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Mathematician

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March 29, 2024 Updated March 31, 2025 17 minute read

Mathematician

A mathematician is a professional who uses mathematical theory, computational techniques, algorithms, and the latest computer technology to solve economic, scientific, engineering, physics, and business problems. At its core, mathematics is the study of quantity, structure, space, and change. Mathematicians explore these concepts, seeking out patterns, formulating new conjectures, and establishing truth through rigorous deduction from appropriately chosen axioms and definitions.

Working as a mathematician can be deeply engaging, offering the chance to tackle complex abstract problems or apply mathematical principles to tangible real-world challenges. The field provides intellectual stimulation through logical reasoning, pattern discovery, and the elegance of mathematical proof. Furthermore, the skills developed are highly transferable, opening doors in diverse sectors like technology, finance, research, and education.

Overview of a Mathematician's Role

Understanding the world through the lens of mathematics involves abstraction, logical reasoning, and quantitative analysis. Mathematicians are experts in these areas, applying their skills across a surprisingly broad spectrum of activities.

Defining the Role and Responsibilities

Mathematicians develop new mathematical principles, deepen understanding of existing principles, and apply mathematical techniques to solve practical problems. Their work often involves creating mathematical models to analyze data, interpret phenomena, and make predictions. Daily tasks might include reading and understanding research papers, developing proofs, writing algorithms, analyzing data sets, collaborating with scientists or engineers, and communicating findings through reports or presentations.

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Salaries for Mathematician

City
Median
New York
$147,000
San Francisco
$166,000
Seattle
$141,000
See all salaries
City
Median
New York
$147,000
San Francisco
$166,000
Seattle
$141,000
Austin
$123,000
Toronto
$153,000
London
£71,000
Paris
€60,000
Berlin
€78,000
Tel Aviv
₪85,000
Singapore
S$133,000
Beijing
¥120,000
Shanghai
¥510,000
Shenzhen
¥246,000
Bengalaru
₹362,000
Delhi
₹352,000
Bars indicate relevance. All salaries presented are estimates. Completion of this course does not guarantee or imply job placement or career outcomes.

Path to Mathematician

Take the first step.
We've curated 24 courses to help you on your path to Mathematician. Use these to develop your skills, build background knowledge, and put what you learn to practice.
Sorted from most relevant to least relevant:

