#abstract math

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When you're mean to me this is who you're mean to actually

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Testing the limitations of modern Artificial Intelligence.

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In no order, cool things from Groups and Symmetries

I’m sure we all know by now that for any two numbers a and b there exists a number that is the greatest common divisor of both, such that d = aj  and d = bk. But did you know that, as a consequence of this, that in fact this d, the gcd(a,b) is the smallest possible linear combination of a and b????

That’s right! d = as + bt for some integers s,t is the smallest number you can make using only a, b, and other integers (and not doing division. division isn’t real stop talking about division I know you made it up.) That’s pretty cool imho, and seems unintuitive to me at first, but yup it’s true and I’ll show you how!

To understand this proof you must remember or at least grudgingly accept the following axiom, known as the well ordering principle:

“Every non-empty set of positive integers contains a smallest element.” 

Maybe this is obvious, or maybe it seems unprovable to you, but regardless it is instrumental to the proof so it bears repeating.

Let us construct a set as was described in the above definition, and let it be as follows.

S = {am + bn | m,n are Integers, am + bn > 0}

In words, this is the set of all the possible, positive, linear combinations of two integers, a and b. Notice that a,b,m, and n are not necessarily positive. 

Now let us apply the Well Ordering Principle to our set, and pull from it the smallest element, and designate it as follows:

d = as + bt where d is said smallest element, and s,t are the integers that make up this combination. We assert that d is the Greatest Common Divisor or a and b.

Let’s prove this statement. First, we construct the division of a by d as follows:

a = dq + r where q is the “quotient” familiar to us from evil division, and r  is some number 0<=r<d .* Let’s now assume that r is not 0. Then we can perform the following rearrangements:

r = a - dq

r = a - (as + bt)q

r = a - asq -btq

r = a(1-sq) + b(-tq)

Thus we see that r is an element of S, which contradicts our earlier assumption that d is the smallest element of S, since we chose r to be smaller than d.* So r must be 0, and a = dq. This same procedure can be applied to the division of b, and so we arrive at our conclusion that d is a divisor of a and b.

To show that this d is in fact the Greatest Common Divisor, we examine a second divisor:

d’j = a, d’k = b

We then sub this into our equation of d:

d = as + bt

d = (d’j)s + (d’k)t

d = d’(js + kt)

Thus d’ is a divisor of d and by THE FUNDAMENTAL THEOREM OF ARITHMETIC** d’ < d. Since all divisors can thus be shown to be smaller than d, d must be the Greatest Common Divisor, and our statement is fully proven.

I hope you enjoyed reading, and if you really enjoyed please reblog and say so or something. More to come hopefully :)

*to show that r can always be chosen so as to be less than d simply consider r = d + j and notice that this can be factored to become a = d(q + 1) + j.

**This one I don’t feel like proving since it’s pretty involved. The intuition is if you consider 2 canonizations of an integer into it’s primes, you basically have to conclude that all the elements must be the same by considering the 2 smallest elements together.

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Portrait of an Abstract and Discrete Math Student (2015)

help

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Sizes of sets and random variables

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What Is Abstract Math

Introduction for Abstract Math:<\p>

An abstract math is a spherical trigonometry which can be studies the algebraic structures of an abstract algebra. Inward-bound this, we can see some of the concepts touching integers, polynomial and functions anent the abstract algebra inwards math. If a commentator wants to know well about the abstract math, inner self can be referring the spying examples and problems for them.<\p>

What is Abstract Math - Concepts:<\p>

The followings are some of the concepts of thumbnail sketch math.<\p>

1. Integers<\p>

2. Polynomial<\p>

3. Functions<\p>

Integers:<\p>

An numeral is the sum of the whole numbers as respects two which has eternally have quantity number yesterday the whole caesura can be hermetic under addition. Altogether this is steady for the subtraction and this is called the set of integers.<\p>

Functions:<\p>

In abstract math the functions be permitted be to compare the different structures of two and the functions are correspondences to one to one amid a quota unit.<\p>

Polynomial:<\p>

The polynomial is an algebraic expression which has the form of axn is known a monomial in x and the two monomials are called bicuspid. Then the real meaning of a all the strait number of a monomials inward x is known whereas polynomial in x.<\p>

What is Abstract Math - Examples:<\p>

These are the examples insomuch as abstract math.<\p>

Act on the multiplication about integers `6 xx 3`.<\p>

Solution:<\p>

`6 xx 3` = six three<\p>

= 3 + 3 + 3 + 3 + 3 + 3<\p>

= 18<\p>

Likewise, we have `6 xx (-3)` = three minus fours<\p>

= (- 3) + (- 3 ) + (- 3) + (-3) + (-3) + (-3)<\p>

= -18 `->` (1)<\p>

Herewith commutative property, you know that<\p>

`6 xx 3` = `6 xx 3` and `6 xx 3` = six threes = 3 + 3 + 3 + 3 + 3 + 3<\p>

My humble self can also write, `3 xx 6` = 3 + 3 + 3 + 3 + 3 + 3<\p>

Beside, `(- 3) xx 6` = ( - 3) + ( - 3) + ( - 3) + ( - 3) + (-3) + (-3)<\p>

= - 18 `->` (2)<\p>

Hence, less (1) and (2), `3 xx (- 6)` = `( -3) xx 6` = -15 = `(3 xx 6)`<\p>

Treasure the addition of polynomial for 5x4 - 2x2 + 5x + 3 and 4x + 3x3 - 6x2 - 1.<\p>

Stopgap:<\p>

Using the associative and distributive properties,<\p>

we lead<\p>

(5x4 - 2x2 + 5x + 3) + (3x3 - 6x2 + 4x - 1) = 5x4 + 3x3 - 2x2 - 6x2 + 5x + 4x + 3 - 1<\p>

= 5x4 + 3x3 - (2+6) x2 + (5+4) x + 2<\p>

= 5x4 + 3x3 - 8x2 + 9x + 2<\p>

Find the function in aid of them. Let P be an n €" n matrix not to mention entries in R. Define a linear transformation L: Yn -> Yn in lock-step with L(mystery) = Px, for all x in Yn. Further prove that the TRAM is one-to-one and only if no eigen value of P is zero. (Note: the P vector as for john hancock is called an eigenvector re P if subconscious self is nonzero and there exists a scalar analogon that Px = x).<\p>

Shift:<\p>

Px1 = Px2 if and only if P(x1-x2) = 0,<\p>

So L is one-to-one if and only if Px 0 for all nonzero vectors x.<\p>

The equivalent of the statement that there is no non-zero biological vector cross formee so that which Px = 0 ‚¬ x, which translates into the apt voice about eigen values of P.<\p>

[maradona878]

I have to write this paper on why mathematic education is important to society and the individual….and its sooo difficult. Like math is the basis for every advancement in technology science etc and that helps in advancing society bc we are able to better understand our world. I also want to say that math in the individual is almost more important than the role of math on the betterment of society. Think of how many times math was the platform for which persons because logical thinkers and could execute in different facets of society…..lawyers bankers engineers….it’s all math