it understands what they are about and what main results it can highlight.

For this I will use the popular now ChatGPTwho somehow suggested that I start this article:

Mathematics is an amazing science that describes the world through numbers, formulas and logical reasoning. However, despite all its achievements and applications in various fields, many of its aspects remain a mystery to humans. In this article, we will look at the world of mathematics through the eyes of artificial intelligence, capable of analyzing and processing huge amounts of data and finding solutions that would be inaccessible to humans. Let’s dive into the amazing world of mathematical concepts and principles that help us understand the world around us and create new technologies and innovations.

And among the main sections distinguished:

  • Mathematical analysis

  • set theory

  • Linear algebra

  • Probability theory

But before moving on to them, a short preface.

Short Preface

All communication followed the same pattern and included the following questions:

  • Main results

  • What are these results about?

  • Additional comprehension question

I also asked for some ideas on how to beautifully illustrate this or that result. It was not always successful, but I used some ideas.

In addition, I did not edit the entire text below in terms of content and meaning, but only translated (there are rumors that he answers better in English). Therefore, there may be errors that I specifically did not fix.

Mathematical analysis

Newton-Leibniz theorem this theorem links the concepts of differentiation and integration. It states that if a function is continuous on a segment, then the definite integral of it on this segment is equal to the difference between the values ​​of the antiderivative of this function at the ends of this segment.

Intermediate value theorem – this theorem states that if a continuous function takes on two different values ​​at two points in an interval, then it must take on every value between those two points at least once.

Taylor series is a method that allows you to represent a function as an infinite sum of terms associated with the derivatives of the function, calculated at a certain point.

Fourier transform is a technique that decomposes a signal or function into its component frequencies. It is the main tool in signal processing, communication systems and many other areas where signal or function analysis is important.

Question about the expansion of a function in a Taylor series

In the last article in the comments there was such an argument.

Let’s see how ChatGPT will solve it.

Function f(x) = e^{-\frac{1}{x^2}} expands into a Taylor series at the pointx = 0.To find the expansion, we calculate the derivatives of the functionf(x)at this point and get the next row

f(x) = -2x^2 + \frac{12x^4}{4!} + \ldots

This is the wrong answer. yes, functionf(x)expands into a Taylor series at the point x=0but the decomposition has the form f(x)=0.

set theory

Cantor’s theorem – this theorem states that the cardinality of any set is strictly less than the cardinality of the set of all its subsets.

Axiom of choice – This is a controversial axiom of set theory which states that for any set of non-empty sets, there is a way to choose one element from each set.

Cardinality of a finite set of all subsets

I decided to ask what is the cardinality of the set of all subsets of the setA,if it is finite and containsnelements.

If A – finite set of powern,then the cardinality of the set of all its subsets is equal to 2^n.

To understand why this is so, you need to pay attention to the fact that in order to construct any subsetAthere are only two choices for each element: either include it in the subset, or not include it.

Because inA Totalnelements, we get that the total number of options 2^n,and each of these options corresponds to a unique subsetA.

Therefore, everything exists 2^nsubsetsA.

Linear algebra

Determinant – is a numeric value associated with a square matrix. It has many important applications in linear algebra, such as determining reversibility, computing eigenvalues ​​and eigenvectors, and solving systems of linear equations.

matrices is a way of representing linear transformations and systems of linear equations. They provide a powerful tool for performing calculations and analyzing the properties of linear systems.

Linear display – is a function that maps vectors from one vector space to another while maintaining certain algebraic properties such as linearity and the preservation of vector addition and scalar multiplication.

Commutativity of matrix multiplication

Here I decided to ask how to show geometrically that matrix multiplication is not commutative.

When we think of matrices as linear mappings, then matrix multiplication corresponds to composition of linear mappings.

To show that matrix multiplication is not commutative, one can consider two linear mappingsAAndb, WhereA – turn to 90degrees counterclockwise, and B– reflection about the axis x.

If we first applyA,and thenb, we get a different transformation than if we first appliedb,and thenA.

This example illustrates why matrix multiplication is not commutative.

Probability theory

Law of Large Numbers – this law states that the average value of the results obtained from a large number of trials of a random event will be close to the mathematical expectation.

Central limit theorem – this theorem states that the sum of a large number of independent random variables has a distribution close to normal.

Markov chain is a random process that satisfies the Markov property, which states that the future state of the system depends only on the current state, and not on any previous state.

Conditional Probability is the probability of an event occurring given that another event has occurred.

The birthday paradox

Here I decided to ask about a popular problem from probability theory.

Let the group 23person. What is the probability that at least two people have the same birthday? A match is a coincidence of a day and a month.

The probability that at least two people have a birthday in a group of 23 person coincides, can be calculated using opposite events

P(A) =1- P(\overline{A}).

Calculate the probability that everyone has different birthdays.

The probability that the first person has a unique birthday is 1.

The probability that the second person has a unique birthday is \frac{364}{365}, because the number of possible options for unique birthdays is left 364.

The probability that a third person has a unique birthday is \frac{363}{365}.

Continuing by analogy, we get that the probability that everyone has different birthdays is equal to

P(\overline{A})=1 \cdot \frac{364}{365} \cdot \frac{363}{365} \cdot \ldots \cdot \frac{343}{365} \approx 0.4927.

Therefore, the probability that at least two people share the same birthday is

P(A) = 1 - P(\overline{A}) = 1 - 0.4927 = 0.5073

Conclusion

In conclusion, I would like to speculate about whether AI can replace a teacher in mathematics or pass an exam in matan there.

I don’t think so, it still can’t. And the point here is that he makes a lot of mistakes, both arithmetic and logical.

Error Example

Various links

To create graphics manimCE was used

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