Reading list

We've selected some books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Mathematician.
Is an excellent starting point for high school students and undergraduates new to number theory. It introduces fundamental concepts like prime numbers, congruences, and Diophantine equations in an accessible and engaging way with numerous examples. It's widely used as a textbook and is great for solidifying basic understanding before moving to more rigorous texts.
Is an excellent starting point for gaining a broad understanding of Fourier Analysis, requiring only basic knowledge of real analysis. It is part of a highly regarded series and is frequently used as a textbook in undergraduate and graduate programs. While providing a solid foundation, it also delves deep enough to be valuable for those seeking to deepen their understanding.
A companion to 'Classical Fourier Analysis,' this book dives into more advanced and contemporary topics in the field. It is essential for graduate students and researchers looking to explore the frontiers of Fourier Analysis and its connections to other areas of mathematics. It requires a strong background in classical Fourier analysis and real analysis.
Considered a standard graduate text in algebraic number theory, this book provides a comprehensive and modern treatment of the subject. It's crucial for students specializing in this area and serves as an excellent reference for researchers.
This graduate-level textbook is excellent for deepening understanding and exploring contemporary topics in Fourier Analysis. It provides a rigorous treatment of the subject, building upon a solid foundation in real analysis. It is considered a standard reference for graduate students and researchers in harmonic analysis.
Is essential for understanding the crucial aspects of accuracy and stability in numerical computations. It delves into the potential pitfalls of floating-point arithmetic and provides rigorous analysis of algorithms. It's a must-read for anyone serious about the reliability of numerical methods, particularly at the graduate level and for professionals.
Focusing specifically on numerical optimization techniques, this book comprehensive and up-to-date resource for graduate students and researchers. It covers a wide range of methods and is considered a leading text in the field of continuous optimization. It is highly relevant for those interested in contemporary topics in numerical methods.
This classic book in algebraic number theory covers topics related to factors and their factorization in algebraic number fields.
A classic in the field of harmonic analysis, this book provides a deep understanding of Fourier analysis in a multi-dimensional setting. It is suitable for graduate students and researchers and foundational text for those pursuing advanced topics in the field. It requires a solid background in real analysis.
This well-written textbook suitable for advanced undergraduates, offering a broad understanding of Fourier series and integrals with a focus on applications in physics and engineering. It covers topics like special functions and generalized functions, making it a valuable reference tool. It provides a solid foundation in the subject and its practical uses.
A comprehensive and popular textbook for undergraduates, this book covers a wide range of topics in elementary number theory, including cryptography and primality testing. Its clear exposition and wealth of exercises make it highly effective for solidifying understanding and it is frequently used in academic settings.
A comprehensive and advanced graduate textbook covering modern analytic number theory. This key reference for researchers and students specializing in this area, providing deep insights and covering a wide range of topics.
Provides a comprehensive overview of exponential functions, including their properties, graphs, and applications in various fields such as science, economics, and finance.
An excellent introduction to elliptic curves, a topic of major importance in modern number theory and cryptography. Suitable for advanced undergraduates and beginning graduate students, this book is crucial for understanding contemporary research areas.
Provides a rigorous and comprehensive treatment of numerical analysis, suitable for advanced undergraduate and graduate students. It delves into the mathematical theory behind the methods and strong resource for deepening understanding. It is often used as a textbook in mathematics departments.
A classic and comprehensive treatise on trigonometric series, this two-volume set foundational work in harmonic analysis. It is suitable for advanced graduate students and researchers seeking a deep and rigorous understanding of the theoretical aspects of Fourier series. It cornerstone reference in the field.
This textbook popular choice for undergraduate numerical analysis courses, offering a balanced introduction to the theory and application of numerical methods. It includes a good selection of topics and is known for its clear presentation, making it suitable for students gaining a broad understanding.
A classic graduate textbook covering the basic material of classical algebraic and analytic number theory. While challenging, it provides a strong foundation for advanced topics and valuable reference for graduate students and professionals.
Key resource for those wanting to deepen their understanding of numerical linear algebra, a critical component of many numerical methods. It covers both theoretical aspects and practical implementation, including the impact of modern computer architectures. It is well-suited for graduate students and researchers.
Presents the real variable methods of Fourier analysis and great introductory book for graduate students to the subject on Euclidean spaces. It builds upon basic knowledge of Lebesgue integrals and functional analysis and is highly recommended for its clear presentation of classical topics and singular integral operators.
Is particularly well-suited for engineering and science students due to its strong emphasis on applications and its integration with MATLAB. It helps solidify understanding by demonstrating how numerical methods are used to solve practical problems. It's a popular textbook in applied fields.
Bridges the gap between elementary and advanced number theory, making it suitable for strong undergraduates and graduate students. It covers a wide array of topics, including algebraic number theory and modular forms, providing a solid foundation for more contemporary areas.
Offers a balanced approach to numerical methods, covering both the theoretical aspects and computational implementation. It is well-regarded for its clear exposition and comprehensive coverage of topics typically found in undergraduate courses. It serves as a good textbook and reference for solidifying understanding.
