Kolmogorov complexity asks whether a string can be outputted by a Turing Machine (TM) whose description is shorter. Analogously, a real number is considered computable if a Turing machine can generate its decimal expansion. The modern -approximation definition of computability, widely used in practical computation, ensures that computable reals are constructively closed under addition. However, Turing’s original 1936 digit-by-digit notion, which demands the direct output of the digit, presents a stark divergence. Though the set of Turing-computable reals is not constructively closed under addition, we prove that a Turing machine capable of computing non-constructively exists. The core constructive computational barrier arises from determining the ones digit of a sum like . This particular example is ambiguous because both and are legitimate decimal representations of the same number. However, if any of the infinite number of 3s in the first term is changed to a 2 (e.g., ), the sum’s leading digit is definitely zero. Conversely, if it is changed to a 4 (e.g., ), the leading digit is definitely one. This implies an inherent undecidability in determining these digits. Recent papers and our work address this issue. Hamkins provides an informal argument, while Berthelette et al. present more complicated formal proof, and our contribution offers a simple reduction to the Halting Problem. We demonstrate that determining when carry propagation stops can be resolved with a single query to an oracle that tells if and when a given TM halts. Because a concrete answer to this query exists, so does a TM computing the digits of , though the proof is non-constructive. As far as we know, the analogous question for multiplication remains open. This, we feel, is an interesting addition to the story. This reveals a subtle but significant difference between the modern -approximation definition and Turing’s original 1936 digit-by-digit notion of a computable number, as well as between constructive and non-constructive proof. This issue of computability and numerical precision ties into algorithmic information and Kolmogorov complexity.

In this paper we study representations of real numbers in a numeral system with the base $a>1$ and alphabet (digits set) $A\equiv\{0,1,...,r\}$, $a-1<r\in N$ given by \[x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{a^n}\equiv \Delta^{r_a}_{\alpha_1\alpha_2...\alpha_n...}, \alpha_n\in A.\] Since the alphabet is redundant the numbers from the interval $[0;\frac{r}{a-1}]$ have not a single representation and can even have a continuous set of different representations. We describe the geometry (topological and metric properties) of such representations (the $r_a$-representations) in terms of cylinders defined by \[\Delta^{r_a}_{c_1c_2...c_m}= \{x: x=\Delta^{r_a}_{c_1c_2...c_ma_1a_2...a_n...}, a_n\in A\},\] We analyze their properties in detail, including the specific nature of overlaps. We present results on the structural, variational, topological, metric and partially fractal properties of the function defined by \[f\left(x=\sum_{n=1}^{\infty}\frac{\alpha_n}{(r+1)^n}\right)= \Delta^{r_a}_{\alpha_1\alpha_2...\alpha_n...},\alpha_n \in A.\] We prove the function is continuous at all points of the interval $[0,1]$ that have a unique representation in the classical numeral system on the base $r+1$ and prove the function is discontinuous at points of a countable everywhere dense set in $[0,1]$. Furthermore, we show that the function is nowhere monotonic and has unlimited variation. In the particular case $r=1$ and $a=\frac{1+\sqrt{5}}{2}$, we specify fractal level sets with Hausdorff--Besicovitch dimension not less than $-\log_a2$.

In the paper we consider a continuum class functions defined by terms of the $Q_s$-representation of real numbers on the segment $[0;1]$, which generalizes the classical $s$-adic representation. The dependence of the $n$-th digit of the $Q_s$-representation of the function value is specified by a finite function $\varphi_n(a_n,\alpha_n)$ of two variables, whose arguments are the corresponding $Q_s$-digits $\alpha_n(x)$ and $a_n(a)$ of the input $x$ and the parameter $a$, respectively. We prove continuity of each function in this class at every $Q_s$-unary number, i.e., at points possessing a unique $Q_s$-representation. Necessary and sufficient conditions for continuity on the entire domain are established. Conditions involving the digits of the parameter $a$ and the sequence of defining functions $(\varphi_n)$, under which the function $f_a$ admits finite or continuum cardinality level sets are obtained. For particular cases ($s=2$), we study integral and differential properties, as well as the fractal properties of the sets of values. Using the self-similarity properties of the function graph and the established connection between the functions under consideration and the inversor of digits of the $Q_2$-representation of numbers, we compute the Lebesgue integral of these functions. Furthermore, we identify a subclass of functions that are piecewise singular or singular on intervals; that is, continuous non-constant functions whose derivative is zero almost everywhere in the sense of Lebesgue measure.

