DeepSeek R1 model for Reasoning
DeepSeek R1 model for Reasoning
Prompts
Please Use Newton method to calculate the exact value of \sqrt{2}. Provide Detailed proof step by step.
Please help prove Riemann Hypothesis and think step by step.
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Prompt: Please Use Newton method to calculate the exact value of \sqrt{2}. Provide Detailed proof step by step.I asked DeepSeek R1 model to to calculate the exact value of \sqrt{2} using Newton method, which is a classic Machine Learning Task. The results from DeepSeek R1 model are pretty good. It even prints out the whole thinking process as part of the LLM output. The thinking took about 101 seconds. And I just listed a few point of the thinking process below: Okay, let's see. The user wants me to use the Newton method to calculate the exact value of sqrt(2). Hmm, Newton method, right? That's the Newton-Raphson method for finding roots. So, first, I need to recall how that works. The idea is to iteratively approach the root of a function by using the tangent line approximations. Alright, so sqrt(2) is the solution to the equation x^2 = 2. To apply Newton's method, I should reformulate this as a root-finding problem. That means defining a function f(x) such that f(x) = 0 when x is sqrt(2). So, let's set f(x) = x^2 - 2. The root of this function is indeed x = sqrt(2). Next, I remember the Newton-Raphson formula: x_{n+1} = x_n - f(x_n)/f'(x_n). I need to compute the derivative of f(x). The derivative f'(x) is 2x. Plugging that into the formula gives x_{n+1} = x_n - (x_n^2 - 2)/(2x_n). Let me simplify that expression. Starting with x_{n+1} = x_n - (x_n^2 - 2)/(2x_n). Let's split the fraction: (x_n^2 - 2)/(2x_n) = x_n/(2) - 1/x_n. Wait, no. Let me actually compute it step by step. Alternatively, maybe combine the terms. x_n - (x_n^2 - 2)/(2x_n) = (2x_n^2 - (x_n^2 - 2)) / (2x_n) = (2x_n^2 - x_n^2 + 2) / (2x_n) = (x_n^2 + 2) / (2x_n). Wait, that doesn't seem right. Wait, let's check the algebra again. Wait, x_{n+1} = x_n - (x_n^2 - 2)/(2x_n). Let's compute this as: x_{n+1} = x_n - (x_n^2)/(2x_n) + 2/(2x_n) = x_n - x_n/2 + 1/x_n = (x_n/2) + (1/x_n). So, simplifying, x_{n+1} = (x_n + 2/x_n)/2. Ah, that's the familiar formula for the square root using Newton's method. So each iteration averages x_n and 2/x_n. Okay, so now I need to choose an initial guess x_0. Since sqrt(2) is between 1 and 2, let's pick x_0 = 1.5 as a reasonable starting point. Wait, or maybe x_0 = 2? Let me think. The method usually converges regardless of the initial guess, but a better guess might converge faster. Let's try both and see. Wait, the user wants the exact value, but Newton's method gives an approximation. Wait, but sqrt(2) is irrational, so we can't get an exact value numerically, but perhaps they want to show the iterative steps converging to sqrt(2). But the exact value can't be computed exactly with Newton's method because it's an iterative approximation. So maybe the question is to show the steps until a certain precision is reached.
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Prompt: Please help prove Riemann Hypothesis and think step by step.DeepSeek R1 models provides detailed thinking process of generating responses of complex math coding problems with surprisingly lower cost. The best part about deepseek is that they even open source their model. Great job done.
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