A widely used and comprehensive textbook for undergraduate number theory courses. It covers a broad range of topics with rigor and clarity, making it excellent for gaining a solid understanding and providing a good foundation for further study.
Focusing on numerical methods for partial differential equations (PDEs), this book covers essential techniques like finite difference, finite element, and finite volume methods. It's a valuable resource for students and researchers in fields where PDEs are central, such as physics and engineering. It's suitable for those looking to apply numerical methods to more complex problems.
A well-regarded introduction to algebraic number theory focusing on number fields. is accessible to advanced undergraduates and beginning graduate students and is praised for its clear exposition and problem sets.
Integrates computational aspects with theoretical number theory and algebra, making it highly relevant for students interested in cryptography and computational mathematics. It's suitable for advanced undergraduates and graduate students and is available for free online.
A concise and influential graduate-level text covering fundamental results in number theory, including quadratic forms and modular forms. It's a classic that is essential for students looking to delve into more advanced and abstract areas of the subject.
Considered a classic in the field, this book provides a rigorous introduction to numerical analysis with a strong theoretical foundation. It's suitable for advanced undergraduates and graduate students looking to deepen their understanding of the mathematical underpinnings of numerical methods. While not the most recent, its depth and clarity make it a valuable reference.
Covers a wide range of topics in algebraic geometry and arithmetic, including factorization of algebraic varieties and curves.
Offers a meaningful introduction to applied mathematics through Fourier Analysis, suitable for gaining a broad understanding. It develops a unified theory of discrete and continuous Fourier analysis and the Fast Fourier Transform (FFT), with applications in various fields. It serves as a valuable reference for students and professionals in science and engineering.
Provides an excellent introduction to abstract harmonic analysis, building upon the concepts of Fourier analysis on Euclidean spaces and groups. It is suitable for graduate students interested in the more abstract and theoretical side of the subject. It requires a background in measure theory and functional analysis.
A classic in abstract harmonic analysis, this book extends Fourier analysis to locally compact abelian groups. It is essential for graduate students and researchers interested in the theoretical underpinnings and generalizations of Fourier analysis. It requires a solid background in functional analysis and abstract algebra.
Takes a broad view of computational science and engineering, integrating numerical methods with applications in various fields. It is known for its clear explanations and covers topics like linear algebra, differential equations, and optimization. It's valuable for gaining a broad understanding of how numerical methods fit into a larger computational context.
Specializes in numerical methods for ordinary differential equations (ODEs), a key area within numerical analysis. It provides a clear and comprehensive treatment of the subject, suitable for students looking to deepen their understanding of this specific domain. It is often used in courses focusing on numerical ODEs.
Considered a monumental classic in number theory, this book offers a vast and deep exploration of the subject, suitable for advanced undergraduates and graduate students. While older, its breadth and rigor make it invaluable for deepening understanding and as a comprehensive reference.
This graduate-level book provides a comprehensive introduction to cyclotomic fields, a key area in algebraic number theory with connections to Fermat's Last Theorem. It's essential for students pursuing advanced topics in the field.
Offers a concise and insightful introduction to spectral methods, which are powerful techniques for solving differential equations. Its use of MATLAB makes it practical for implementation. It's suitable for graduate students and researchers interested in advanced numerical techniques. It provides a good entry point into a more specialized area of numerical methods.
As the title suggests, this book aims to be accessible to students new to the subject. It provides a clear and gentle introduction to the core concepts of numerical analysis, making it suitable for high school or early undergraduate students seeking a broad understanding.
A graduate-level text focusing on modular forms and Dirichlet series, key tools in analytic number theory. is essential for students delving deeper into the analytic aspects and their connections to other areas of mathematics.
Focuses on the practical implementation of numerical methods using Python. It's excellent for students and professionals who want to translate theoretical knowledge into working code. It complements theoretical texts and is highly relevant given the prevalence of Python in scientific computing.
Provides a clear introduction to the methods of analytic number theory, suitable for advanced undergraduates and graduate students. It covers topics like Dirichlet series, zeta functions, and the distribution of primes, essential for understanding this branch of number theory.
This advanced textbook provides a comprehensive treatment of analytic number theory and covers topics related to the distribution of factors.
Explores the relationship between algebraic geometry and arithmetic curves and covers topics related to factorizations of curves.
Explores the properties and applications of exponential functions and inequalities, with a focus on solving complex problems.
A true classic in the field, this book emphasizes the 'why' behind numerical methods, focusing on gaining insight rather than just numbers. While older, its fundamental principles and unique perspective remain highly relevant and valuable for anyone seeking a deep understanding. It's more valuable as additional reading for historical context and foundational concepts.
While primarily a cryptography book, this text provides a solid foundation in the number theory essential for modern cryptographic systems like RSA. It's highly relevant for students interested in applied number theory and assumes a certain mathematical maturity, suitable for advanced undergraduates and graduate students.
This textbook covers the fundamentals of numerical analysis and its applications, suitable for undergraduate students in science and engineering. It provides a detailed discussion on topics including difference equations, Fourier series, and finite element methods, offering a broad understanding of the subject.
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