Numeracy in Large Language Models: Fundamental Limitations and Paths to Improvement

Large language models (LLMs) achieve strong results on mathematical reasoning benchmarks yet remain unreliable on elementary numerical tasks, including magnitude comparison, large-integer arithmetic, fractions, and scientific notation. This survey examines basic numerical understanding as a capability distinct from high-level mathematical reasoning. We propose the Numerical Grounding Framework (NGF), which decomposes numeracy into Representational Grounding (RG), mapping numeral forms to value, magnitude, and equivalent representations, and Procedural Grounding (PG), executing arithmetic operations in accordance with their mathematical definitions. Using NGF, we organize recent diagnostic benchmarks, failure modes, structural explanations, and mitigation strategies. We review evidence concerning tokenization, positional encoding, embedding geometry, and pretraining-data distribution. We also apply NGF in a coordinated evaluation of three frontier model families across Number Cookbook, NumericBench, and GSM-Symbolic, comparing atomic, contextual, and reasoning-assisted numeracy. Architectural interventions such as digit-aware tokenization and Abacus Embeddings can improve models trained from scratch but are generally unavailable to users of pretrained systems, for whom supervised fine-tuning, reasoning scaffolds, and external tools are more practical. We conclude with deployment recommendations and research directions for more reliable numerical behavior in foundation models.

Nothing from Something: Can a Language Model Discover 0?

AI systems based on artificial neural networks are being developed with aspirations of pushing the boundary of human mathematical knowledge. A key question for these systems is how much they can reach beyond their training data. Mathematical discovery requires a strong form of out of distribution generalization; the ability to hypothesize genuinely new - and potentially logically more powerful - mathematical structures. It has been hypothesized that language abilities support such generalizations in human cognition. In this work, we use simple arithmetic as a case study for examining how modern AI models could expand their mathematical horizons, evaluating whether these models can independently discover the concept of "zero". We show that (1) language models of a GPT-2 size are unable to perform this generalization at test time regardless of language pretraining, but (2) models can improve substantially after training on tens or hundreds of examples of zero. Additionally, we find that language pretraining reduces the number of required examples by approximately $50\%$, showing that language abilities can scaffold mathematical discovery in neural models.

Quotient Tree Arithmetic: Deferred-Division Computation with Bounded Symbolic Depth and Cross-Subtree Cancellation

We introduce Quotient Tree Arithmetic (QTA), a computational substrate in which values are represented as deferred quotient pairs (N, D) whose ratio is evaluated lazily at a designated materialization boundary. The framework applies to any domain: IEEE 754 doubles used as exact integer containers give exact rational arithmetic within the 2^53 exactness window; arbitrary IEEE doubles extend coverage to transcendental values including machine learning activations such as exp(x) and sqrt(x). Three structural theorems underpin QTA. (1) Bounded Depth Growth: each arithmetic operation increases tree depth by at most 1, giving O(m) tree size after m operations with no combinatorial explosion. (2) Cross-Subtree Cancellation: subtrees appearing in both numerator and denominator positions cancel via reference identity without arithmetic, including transcendental values computed once and shared. (3) Deferred Stability: a single IEEE division at the materialization boundary introduces at most one-half ULP of rounding error, versus O(m) ULP for eager evaluation. For machine learning training, QTA provides: structural prevention of gradient underflow to zero; O(1)-cost gradient computation via chain-rule tape collapse when intermediate activations are reference-identical; shared-weight batch compression reducing DAG storage from O(BLd) to O(L+Bd) for a batch of B examples through L layers; and tracked factor cancellation replacing O(log n) GCD with O(1) trial division when denominators are known. We propose a vectorized hardware normalization instruction (RatCleanup) for SIMD-parallel rational pair reduction. The algebraic foundation is the localization of a ring at its multiplicative set, connecting QTA to algebraic structure theory while grounding it in hardware-native IEEE arithmetic.