The definition of a good mathematical problem is the mathematics it generates rather than the problem itself. — Andrew Wiles
Pure mathematicians just love to try unsolved problems - they love a challenge. — Andrew Wiles
Very often in mathematics the crucial problem is to recognize and discover what are the relevant concepts; once this is accomplished the job may be more than half done. — Israel Nathan Herstein
It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts. — Paul Halmos
Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field. — Paul Dirac
The problems of puzzles are very near the problems of life. — Erno Rubik
Do not worry about your difficulties in Mathematics. I can assure you mine are still greater. — Albert Einstein
Problems are challenges to creative minds. Without problems, there would be little reason to think at all. — Earl Nightingale
Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country. — David Hilbert
Mathematics my foot! Algorithms are mathematics too, and often more interesting and definitely more useful. — Doron Zeilberger
Pure mathematics is, in its way, the poetry of logical ideas. — Albert Einstein
Mathematicians aren't satisfied because they know there are no solutions up to four million or four billion, they really want to know that there are no solutions up to infinity. — Andrew Wiles
Programming is one of the most difficult branches of applied mathematics; the poorer mathematicians had better remain pure mathematicians. — Edsger Dijkstra
Mathematics is the supreme judge; from its decisions there is no appeal. — Tobias Dantzig
Short Mathematical Problems Quotes
The only way to learn mathematics is to do mathematics. — Paul Halmos
For the things of this world cannot be made known without a knowledge of mathematics. — Roger Bacon
Mathematics allows for no hypocrisy and no vagueness. — Stendhal
Mathematics is as much an aspect of culture as it is a collection of algorithms. — Carl Benjamin Boyer
A mathematician is a device for turning coffee into theorems. — Paul Erdos
The more I work and practice, the luckier I seem to get. — Gary Player
Logic is the hygiene the mathematician practices to keep his ideas healthy and strong. — Hermann Weyl
The biggest communication problem is we do not listen to undrstand. We listen to reply.
The introduction of numbers as coordinates by reference to the particular division scheme of the open one-dimensional continuum is an act of violence whose only practical vindication is the special calculatory manageability of the ordinary number continuum with its four basic operations. — Hermann Weyl
To arrive at the simplest truth, as Newton knew and practiced, requires years of contemplation. Not activity Not reasoning. Not calculating. Not busy behaviour of any kind. Not reading. Not talking. Not making an effort. Not thinking. Simply bearing in mind what it is one needs to know. — George Spencer-Brown
We need to tap the resource of current and retiring science and math professionals that have both content mastery and the practical experience to serve as effective teachers. — Cathy McMorris Rodgers
The problem with the world is that the intelligent people are full of doubts while the stupid ones are full of confidence.
It can be of no practical use to know that Pi is irrational, but if we can know, it surely would be intolerable not to know. — Edward Charles Titchmarsh
Affirmative action makes employers think, 'Black woman nuclear physicist? Hah! Probably let her into Harvard 'cause they were looking for a twofer. Bet she got C's in high school practical math. Give her a job in personnel.' — P. J. O'Rourke
Speaking of human computers, there is a guy named Art Benjamin, he's a human calculator. He says it's a skill he learned as a kid. Now he's a math professor at Harvey Mudd. He can find the square root of a six digit number in a few seconds. Practice. — Bill Nye
Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle. — Jacques Hadamard
Math Problem Quotes
It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts. — Arthur Conan Doyle
It is better to solve one problem five different ways, than to solve five problems one way. — George Polya
There are no solved problems; there are only problems that are more or less solved. — Henri Poincare
Be selective in your battles, don't make every problem a war.
Science, in the very act of solving problems, creates more of them. — Abraham Flexner
All the mathematical sciences are founded on relations between physical laws and laws of numbers. — James Clerk Maxwell
If there is a problem you can't solve, then there is an easier problem you can't solve: find it. — George Polya
Life is a journey to be experienced, not a problem to be solved.
So what should we say when children complete a task—say, math problems—quickly and perfectly? Should we deny them the praise they have earned? Yes. When this happens, I say, “Whoops. I guess that was too easy. I apologize for wasting your time. Let’s do something you can really learn from! — Carol S. Dweck
One must acknowledge with cryptography no amount of violence will ever solve a math problem. — Jacob Appelbaum
Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems. — Carl Pomerance
In mathematics, the art of asking questions is more valuable than solving problems. — Georg Cantor
Math Problems Quotes
Dealing with failure is easy: Work hard to improve. Success is also easy to handle: You've solved the wrong problem. Work hard to improve. — Alan Perlis
The aim (of education) must be the training of independently acting and thinking individuals who, however, see in the service to the community their highest life problem. — Albert Einstein
I think there are dozens or hundreds of different forms of creativity. Pondering science and math problems for years is different from improvising jazz. Something which seems to me remarkable is how unconscious the creative process is. You encounter a problem, but can't solve it. — Oliver Sacks
If it can be solved, theres no need to worry, and if it can't be solved, worry is of no use.