The article presents a methodology for designing a benchmark to assess numerical reasoning skills in Large Language Models (LLMs). In the context of LLMs, numerical reasoning is defined as a model’s ability to correctly interpret, process, and utilize numerical information in text, including understanding magnitudes and relations between numbers, performing arithmetic operations, and generating numerals accurately in its outputs. The proposed methodology is based on decomposing applied tasks and enables targeted evaluation of specific facets of numerical reasoning using tasks that involve numerals. Particular attention is paid to the representation of numbers in textual prompts to LLMs, as this factor directly affects the quality of the final output. The need for rigorous assessment of LLMs’ numerical reasoning stems from its critical role across a wide range of text-centric applications, including automated summarization, generation of analytical reports, extraction and interpretation of quantitative data, and conversational systems operating on financial, scientific, or technical information. Drawing on an analysis of state-of-the-art LLM evaluation approaches, core principles for constructing evaluation benchmarks are formulated with an emphasis on generality and real-world applicability. In accordance with the proposed methodology, the MUE (Math Understanding Evaluation) benchmark is introduced; it comprises five test suites, each designed to assess a distinct aspect of LLM numerical reasoning. A comparative evaluation of popular LLMs is conducted, leading models are identified, and the strengths and weaknesses of their numerical reasoning are characterized. The findings are intended to inform LLM developers in refining architectures and training strategies, and to guide end users and integrators in selecting an optimal model for applied projects.

The Board of Educational Affairs is pleased to bestow the Award for Distinguished Contributions of Applications of Psychology to Education and Training. This award acknowledges psychologists who contribute to new teaching methods or solutions to learning problems through the use of research findings or evidence-based practices. The 2025 recipient of the APA Award for Distinguished Contributions of Applications of Psychology to Education and Training is Clarissa A. Thompson. Thompson's long-time passion for mathematics education and the translational applications of best practice principles and educational theory to support teacher training, parental support, and student understanding of mental representations of numbers makes her a fitting recipient of this award. Her generative and far-reaching research agenda spans multiple social sciences disciplines-cognitive, developmental, and educational psychology-whilst also prioritizing collaboration with scholars in special education, health psychology, and sociology to maximize impact across fields. Through consistent success in securing extramural funding from government entities such as the U.S. Department of Education, Institute of Education Sciences and the National Science Foundation, Thompson develops workshops and educational intervention projects that prioritize the needs of children and adult learners in the classroom, addressing their attitudes, anxieties, and confidence in mathematics learning, numerical cognition, and problem-solving. Beyond exemplary research, Thompson's influence extends through her leadership in the scientific community through journal editorship, and mentorship commitment to early-career and diverse scholars in educational and child psychology. She strives to create meaningful change in educational practice research whilst also nurturing the next generation of researchers. (PsycInfo Database Record (c) 2025 APA, all rights reserved).

Abstract To enrich curricular materials for quantum computing education, we show how Grover's algorithm can be used to find approximate solutions to algebraic and transcendental equations. This significantly expands the range of exercises based on Grover's algorithm, which typically solves problems with exact binary solutions. These exercises teach and unify a diverse skillset, including binary representation of fractional and negative numbers, implementation of Boolean operations with quantum gates, and comprehensive design of quantum circuits to implement Grover's algorithm.

The ability of a machine to store numbers and perform calculations on them is fundamental to computing. Many novice and expert users wrongly assume that number representation is a long-solved problem. Modern computers use binary encodings, and nothing more needs to be said, right? Wrong! For example, irrational numbers like pi are approximated; rational numbers like 2.0 may overflow when multiplied by a larger number; and (1/x)*x should equal 1.0 but often does not! Modern floating point evolved rather than being designed! John Gustafson has studied this and related problems of computing for many decades. He is the inventor of posits---a superior way to represent numbers in binary. In this two-part article, Gustafson surveys the history of machine number representations and illustrates by example how even the most modern number representations used today often fail. The IEEE 754 floating point standard is on shaky grounds, lacking a sound basis, and yet we persist in using it. How did we get here, and where might it lead? This is the fascinating story of computer numbers. --- Ted G. Lewis , Ubiquity senior editor

Neuromorphic hardware systems allow efficient implementation of artificial neural networks (ANNs) across various applications that demand high data throughput, reduced physical size, and low energy consumption. Field-Programmable Gate Arrays (FPGAs) possess inherent features that can be aligned with these requirements. However, implementing ANNs on FPGAs also presents challenges, including the computation of the neuron activation functions, due to the balance between resource constraints and numerical precision. This paper proposes a resource-efficient hardware approximation method for the sigmoid function, utilizing a combination of first- and second-degree polynomial functions. The method aims mainly to minimize the approximation error. This paper also evaluates the obtained results against existing techniques and discusses their significance. The experimental results showed that, although the proposed method mainly aimed to minimize the approximation error, it also had lower hardware resource usage than several of the most closely related works. Using 16-bit fixed-point number representation, the absolute mean error was 1.66×10−3 by using 0.04% of the logic blocks and 3.21% of the DSP blocks in a Ciclone V 5CGXFC7C7F23C8 FPGA Device.

