To test OpenAI o1 model's ability of graduate-level machine learning & math, I tried to ask OpenAI o1 to solve the problem of "Use Newton method to calculate the exact value of \sqrt{2}. Provide Detailed proof step by step." OpenAI o1 actually gives me a very long detailed step-by-step proof, which is much detailed and longer than GPT4o or GPT4o-mini. The complete response is very long and I just summarized the detailed steps of the reasoning here. Overall, I think OpenAI o1 can provide more accurate responses than previous GPT4o' answers. But the response usually is very long and there are a lot of top-down break-down, which makes the final result correct but sometimes hard to understand and follow. OpenAI o1 breaks down the machine learning math problem into these steps: ### Step 1: Define the Function and Its Derivative ### Step 2: Derive the Newton's Method Iteration Formula ### Step 3: Choose an Initial Approximation ### Step 4: Perform Iterations ### Step 5: Understanding the Result Each iteration yields a rational approximation that gets closer to ( \sqrt{2} ): - Iteration 1: ( x_1 = \dfrac{3}{2} = 1.5 ) - Iteration 2: ( x_2 = \dfrac{17}{12} \approx 1.4167 ) - Iteration 3: ( x_3 = \dfrac{577}{408} \approx 1.41422 ) - Iteration 4: ( x_4 = \dfrac{667,217}{471,632} \approx 1.41421 ) ### Conclusion Using Newton's method with the function ( f(x) = x^2 - 2 ) and starting approximation ( x_0 = 1 ), we generate a sequence ( { x_n } ) defined by: [ x_{n+1} = \frac{1}{2} \left( x_n + \frac{2}{x_n} \right) ] This sequence converges to ( \sqrt{2} ) as ( n ) approaches infinity. ### Final Answer By applying Newton's method to the equation ( x^2 - 2 = 0 ), we find that ( \sqrt{2} ) is the limit of the sequence: [ x_{n+1} = \dfrac{1}{2} \left( x_n + \dfrac{2}{x_n} \right) ] Starting from ( x_0 = 1 ), each iteration gives a rational approximation of ( \sqrt{2} ). Although we cannot express ( \sqrt{2} ) exactly using Newton's method (since it's irrational), the method provides increasingly accurate approximations.

Finally, OpenAI released o1 mdoel with stronger reasoning ability. And I looked through the detailed comparison of a math solving results on their website and the comparison between GPT4o vs OpenAI o1-preview on this Algebra problem. For the math question as the in the prompt, o1 uses a chain of thought when attempting to solve a problem, which is similar to how a human may think for a long time before responding to a difficult question. o1 response actually break down the question into a few steps: "Understanding the Given Information", "Defining a New Polynomial", "Properties of q(x)", "Constructing s(x)", "Matching Coefficients". Finding Additional Solutions and finally reaching Conclusion. The additional real numbers x satisfying p(1/x) = x^{2} are x=\frac{1}{n!} or -\frac{1}{n!}. Overall, the reasoning ability is quite complex compared to previous version, so it's helpful and the answers are correct.

Truthfully, I am a little bit surprised by the math proof ability and reasoning ability of ChatGPT. To solve the topological math problem, ChatGPT firstly gives a outline of the proof and provides more details to support the proof. The detailed response is as follows: To prove that the sum of two compact sets in a topological vector space X, X is compact, we need to show that if A and B are compact subsets of X, then A+B is also compact. Proof: 1. Recall Definitions and Properties 2. Consider the Projection Maps 3. Use the Compactness of A and B 4. Image of a Compact Set Under a Continuous Map 5. Conclusion: Thus, we have shown that if A and B are compact subsets of a topological vector space X, then A+B is also compact.

ChatGPT gives the correct answer to the differentiation math problem "What's the differentiation of function f(x) = e^x + log(x) + sin(x)?". It uses the step-by-step thinking and applies the derivative rules for each term, and finally get the overall differentiation result of the function. It' quiet helpful.

ChatGPT gives the correct answer to the math problem- a quadratic equation x^2+5x+6=0 and gives the exact answer of x=-2 pr x=-3. It quotes the method of factoring (x+2) (x+3) =0 and gives the detailed steps to solve the question. Even though the method is different from other AI replies using the exact solution of quadratic equation, it's simple and helpful!

I asked ChatGPT the math question which is bigger number and generate the answers using both GPT-4o and GPT-4o Mini. The results are almost exactly the same. And the speed of GPT-4o is almost the same as GPT-4o Mini without feeling much faster? Maybe the answer is too short?

I asked Gemini to act as an expert in mortgage help me make some calculation of the money that I might lose. The prompts is detailed as I provide as much information as possible, including the original value of the house, how much I already paid, what's the mortgage length and what's the interest rate. But the results are a little bit confusion and the math calculation of the potential money to lose is not very trustworthy. So more improvements are needed. Prompts: 1. You are an expert in mortgage, and I will consult you a few questions about the mortgage of my house. The questions include "I bought my current house in California for about 1 million dollars 4 years ago and paid 10% fees. The mortgage will last 30 years with fixed rate 3%. I've paid the mortgage timely to the bank for 4 years already and I just lost my job. How much money will I lose if I stop paying the mortgage to the bank? 2. Follow up question: "Can you help me make some calculation on how much money will I lose. Do you need some more information?". 3. Third question: "Original loan amount: 0.9 million dollars, Current loan balance: 0.8 million dollars, Estimated property value: 1.1 million dollars"

I asked ChatGPT to act as an expert in mortgage and help me make some calculation on how much money will I lose if I stop paying the mortgage after 4 years of payment. Overall, I think the interaction is not very helpful. I tried to provide a lot of information in advance, including: Original price of the house is 1 million, I already paid 10%, the mortgage lasts 30 years for 3% interest rate, I've paid 4 years already. " Still, After three rounds of conversation, I still can't get the estimated amount of lost money. So the ability is still a little bit far from satisfactory. The math is a little bit strange and I am not quite sure whether it's correct. Prompts: 1. You are an expert in mortgage, and I will consult you a few questions about the mortgage of my house. The questions include "I bought my current house in California for about 1 million dollars 4 years ago and paid 10% fees. The mortgage will last 30 years with fixed rate 3%. I've paid the mortgage timely to the bank for 4 years already and I just lost my job. How much money will I lose if I stop paying the mortgage to the bank? 2. Follow up question: "Can you help me make some calculation on how much money will I lose. Do you need some more information?". 3. Third question: "Original loan amount: 0.9 million dollars, Current loan balance: 0.8 million dollars, Estimated property value: 1.1 million dollars"