Learning through the arts reinforces critical academic skills in reading, language arts and math, and provides students with the skills to creatively solve problems. — Michelle Obama
The life of a mathematician is dominated by an insatiable curiosity, a desire bordering on passion to solve the problems he is studying. — Jean Dieudonne
A GREAT discovery solves a great problem but there is a grain of discovery in any problem. — George Polya
Be grateful for small things, big things, and everything in between. Count your blessings, not your problems.
Dear Math, please grow up and solve your own problems. I'm tired of solving them for you.
Always remember that you are absolutely unique. Just like everyone else. — Margaret Mead
We've lost something that's been with us for so long, and something that drew a lot of us into mathematics. But perhaps that's always the way with math problems, and we just have to find new ones to capture our attention. — Andrew Wiles
One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. — Leon Henkin
This paper gives wrong solutions to trivial problems. The basic error,however, is not new. — Clifford Truesdell
Mathematical Equations Quotes
Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality. — Nikola Tesla
In order to solve this differential equation you look at it until a solution occurs to you. — George Polya
The mathematical sciences particularly exhibit order symmetry and limitations; and these are the greatest forms of the beautiful. — Aristotle
Anyone can run away, it's super easy. Facing problems and working through them, that's what makes you strong.
It is more important to have beauty in one's equations than to have them fit experiment. — Paul Dirac
Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding. — William Thurston
If equations are trains threading the landscape of numbers, then no train stops at pi. — Richard Preston
Running from your problems is just adding another one to the list.
If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries. — Carl Friedrich Gauss
The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved. — Paul Dirac
Geometry is the science of correct reasoning on incorrect figures. — George Polya
Love and you shall be loved. All love is mathematically just, as much as the two sides of an algebraic equation. — Ralph Waldo Emerson
Problem Solving Quotes
A clever person solves a problem. A wise person avoids it. — Albert Einstein
Leadership is solving problems. The day soldiers stop bringing you their problems is the day you have stopped leading them. They have either lost confidence that you can help or concluded you do not care. Either case is a failure of leadership. — Colin Powell
Intelligence is what you use when you don't know what to do. — Jean Piaget
The distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers — Lev S. Vygotsky
Life is not a problem to be solved, but an experience to be had. — Alan Watts
If you only focus on your problems, that's all you'll ever see.
You may not always have a comfortable life and you will not always be able to solve all of the world's problems at once but don't ever underestimate the importance you can have because history has shown us that courage can be contagious and hope can take on a life of its own. — Michelle Obama
I would like to remind the black ministry and indeed all black people that God is not in the habit of coming down from heaven to solve people's problems on earth. — Steven Biko
We cannot solve our problems with the same thinking we used when we created them. — Albert Einstein
No problem can be solved from the same level of consciousness that created it. — Albert Einstein
Studying Mathematics Quotes
It is with children that we have the best chance of studying the development of logical knowledge, mathematical knowledge, physical knowledge, and so forth. — Jean Piaget
Your most important skill isn’t even what you majored in or even what you studied, it’s just knowing how to learn. If you have a good grasp of mathematics and if you like to read, there’s nothing you can’t learn on your own. — Naval Ravikant
I must study politics and war that my sons may have liberty to study mathematics and philosophy. — John Adams
Stay away from negative people. They have a problem for every solution,
I have to study politics and war so that my sons can study mathematics, commerce and agriculture, so their sons can study poetry, painting and music. — John Quincy Adams
Technology is applied science. Science is the study of nature. Mathematics is the language of nature. Philosophy is the root of mathematics. All tightly interrelated. — Naval Ravikant
Study microeconomics, game theory, psychology, persuasion, ethics, mathematics, and computers. — Naval Ravikant
It is a well-known experience that the only truly enjoyable and profitable way of studying mathematics is the method of "filling in details" by one's own efforts. — Cornelius Lanczos