While numeration systems are found in almost every human society, they also vary strikingly around the globe. One important feature of many systems is being structured around a base. The presence, format and size of a base have implications for how representations of numbers are composed, conceptualized and used. The numerical cognition literature is rife with claims about which bases prevail, with sweeping generalizations on their origins and evolution. Yet these claims are rarely scrutinized, and research on numeration systems is plagued by a surprising lack of consensus on what a base is. This theme issue brings together scholars from the cognitive, social and behavioural sciences for a comprehensive overview of bases, aimed at creating common ground for communication and collaboration across disciplines. Contributions include (i) proposals for conceptual clarification and consistency, aided by a tool for visualizing system structure; (ii) reports on how bases are realized across different representational formats, cultures and time, in search of patterns and evolutionary trajectories; and (iii) analyses of cognitive implications and cultural imprints of bases. The main goal of this endeavour is to help build an integrative synthesis of both theorizing and findings on the cultural evolution of a key cognitive tool.This article is part of the theme issue ‘A solid base for scaling up: the structure of numeration systems’.

Hybrid Number Triplets and a New Polar Representation of Hybrid Numbers

Abstract Gcd-graphs over the ring of integers modulo n are a natural generalization of unitary Cayley graphs. The study of these graphs has foundations in various mathematical fields, including number theory, ring theory, and representation theory. Using the theory of Ramanujan sums, it is known that these gcd-graphs have integral spectra; i.e., all their eigenvalues are integers. In this work, inspired by the analogy between number fields and function fields, we define and study gcd-graphs over polynomial rings with coefficients in finite fields. We establish some fundamental properties of these graphs, emphasizing their analogy to their counterparts over ${\mathbb {Z}}.$

The subject matter of the article is the analysis of the artificial intelligence models for generating 3D objects based on a text query with different formats for representing numerical parameters. The goal of the work is to evaluate and compare the effectiveness of the Hunyuan3D 2.0, Michelangelo and SDFusion models depending on the format of the text input queries using the following metrics: generation time, CLIP-Similarity and Chamfer Distance. The tasks to be solved are: 1) to conduct a systematic review of modern generative 3D models regarding their ability to process queries with numerical parameters presented in different formats (decimals, verbal representations, fractions); 2) to conduct a quantitative and qualitative evaluation of generative 3D models using the metrics of generation time, CLIP-Similarity and Chamfer Distance, taking into account the visual similarity to text descriptions and geometric similarity to Ground Truth objects; 3) to analyse the obtained results of generative text-to-3D models evaluation for parameterized 3D objects generation tasks for further implementation in user interface applications. The methods used in this work are machine learning methods, methods of vector representation of text and images, statistical methods of evaluating results, and a heuristic method of forming text queries. The results show that there is no universal generative 3D model that is capable of creating objects that fully correspond to a parameterised text query. The proposed methodology enables the formation of input text sequences containing such numerical representations, including decimals, verbal representations or fractions, for further analysis of the accuracy of object generation. Conclusions. Similar efficiency of training 3D generative models in the joint latent space of text features containing numerical parameters and using datasets with precisely defined geometric characteristics of objects was confirmed. The results show that the Hunuyan3D-2.0 model is suitable for further research and modification to adapt the used methods to create personalised 3D objects with given numerical parameters, such as a prosthetic cover.

Abstract Plant‐available water and adequate soil aeration are two fundamental requirements for successful plant growth. These prerequisites have generally been assessed independently in relation to plant growth, with limited focus on their complementary and competing behavior in a soil–water matrix. In this study we introduce a corequisite index adopted from a complex number representation by linking soil–water (the real component) and soil–air (the imaginary component) as the orthogonal counterparts in an Argand diagram. The new corequisite index constitutes a soil–water component defined based on field capacity and permanent wilting point, and a soil–air component defined based on critical soil–gas diffusivity. To calibrate model parameters, the soil–water characteristics were measured in vadose soil profiles (0‐ to 60‐cm depth) from 48 replicate sites. Results revealed that the corequisite index, with its magnitude (0.5–1) and corequisite angle (0–30°) in the given range, provided the best combined soil water and aeration status for the selected soil. The majority of the selected soils were affected by insufficient aeration (gas diffusivity < 0.01) when at field capacity (drained to −10 kPa), requiring the soils to drain further (−50 to −100 kPa) to satisfy the corequisites. The derived soil aeration parameters showed promising relationships with measurable soil physical properties. We further recommend adopting a 15% volumetric soil air content as a general threshold for minimum soil aeration in the absence of measured soil–gas diffusivity data.

