Popular Problems in Math 1

8/11/2019 Popular Problems in Math 1

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PopularAlgebraicProblemsMath 1 T


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SOLVED ALGEBRAICPROBLEMS


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SOLVED ALGEBRAIC PROBLEM

I. Statement of the problem:

Jane spent $42 for shoes. This was $14less than twice what she spent for a blouse.How much was the blouse?


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SOLVED ALGEBRAIC PROBLEMII. Solution:Let x be the answer to the question

2x 14 = 42.

2x = 42 + 14

2x = 56.

x = 562

= 28.


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SOLVED ALGEBRAIC PROBLEM

Other examples:

- Age problem

- Mixture problem

- Distance problem

- Work problem


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AGE PROBLEM

PROBLEM

Jam is twice as old as Jake.

The sum of their ages is 5times Jakesage less 48.

How old are Jan and Jake?


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AGE PROBLEM


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UNSOLVED ALGEBRAICPROBLEMS


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UNSOLVED ALGEBRAIC PROBLEM

Perfect Numbers

A perfect number is one which equals thesum of its proper divisors.

6 is a perfect number since 6 is equal to thesum of its divisors (1, 2, and 3).

6 = 1+ 2 + 3

Perfect numbers have been studied for a

long time. It is known that all even perfectnumbers are of the form


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Perfect Numbers

2p1 is prime. Primes of that form are called

Mersenne Primes. It can be shown that pmust be prime for 2p1 to be prime. As ofDecember 2002, 39Mersenne Primesareknown. There are thus 39 known even

perfect numbers. The following questionshave not been answered to date:


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Perfect Numbers

The following questions have not beenanswered to date:

Problem:

1. How many even perfect numbers arethere? More precisely, are there infinitely

many, or finitely many? In the latter case,just how many are there?

2. Are there any odd perfect numbers?


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THE P VERSUS NP PROBLEM

The problem of deciding whether theanswer to the problem below can be quicklychecked can really take much longer to solve,no matter how clever a program we write, is

considered one of the outstanding problems inlogic and computer science. It was formulatedby Stephen Cook in 1971.


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UNSOLVED ALGEBRAIC PROBLEMTHE P VERSUS NP PROBLEM

It is Saturday evening and you arrive at a big

party. Feeling shy, you wonder how many peopleyou already know in the room? Your host proposesthat you must certainly know Rose, the lady in thecorner next to the dessert tray. In a fraction of asecond you are able to cast a glance and verifythat your host is correct. However, in the absence ofsuch a suggestion, you are obliged to make a tour ofthe whole room, checking out each person one byone, to see if there is anyone you recognize.


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This is an example of the general

phenomenon that generating a solution to aproblem often takes far longer than verifying a givensolution is correct. Similarly, if someone tells you thatthe number 13,717,421 can be written as theproduct of two smaller numbers, you might not knowwhether to believe him, but if he tells you that it canbe factored as 3607 times 3803 then you can easilycheck that it is true using a hand calculator.


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P is the general class of questions for which

some algorithms can provide an answer.

NP is the class of question for which answercan be verified.


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Informally, it asks whether any problem whosesolution can be quickly verified by a computer can

also be quickly solved by a computer.

Consider the subset sum problem, anexample of a problem that is easy to verify butwhose answer is somehow difficult to compute.Given a set of integers, does some of nonempty

subset of them sum to 0? The answer can quickly beverified through series of addition. However, there isno known algorithm to find such a subset, hence thisproblem is in NP(quickly checkable) but notP(quickly solvable).


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HODGE CONJECTURE

The problem says that certain de Rhamcohomology classes are algebraic, that is, theyare sums of Poincare duals of the homologyclasses of sub varieties. This is one of the ClayMathematics Institutes Millennium Prize

Problems with a prize of $1,000,000.00 forwhoever can prove or disprove the HodgeConjecture using someargument.


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RIEMANN HYPOTHESIS

A conjecture about the location ofnontrivial zeros of the Riemann Zeta Functionwhich states that all non trivial zeros have realpart . This hypothesis implies results about thedistribution of prime numbers that are in someways as good as possible. Along with suitable

generalizations, it is considered by somemathematicians the most important unsolvedproblem in pure Mathematics (Bombieri, 2000).


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YangMills Existence and Mass Gap

Prove that for any compact simplegauge group G, a non-trivial quantum YangMills theory exists on and has a mass gap > 0.

YangMills theory is the (non-Abelian)quantum field theory underlying the Standard

Model of particle physics; is Euclidean 4-space;the mass gap is the mass of the least massiveparticle predicted by the theory.


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UNSOLVED ALGEBRAIC PROBLEMBirch and Swinnerton-Dyer Conjecture

An open problem in the field of number theory.

Its status as one of the most challenging mathematicalquestions has become widely recognized.

The conjecture relates arithmetic data associated toan elliptic curve E over a number field K to thebehaviour of the Hasse-Weil L-function L(E, s) of E at s =1. More specifically, it is conjectured that the rank ofthe abelian group E(K) of points of E is the order of thezero of L(E, s) at s = 1, and the first non-zero coefficientin the Taylor expansion of L(E, s) at s = 1 is given by morerefined arithmetic data attached to E over K (Wiles2006).


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NavierStokes Existence and Smoothness

Problem is concerned with the mathematicalproperties of solutions to the NavierStokes

equations, one of the pillars of fluid mechanics (suchas with turbulence). These equations describe themotion of a fluid (that is, a liquid or a gas) in space.Solutions to the NavierStokes equations are used inmany practical applications. However, theoreticalunderstanding of the solutions to these equations isincomplete. In particular, solutions of the NavierStokes equations often include turbulence, whichremains one of the greatest unsolved problems inphysics, despite its immense importance in scienceand engineering.

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Popular Problems in Math 1