It is clear that the chief end of mathematical study must be to make the students think. — John Wesley Young
Everything you've learned in school as "obvious" becomes less and less obvious as you begin to study the universe. For example, there are no solids in the universe. There's not even a suggestion of a solid. There are no absolute continuums. There are no surfaces. There are no straight lines. — Richard Buckminster Fuller
In school, my favorite subject was math. That's where I learned to count money. — French Montana
I never teach my pupils, I only attempt to provide the conditions in which they can learn. — Albert Einstein
I learned that despite having years and years of experience in math and computer science and so on, I didn't really know how to code until I formed a company. — Balaji Srinivasan
Geometry is the foundation of all painting. — Albrecht Durer
I'd never been a teacher before, and here I was starting my first day with these eager students. There was a shortage of teachers, and they had been without a math teacher for six months. They were so excited to learn math. — Andrew Shue
The only way to learn mathematics is to do mathematics. That tenet is the foundation of the do-it-yourself, Socratic, or Texas method. — Paul Halmos
The purpose of computing is insight, not numbers. — Richard Hamming
Obvious is the most dangerous word in mathematics. — Eric Temple Bell
A circle is a round straight line with a hole in the middle. — Mark Twain
Mathematical Quotes
The laws of nature are written by the hand of God in the language of mathematics. — Galileo Galilei
To make a great dream come true, the first requirement is a great capacity to dream; the second is persistence. — Cesar Chavez
The laws of nature are but the mathematical thoughts of God. — Euclid
No, it is a very interesting number, it is the smallest number expressible as a sum of two cubes in two different ways. — Srinivasa Ramanujan
Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty. — Archimedes
Mathematics is the language with which God has written the universe. — Galileo Galilei
If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. — John Von Neumann
I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind. — Lord Kelvin
Young man, in mathematics you don't understand things. You just get used to them. — John Von Neumann
We are all concerned about the future of American education. But as I tell my students, you do not enter the future - you create the future. The future is created through hard work. — Jaime Escalante
Complex Problems Quotes
Men will surrender to the spirit of the age. They will say that if they had lived in our day, faith would be simple and easy. But in their day, they will say, things are complex; the Church must be brought up to date and made meaningful to the day's problems. — Anthony of Padua
When you first start off trying to solve a problem, the first solutions you come up with are very complex, and most people stop there. But if you keep going, and live with the problem and peel more layers of the onion off, you can often times arrive at some very elegant and simple solutions. — Steve Jobs
For every problem, there is a solution that is simple, neat, and wrong. — H. L. Mencken
Moreover, such a body freed from nervous tension and over-fatigue is the ideal shelter provided by nature for housing a well-balanced mind that is always fully capable of successfully meeting all of the complex problems of modern living. — Joseph Pilates
Even the most complex problem is resolved in the end. Because if there's a question, there also is an answer. — Kim Joon
Some problems are so complex that you have to be highly intelligent and well informed just to be undecided about them. — Laurence J. Peter
The problem is that we attempt to solve the simplest questions cleverly, thereby rendering them unusually complex. One should seekthe simple solution. — Anton Chekhov
Like snowflakes, the human pattern is never cast twice. We are uncommonly and marvelously intricate in thought and action, our problems are most complex and, too often, silently borne. — Alice Childress
The older I get the more wisdom I find in the ancient rule of taking first things first. A process which often reduces the most complex human problem to a manageable proportion. — Dwight D. Eisenhower
People who work together will win, whether it be against complex football defenses, or the problems of modern society. — Vince Lombardi
Math Education Quotes
The most important thing we can do is inspire young minds and to advance the kind of science, math and technology education that will help youngsters take us to the next phase of space travel. — John Glenn
Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency. — Rene Descartes
Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful.
If you don’t see why, someone can’t tell you. I know numbers are beautiful.