The article presents an analysis of the representation of the family images and family fertility in commercial advertising. In sociology the terms “need for children” and “desire to have children”, and the concepts of the average expected, the average ideal and the average desired number of children were introduced. Commercial advertising, as one of the forms of mass media, can provide a limited impact on the desire to have children through moderating the social norm. On the basis of the concept of the limited impact of mass media on the values and group ideas about the social norms, the representation of family models and number of children in modern commercial advertising, presented on most popular Russian TV channels were analyzed. On the basis of the empirical research, the models of family fertility in commercial advertising were presented. The family models demonstrated in advertising can be integrated into collective consciousness as the examples of group norms.

It is debated whether there is an abstract, format-independent representation of number in the human brain, eg whether "four" shares a neural representation with "4." Most previous studies have used magnitude to investigate this question, despite potential confounds with relative quantity processing. This study used the numerical property of parity. Electroencephalogram recordings were collected from participants performing a fixation-cross task, while viewing 20-s sequences of alternating even and odd Arabic numerals presented at 7.5Hz: responses to parity were selectively tagged at the asymmetry frequency of 3.75Hz. Parity asymmetry responses emerged significantly over the occipito-temporal (OT) cortex, and were larger than control asymmetry responses to isolated physical stimulus differences, replicating a previous study. Following 20-s adaptation to cross-font even numerals, larger parity responses were recorded over the right OT cortex, further supporting distinct representations of even/odd numbers; there was no corresponding control adaptation effect. Interestingly, adaptation to even canonical dot stimuli also produced significantly larger parity asymmetry responses; adaptation to even number words trended non-significantly. These results are in line with parity being processed automatically, even across formats. More generally, they suggest that parity is a useful means for probing abstract representation of number in the human brain.

The influence of embodied cognition on the perception and representation of numbers: A meta-analysis.

For two unlimiteWe say that two unlimited natural numbers x and y have the same order if x/y is appreciable. As a continuation of our subsequent paper [4], we study the representation of unlimited natural numbers as the sum of a limited integer and the product of at least two coprime unlimited natural numbers having the same order.

Brain-wide decoding of numbers and letters: Converging evidence from multivariate fMRI analysis and probabilistic meta-analysis.

It is demonstrated that neuromorphic materials designed for computational tasks can be effectively implemented by drawing an analogy with trigger-based systems built upon classical binary elements. Among the most promising approaches in this context are systems that perform computations based on the Residue Number System (RNS). A specific implementation of a trigger-based adder employing the proposed methodology is presented and tested through simulation modeling. This adder utilizes the representation of natural numbers as elements of a subtraction ring modulo P, where P is the product of Mersenne prime numbers. This configuration enables component-wise, independent execution of arithmetic operations. It is further shown that analogous trigger-based systems can be realized using recurrent neural network analogs, particularly those implemented with neuromorphic materials. The study emphasizes that it is possible to construct a neural network, especially one based on neuromorphic substrates, that can perform logical operations equivalent to those executed by conventional binary circuitry. A key challenge in the proposed approach lies in implementing an operation analogous to the carry mechanism employed in classical binary adders. An algorithm addressing this issue is proposed, based on the transition from computations modulo P to computations modulo 2P.

Early math difficulties can stem from children's failure to link nonsymbolic and symbolic representations of number. Well-designed math games have been consistently shown to help children make this link (Balladares et al., 2024), but research using fingers as numerical representations is more mixed (e.g., Moeller et al., 2011). In the present study, 95 kindergarteners (5-7 years old; estimated demographics based on publicly available data: 81% White; 7% Hispanic/Latinx; 2% Black; 1% Asian/Pacific Islander; 9% multiracial) participated in a 2-week intervention that targeted quantitative skills through one of three approaches to math games: finger-based games, manipulative-based games, or a combination of fingers and manipulatives. There was also a nonmath control group. Across all math-based conditions, children improved significantly from pretest to posttest in measures of early numeracy. This pattern of improvement was moderated by children's initial performance; children performing at or above grade level at pretest derived the most benefit from the manipulative-based games and combined conditions, while their peers who began the study below grade-level performance saw the largest improvements in the finger-based condition. The findings add nuance to a growing literature that seeks to understand the benefits of finger-based activities or numerical games on quantitative skills in 5- to 7-year olds. We underscore the practical and theoretical importance of considering student's baseline understanding of number and argue that fingers are an especially appropriate tool for children who are at the beginning of their numeracy journey, perhaps because they embody the connection between symbolic and nonsymbolic number in a single, pared-down representation. (PsycInfo Database Record (c) 2025 APA, all rights reserved).