On this article, learn the way GPT-5 handles intermediate to superior mathematical reasoning, akin to fixing methods of equations and developing clear textbook-style proofs.
Matters lined embody:
A fast and systematic warm-up for fixing 2×2 linear methods. Proofs concerning monotonicity and bounds of capabilities are clearly and rigorously written. Dialogue about high quality of response, tone, and the place the mannequin nonetheless feels mechanical.
Let’s get began.
Can ChatGPT-5 present superior math proofs?
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introduction
One of many claims about OpenAI’s newest mannequin, GPT-5, is that it’s a breakthrough in mathematical and logical reasoning, with the flexibility to “suppose” extra deeply when prompts profit from cautious evaluation. This contains complicated, human-like reasoning settings akin to coding, asking scientific questions, synthesizing data, and analyzing monetary information. In arithmetic, the mannequin was reported to have achieved a 94.6% success price in fixing arithmetic issues on the AIME 2025 worldwide convention with out counting on exterior instruments, exhibiting vital enchancment over earlier variations.
On this article, we are going to show by means of intermediate to superior mathematical reasoning examples how the newest model of ChatGPT tackles complicated mathematical issues with excessive precision and a very systematic analytical strategy. We additionally briefly clarify the conduct exhibited by the responses obtained.
Disclaimer: The intention right here is to not go as deep as proven on this Reddit publish which claims that ChatGPT-5 has been innovated by creating “new arithmetic”, however to briefly analyze the mannequin’s potential to assemble elaborate and stable proofs for numerous issues.
Heat-up: Fixing simultaneous linear equations
Earlier than we transfer on to harder issues, particularly fixing a system of two linear equations with two variables x and y, let’s begin with one thing gentle and purely methodological.
The system is as follows (taken from these lecture notes):
5x − 2y = 7 −2x + y = 3
5 occasions − 2 years outdated = 7
−2x + y = 3
The prompts are easy and the mannequin is free to decide on its inference strategy.
Resolve the next system of linear equations: (adopted by the system of equations above)
The response ChatGPT returns is:
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This mannequin not solely accurately solved the system with x = 13 and y = 29 because the true resolution, but additionally offered a transparent step-by-step technique (e.g., substitution or elimination) that saved the algebra orderly.
Curiously, if you happen to ask ChatGPT to decide on essentially the most environment friendly technique out of the obtainable strategies to resolve the system and justify its alternative, it could generate two doable solutions and permit the consumer to decide on the strategy they need (because it did in my case).
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Each solutions depend on the identical technique: elimination. This reveals how ChatGPT is just not “inventive” for its personal sake, however how prompt-oriented and logically constant it’s (e.g. by deliberately creating two totally different strategies). Step-by-step reasoning can also be related in each instances, however the primary distinction is the fashion used to justify the method of elimination.
Just a little more difficult: Examples of mathematical proofs
Transferring on to extra superior college-level arithmetic, I requested ChatGPT to assemble a proof with out offering particular operate examples.
Show that the composition of any two reducing capabilities is rising.
With out explicitly activating “Lengthy Suppose Mode,” one in all ChatGPT’s latest modalities, the app offered convincing responses that appeared like self-contained proofs.
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That GPT-5 efficiently dealt with this problem may be simply verified by means of options obtainable on the net, akin to this instance.
Here is one other instance.
Let g(x) = 2x + 3x for |x|. ≤ 1. (that’s, the area of g is [−1, 1].) Show that the vary of g is correct. [5/6, 5].
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And the proof is, certainly, disgustingly right. Nothing is mistaken and there are only a few small particulars. General, the proof construction is full and flows logically. Moreover, we precisely establish the necessary properties of g(x): monotonicity, continuity, and differentiability. If I needed to say, the narrative stays considerably mechanical and unappealing (for instance, it might embody pleasant signposts like “Here is the arduous half” or “The following step is simple to know”). Nevertheless, to be truthful, a proper, impartial tone is usually applicable when presenting proof. Other than the tone, there may be little query from a mathematical viewpoint.
abstract
This text lined intermediate to superior mathematical reasoning and downside fixing with OpenAI’s newest mannequin, GPT-5. The accuracy and systematic depth of the mannequin was showcased by means of a number of examples, adopted by a quick dialogue of the outcomes and the strategy used to generate them.
