Difference Between A Postulate, Corollary and a Theorem

THEOREM : A mathematical statement that is proved using extremely careful and thoughtful mathematical reasoning. 

POSTULATE:  A statement that is assumed to be true without proof.

COROLLARY: A result in which proof is heavily on a given theorem.

What are the differences between theorems, corollaries, propositions, ?

A proposition is a statement which is offered up for investigation as to its truth or falsehood. The term theorem is used throughout the whole of mathematics to mean a statement which has been proved to be true from whichever axioms relevant to that particular branch. A corollary is a proof which is a direct result, or a direct application, of another proof.It can be considered as being a proof for free on the back of a proof which has been paid for with blood, sweat and tears.The word is ultimately derived from the Latin corolla, meaning small garland, or the money paid for it. Hence has the sense something extra.

chehada assaf

a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory.

postulate: A postulate is a statement that is accepted without proof.Axiom is another name for postulate. For example, if you know that Pam is 5-feet tall and all her siblings are taller than her, you would believe her if she said that all of her siblings is at least 5.1'. Pam just stated a postulate, and you just accepted it without grabbing a tape measure to verify the height of her siblings.

corollary: A special case of a more general theorem which is worth noting separately. For example, the Pythagorean theorem is a corollary of the law of cosines.

proposition: A proposition is a declaration that can be either true or false, but not both. For example, “Today is Friday” is a proposition. This statement can be true or false, but not both.

Term 1- Assignment

Dear all;
Here is your 2nd assignment for term 1 .

Write down the difference between a theorem , a postulate, a corollary, and proposition ( if there is any)

Thanks in Advance

Keep the Good Work up :) 

3 examples:


--Factoring trinomials can be used to find out how long it takes for a ball that was dropped from a top of a building, to reach a specific height in a specific time


--Factoring polynomials can be used in architecture to reduce the numbers, by factoring them, and make scaling easier.

--It could be used also for seeing how longs things take. For example if it takes three different people three different times to do some thing by themselves, you can use factoring of polynomials to see how long it will take for them to do it together. Since they are three it will be a trinomial.

Factoring refers to the separation of a formula, number or matrix into products. For example, 49 can be factored into two 7s, or x^2 - 9 can be factored into x - 3 and x + 3. This is not a procedure used commonly in everyday life. Part of the reason is that the examples given in algebra class are so simple and that equations do not take such simple form in higher-level classes. Another reason is that everyday life does not require use of physics and chemistry calculations, unless it is your field of study or professiok

High School Science

Second-order polynomials--e.g., x^2 + 2x + 4--are regularly factored in high school algebra classes, usually in ninth grade. Being able to find the zeros of such formulas is basic to solving problems in high school chemistry and physics classes in the following year or two. Second-order formulas come up regularly in such classes.

Quadratic Formula

However, unless the science instructor has heavily rigged the problems, such formulas will not be as neat as they are presented in math class when simplification is used to help focus students on factoring. In physics and chemistry classes, the formulas are more likely to come out as 4.9t^2 + 10t - 100 = 0. In such cases, the zeros are no longer mere integers or simple fractions as in math class. The quadratic formula must be used to solve the equation: x = [-b +/- ?(b^2 - 4ac)] / [2a], where +/- means “plus or minus.” This is the messiness of the real world entering into mathematical application, and because the answers are no longer as neat as you find in algebra class, more complex tools must be used to deal with the added complexity.

Finance

In finance, a common polynomial equation that comes up is the calculation of present value. This is used in accounting when the present value of assets must be determined. It is used in asset (stock) valuation. It is used in bond trading and mortgage calculations. The polynomial is of high order, for example, with an interest term with exponent 360 for a 30-year mortgage. This is not a formula that can be factored. Instead, if the interest needs to be calculated, it is solved for by computer or calculator.

Numerical Analysis

This brings us into a field of study called numerical analysis. These methods are used when the value of an unknown can’t be solved for simply (e.g., by factoring) but must instead be solved for by computer, using approximation methods that estimate the answer better and better with each iteration of some algorithm such as Newton’s method or the bisection method. These are the sorts of methods used in financial calculators to calculate your mortgage rate.

