Every mathematician should know how to think logically, solve problems, and communicate mathematical ideas effectively.
An expanded response to your question
Mathematics is a vast subject with everevolving concepts and theories. However, there are several fundamental skills and knowledge that every mathematician should possess to be successful in their field. In addition to logic, problemsolving, and communication skills, a mathematician should have:

Strong foundation: Every mathematician should have a strong foundation in basic mathematical concepts such as algebra, geometry, trigonometry, and calculus. These concepts are essential for advanced mathematics and theoretical research.

Critical thinking: A mathematician should be able to analyze, interpret, and evaluate data to develop new theories and solve complex problems. Critical thinking allows them to approach new problems with a fresh perspective and develop innovative solutions.

Research skills: Like other scientists, mathematicians rely on research to develop new theories and models. They should have the necessary skills to conduct research, analyze data, and interpret results.

Programming skills: In today’s digital age, programming skills are essential for mathematicians. Programming allows mathematicians to develop algorithms and models, perform simulations and analyze large amounts of data.

Computational skills: Mathematics often involves complex calculations and computations. Mathematicians should have a solid understanding of computational methods and tools to aid them in their research.
In the words of mathematician John von Neumann, “In mathematics, you don’t understand things, you just get used to them.” Mathematics is not just about memorizing formulas and equations but about developing a deep understanding of concepts and theories to apply them in solving new problems.
Here is a table summarizing the essential skills every mathematician should possess:
Skills  Description 

Logical reasoning  Ability to think logically and critically 
Problemsolving  Ability to solve complex problems through analysis and logic 
Effective communication  Ability to effectively communicate mathematical ideas 
Strong foundation  Solid understanding of basic mathematical concepts 
Research skills  Ability to conduct research, analyze data, and interpret 
Programming skills  Ability to program and develop algorithms 
Computational skills  Ability to perform complex calculations and computations 
In conclusion, every mathematician should possess a strong foundation in basic mathematical concepts and skills like critical thinking, research, programming, and effective communication. These skills are essential to succeed in the field and develop new theories and models.
You might discover the answer to “What should every mathematician know?” in this video
The Map of Mathematics video explains the interconnectedness of different areas of mathematics and how they are applied to solve problems in other fields. It also discusses the foundations of mathematics and how it does not have a complete and consistent set of axioms.
Other answers to your question
aspects of mathematical thought: Geometry, Algebra, Computation, and The mathematical intuition(s) derived from Physics.
I think every mathematician should know the following (in no particular order):
 Pythagorean Theorem.
 Summing ∑ k = 1 n k using Gauss’ triangle trick.
 Irrationality of 2 by proof without words.
 Niven’s proof of the irrationality of π.
 Uncountability of the Reals by Cantor’s Diagonal Argument.
 Denumerability of the Algebraics by Heights and Counting Roots.
 Infinitude of primes by both Euclid’s proof and Euler’s proof.
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What do you need to know to be a mathematician?
As a response to this: Most colleges and universities have bachelor’s degree programs in mathematics. Courses usually include calculus, differential equations, and linear and abstract algebra. Mathematics students also commonly take courses in a related field, such as computer science, physics, or statistics.
Who is the No 1 mathematician in the world?
The answer is: 1. Pythagoras. The life of the famous Greek Pythagoras is somewhat mysterious. Probably born the son of a seal engraver on the island of Samos, Pythagoras has been attributed with many scientific and mathematical discoveries in antiquity.
What makes a mathematician a good mathematician?
As a response to this: To develop fluency in performing written and mental calculations and mathematical techniques. To develop the ability to reason, generalise and make sense of solutions through explanations. To be able to develop a broad range of skills in using and applying mathematics.
What is the most important skill in math?
Answer: Mental Arithmetic is undoubtedly the most important mathematical skill for everyday life as it is the process of doing calculations in your head, without any form of help from external tools such as paper and pencil, calculator, etc.
What should a mathematician know?
As an answer to this: I think every mathematician should know the following (in no particular order): Pythagorean Theorem. Summing ∑ k = 1 n k using Gauss’ triangle trick. Irrationality of 2 by proof without words. Niven’s proof of the irrationality of π. Uncountability of the Reals by Cantor’s Diagonal Argument.
Should I read men of mathematics?
Response: Men of Mathematics by E. T. Bell; the best biographies on mathematicians that I have ever read, this two volume book is still in print and is still a classic. You should definitely read this.
How can I improve my math skills in real life?
Simply drawing a diagram of what’s happening, or organizing the information in a chart or table will give you a better understanding of the problem, and possibly help you spot patterns or information that you otherwise would have missed. 3. Estimate In terms of real life application, this is one of the most overlooked math skills.
Which math book should I read?
Answer: You should definitely read this. What is Mathematics? by Corant and Robbins; written by two master mathematicians, this book contains everything that you have ever encountered in math and much more. Ideally suited for any young student eager to see the world beyond the school curriculum.
What should a mathematician know?
I think every mathematician should know the following (in no particular order): Pythagorean Theorem. Summing ∑ k = 1 n k using Gauss’ triangle trick. Irrationality of 2 by proof without words. Niven’s proof of the irrationality of π. Uncountability of the Reals by Cantor’s Diagonal Argument.
Should I read men of mathematics?
Answer will be: Men of Mathematics by E. T. Bell; the best biographies on mathematicians that I have ever read, this two volume book is still in print and is still a classic. You should definitely read this.
What skills do you need to be a good math student?
Response will be: Reading Comprehension — Understanding written sentences and paragraphs in work related documents. Active Learning — Understanding the implications of new information for both current and future problemsolving and decisionmaking. Mathematical Reasoning — The ability to choose the right mathematical methods or formulas to solve a problem.
How can I improve my math skills in real life?
Simply drawing a diagram of what’s happening, or organizing the information in a chart or table will give you a better understanding of the problem, and possibly help you spot patterns or information that you otherwise would have missed. 3. Estimate In terms of real life application, this is one of the most overlooked math skills.