If they aren’t beautiful, nothing is. — Paul Erdos
When you can measure what you are speaking about, and express it in numbers, you know something about it. — Lord Kelvin
One's work may be finished someday, but one's education never. — Alexandre Dumas
But just as much as it is easy to find the differential [derivative] of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not. — Johann Bernoulli
Beauty in mathematics is seeing the truth without effort. — George Polya
Scientific research and other studies have demonstrated that arts education can enhance American students' math and language skills and improve test scores which in turn increase chances of higher education and good jobs in the future. — Thad Cochran
The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization. — John G. Kemeny
In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generations adds a new story to the old structure. — Hermann Hankel
In mathematics, if I find a new approach to a problem, another mathematician might claim that he has a better, more elegant solution. In chess, if anybody claims he is better than I, I can checkmate him. — Emanuel Lasker
I tried to fit it in with some previous broad conceptual understanding of some part of mathematics that would clarify the particular problem I was thinking about. — Andrew Wiles
Thus you see, most noble Sir, how this type of solution to the Königsberg bridge problem bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle. — Leonhard Euler
Bitcoin is a solution out of that problem [money printing]. It’s a system change. — Jeff Booth
Neumann, to a physicist seeking help with a difficult problem: Simple. This can be solved by using the method of characteristics. Physicist: I'm afraid I don't understand the method of characteristics. Neumann: In mathematics you don't understand things. You just get used to them. — John Von Neumann
All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers. — James C. Maxwell
A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution. — David Hilbert
Can’t be starting all these problems if you cannot solve them. — Cardi B
In the meantime, most noble Sir, you have assigned this question to the geometry of position, but I am ignorant as to what this new discipline involves, and as to which types of problem Leibniz and Wolff expected to see expressed in this way. — Leonhard Euler
The problem with cinema nowadays is that it's a math problem. People can read a film mathematically; they know when this comes or that comes; in about 30 minutes, it's going to be over and have an ending. So film has become a mathematical solution. And that is boring, because art is not mathematical. — Nicolas Winding Refn
The first and foremost duty of the high school in teaching mathematics is to emphasize methodical work in problem solving...The teacher who wishes to serve equally all his students, future users and nonusers of mathematics, should teach problem solving so that it is about one-third mathematics and two-thirds common sense. — George Polya
If you think it's simple, then you have misunderstood the problem. — Bjarne Stroustrup
If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution. There is nothing like having somebody else beside you, because he can usually peer round the corner. — Michael Atiyah
The older I get, the more I believe that at the bottom of most deep mathematical problems there is a combinatorial problem. — Israel Gelfand
The broader the chess player you are, the easier it is to be competitive, and the same seems to be true of mathematics - if you can find links between different branches of mathematics, it can help you resolve problems. In both mathematics and chess, you study existing theory and use that to go forward. — Viswanathan Anand
Pi is not merely the ubiquitous factor in high school geometry problems; it is stitched across the whole tapestry of mathematics, not just geometry's little corner of it. Pi occupies a key place in trigonometry too. It is intimately related to e, and to imaginary numbers. Pi even shows up in the mathematics of probability — Robert Kanigel
There is no problem that cannot be solved. — Francois Viete
Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity-- to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs-- you deny them mathematics itself. — Paul Lockhart
The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill. — Albert Einstein
Seek simplicity, and distrust it. — Alfred North Whitehead
What good your beautiful proof on the transcendence of Pi: Why investigate such problems, given that irrational numbers do not even exist? — Leopold Kronecker
As a single atom man is an enigma: as a whole he is a mathematical problem. As an individual he is a free agent, as a species the offspring of necessity. — William Winwood Reade
It is well known that the central problem of the whole of modern mathematics is the study of transcendental functions defined by differential equations. — Felix Klein
Don’t try to make life a mathematics problem with yourself in the center and everything coming out equal. When you’re good, bad things can still happen. And if you’re bad, you can still be lucky. — Barbara Kingsolver
The classes of problems which are respectively known and not
known to have good algorithms are of great theoretical interest. [...]
I conjecture that there is no good algorithm for the traveling
salesman problem. My reasons are the same as for any mathematical
conjecture: (1) It is a legitimate mathematical possibility, and
(2) I do not know. — Jack Edmonds
The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, refuge from the goading urgency of contingent happenings, and the sort of beauty changeless mountains present to senses tried by the present day kaleidoscope of events. — Morris Kline
Some mathematics problems look simple, and you try them for a year or so, and then you try them for a hundred years, and it turns out that they're extremely hard to solve. There's no reason why these problems shouldn't be easy, and yet they turn out to be extremely intricate. Fermat's Last Theorem is the most beautiful example of this. — Andrew Wiles
The mere formulation of a problem is far more essential than its solution, which may be merely a matter of mathematical or experimental skills. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advances in science. — Albert Einstein
It's fine to work on any problem, so long as it generates interesting mathematics along the way - even if you don't solve it at the end of the day. — Andrew Wiles
Even when I was studying mathematics, physics, and computer science, it always seemed that the problem of consciousness was about the most interesting problem out there for science to come to grips with. — David Chalmers
It is obvious that mathematics needs both sorts of mathematicians, theory-builders and problem-solvers. — Timothy Gowers
To parents who despair because their children are unable to master the first problems in arithmetic I can dedicate my examples. For, in arithmetic, until the seventh grade I was last or nearly last. — Jacques Hadamard
It is impossible to overstate the imporance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps... Every new discovery in mathematics, results from an attempt to solve some problem. — Howard Whitley Eves
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