Matrix Factorization

Speaking of numerical analysis, one use of factorization is in numerical computations to split a matrix into two product matrices. This is done to solve not a single equation but instead a group of equations simultaneously. The algorithm to perform the factorization is itself far more complex than the quadratic formula.

The Bottom Line

Factorization of polynomials as it is presented in algebra class is effectively too simple to be used in everyday life. It is nevertheless essential to completing other high school classes. More advanced tools are needed to account for the greater complexity of equations in the real world. Some tools can be used without understanding, e.g., in using a financial calculator. However, even entering the data in with the correct sign and making sure the right interest rate is used makes factoring polynomials simple by comparison.

For polynomials with integer coefficients the question is roughly the same as "what are the practical applications of algebraic number theory". The usual answers are coding theory and cryptography where factorization (and related operations such as testing whether a polynomial can be factorized) is part of the basic infrastructure from which systems are built or broken. Coding is necessary for digital communication (including telephone, video and satellites) and cryptography has become a basic feature of everyday computer use and commerce.

For polynomials with real coefficients there is partial fraction expansion used in calculus to compute integrals.

For polynomials with complex numbers as coefficients the factorization is into linear factors so that factoring is practically the same as numerical root finding (and this is in part true for real numbers as well). Problems in engineering where the location of complex roots of a polynomial determines the behavior of the system are common. For example, stability or instability can be decided by whether all the roots are inside the unit circle, or have positive real part, or other location-based criteria. Oscillations might be periodic if roots are n$n$'th roots of 1$1$ for some n$n$, or quasiperiodic behavior if roots are on the unit circle but not all at roots of 1$1$. A system governed by a partial differential equation would show diffusion (like heat) or wave-like behavior based on the factorization of an associated "differential operator", which is essentially a polynomial.

In general, many phenomena are decomposable into components, pieces or subsystems in a way that (when the systems are modeled mathematically) appears as a multiplicative decomposition of I don't have any idea what this fraction 1848 / 16632 means.  

   The numbers are too large.  But if I factor the numerator and    
   denominator I get:  (3)(8)(7)(11) / (8)(11)(7)(9) and I can 
   see that the 3, the 8, the 7, and a 5 are factors of both the  
   numerator and denominator.  Since 3/3, 8/8, 7/7 and 11/11 are all  
   equal to 1 the fraction reduces to 1/3, which is a lot easier to  
   understand and to compute with.

It's the same with complex algebraic fractions. (x^4 - x^2)/(x^3+x^2) 
looks REALLY complicated, but the numerator factors into x^2(x-1)(x+1) 
and the denominator factors into x^2(x+1), and since x^2/x^2 and 
(x+1)/(x+1) are both equal to 1, this fraction simplifies to x-1 . . . 
a LOT easier to work with.

A second important use of factoring is in solving equations. You don't 
need to factor to solve 2x+3 = 5 ... linear equations use a different 
method. And you don't need to factor second degree equations because 
you can use the Quadratic Formula (although factoring is often MUCH 
easier!). But if you need to solve equations where the degree of the 
highest term is more than 2 then you really have no choice at all 
because you don't have formulas for most of them.

Here is an example of a hard equation to solve:  

   x^5 - 4x^4 - 12x^3 = 0

But if you factor it completely you get (x^3)(x-6)(x+2) = 0 and now it 
is easy, because this says that a product of things turns out to be 
equal to zero. If you multiply, the only way to get zero as an answer 
would be if you multiplied by zero. So one of the three factors has 
to be zero.

    If x^3 = 0 then x = 0
    If x-6 = 0 then x = 6
    If x+2 = 0 then x = -2

So the solutions to this equation are x = 0 or 6 or -2.

How can factoring polynomials be used in real life?

The factoring of a polynomial refers to finding polynomials of lower order (highest exponent is lower) that, multiplied together, produce the polynomial being factored. For example, x^2 - 1 can be factored into x - 1 and x + 1. When these factors are multiplied, the -1x and +1x cancel out, leaving x^2 and 1.

Unfortunately, factoring is not a powerful tool, which limits its use in everyday life and technical fields. Polynomials are heavily rigged in grade school so that they can be factored. In everyday life, polynomials are not as friendly and require more sophisticated tools of analysis. A polynomial as simple as x^2 + 1 isn't factorable without using complex numbers--i.e., numbers that include i = √(-1). Polynomials of order as low as 3 can be prohibitively difficult to factor. For example, x^3 - y^3 factors to (x - y)(x^2 + xy + y^2), but it factors no further without resorting to complex numbers.

 Factoring can be used in laying a tile floor or packing various things into boxes. Also somewhat useful when changing screen resolutions. There is limited use in code breaking. Some things are useful as building blocks to bigger stuff. Other things will become useful later on, Microsoft Vista could be one of them. It is not used that much, not nearly as much as finding the greatest common divisor. We may find more uses later on, or not.

Factoring in real life

If you model some phenomenon with a polynomial, it's often of interest to determine when the polynomial evaluates to zero. One of the tools used in deciding when this happens is factoring.

For example, simple trajectory can be modeled with a quadratic function. If you think of time as the input and height as the output, then the positive time for which the polynomial evaluates to zero is precisely the time when the object hits the ground.

Factors in algebra are used almost all the time outside school. because factors tell you which numbers to multiply to give any  number you want. one simple example is  deciding how many soccer ball teams you can make from a group of 35 you can use the fact that 5 & 7 are two factors that make 35. In fact, any problem solving that requires you to group, divide and even rearrange shapes will involve knowing the factors that make up the numbers. The biggest advantage is that every number can be written as a multiplication  of  its factors. another way factors can be used is 

thank you

Yousef Shahin

ps; my fist post got deleted because i thought i could copy paste

Factoring polynomials in Real Life

Factoring Polynomials 

in Real Life

Three Examples:

1. You can use the concept of factoring binomials to figure out how long it takes a ball dropped from the top of a building to reach a certain height in a certain amount of time.

2. A good real world situation involving polynomials would be architecture. Factoring the polynomials can help to reduce the numbers you have to work with and make scaling the building a lot easier.

3. Say you own a painting company. You are asked to get a conference room painted in about 12 hours. Say Benjamin can do it in 12, Sally can do it in 10, and Alex can do it in 8 1/2. If you are good at factoring polynomials, you can figure out how long it should take them to get the room painted if they all work together. If you can't factor, they may take advantage of your math ignorance when you set them to work.

Factoring polynomials in 

real life   

Quadratics do play a role, for example, in area and construction problems.

Example: Suppose that you were told that a piece of land had a width that was 15 feet wider than its length, and that the area was 5800 square feet.

With that problem, you'll be ending up solving the equation  or further worked out,  which you would need to factor to get . 

Suppose that you have a bus, and you're renting it to organizations. Your bus can seat 80 people. You decide to charge the first person $30.00. If that person brings another person, you charge both of them $29.75 each. If there are three people, you charge $29.50 each. In other words, You charge EVERYBODY $0.25 less for every person who joins in. The thing is, there will come a point when you've got enough people to ride that you'll start losing profit if you add more people to the deal. With your starting price and discount price per additional person, will you: A) maximize your profit on the 80th person who rides? B) Be losing profit before you fill your bus, or C) would not yet reach your maximum profit on the 80th passenger?

So the total charge is dependent on how many people there are, how much you charge them, and what type of incentive you give depending on how many people there are. So, if there's one person, your total profit is $30.00. If there are two people, that's 2 people * ($30.00 - 0.25(1)). If there's 3 people, your total charge would be 3 people * ($30.00 - 0.25(2)).... Pretty soon, you'll see that your profit will be  or if simplified,  

Assignment1

Hello Dear Students,

Hope you all a nice weekend ahead.

Our First Assignment is:

How Can we use Factoring Polynomials in Real Life ?
Give examples,(case studies) and explain these examples.
You can either post on the blog your research resultsor comment on this post(if you are not able to post)
I  appreciate some Feedbacks on each others posts

Thank you in Advance  :)

Miss Roba Hatoum ;