Partial fractions video for funnnnnnn

Hi since next week is the math final, I did not put anything unrelated to the final for our fun post.

However, instead, I found a YouTube video for partial fractions that really helped me when I do partial fractions.

So check it out!

http://youtu.be/cyYJ4HP2RFw

Vectors in space

Vectors in Space

Solving a Unit Vector

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The equation for a unit vector is:

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A vector space  is a set that is closed under finite vector addition and scalar multiplication. The basic example is -dimensional Euclidean space , where every element is represented by a list of  real numbers, scalars are real numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately.

For a general vector space, the scalars are members of a field , in which case  is called a vector space over .

Euclidean -space  is called a real vector space, and  is called a complex vector space.

In order for  to be a vector space, the following conditions must hold for all elements  and any scalars :

1. Commutativity:

(1)

2. Associativity of vector addition:

(2)

3. Additive identity: For all ,

(3)

4. Existence of additive inverse: For any , there exists a  such that

(4)

5. Associativity of scalar multiplication:

(5)

6. Distributivity of scalar sums:

(6)

7. Distributivity of vector sums:

10.5 Parametric Equations

Definition of a plane curve

If f and g are continuous functions of t on an interval l, the set of ordered pairs (f(t),g(t)) is a plane curve C. The equations:

 

        x=f(t)      and     y=g(t)

are parametric equations for C, and t is the parameter.

 

Eliminating the Parameter

1. Parametric equations

2. solve for t in one equation

3. substitute into second equation

4.rectangular equation (only 2 variable)

 

Example:

1.

x=t+3, y= -t^2

step 1:  subtract both side of x=t+3 by three:

                                x-3=t

then, take x-3 into another equation

                   get   y= - (x-3)^2

 

Aristotle and logic : creating something from nothing

Aristotelian logic, after a great and early triumph, consolidated its position of influence to rule over the philosophical world throughout the Middle Ages up until the 19th Century.  All that changed in a hurry when modern logicians embraced a new kind of mathematical logic and pushed out what they regarded as the antiquated and clunky method of syllogisms.  AlthoughAristotle’s very rich and expansive account of logic differs in key ways from modern approaches, it is more than a historical curiosity.  It provides an alternative way of approaching logic and continues to provide critical insights into contemporary issues and concerns.  The main thrust of this article is to explain Aristotle’s logical system as a whole while correcting some prominent misconceptions that persist in the popular understanding and even in some of the specialized literature.  Before getting down to business, it is important to point out that Aristotle is a synoptic thinker with an over-arching theory that ties together all aspects and fields of philosophy.  He does not view logic as a separate, self-sufficient subject-matter, to be considered in isolation from other aspects of disciplined inquiry.  Although we cannot consider all the details of his encyclopedic approach, we can sketch out the larger picture in a way that illuminates the general thrust of his system.  For the purposes of this entry, let us define logic as that field of inquiry which investigates how we reason correctly (and, by extension, how we reason incorrectly).  Aristotle does not believe that the purpose of logic is to prove that human beings can have knowledge.  (He dismisses excessive scepticism.)  The aim of logic is the elaboration of a coherent system that allows us to investigate, classify, and evaluate good and bad forms of reasoning.

Mathematical Induction

This is the section of my presentation!!

 1. Definition:

mathematical induction is a method for proving that statements involving natural numbers are true for all natural numbers.  

2. Theorem:

suppose that 他the following two conditions are satisfied with regard to a statement bout natural numbers:

CONDITIONＩ:The statement is true for  natural number 1

Condition II: if the statement is true for some natural number k, it is also true for k+1

Thus the statement is true for all numbers

3. sums of powers of integers

Examples:

1.

For any positive integer n, 1 + 2 + ... + n = n(n+1)/2.

           n=1  1 = 1(2)/2 which is clearly true.

          Here we must prove the following assertion: "If there is a k such that P(k) is true, then (for this same k) P(k+1) is true."

        assume there is a k such that 1 + 2 + ... + k = k (k+1)/2.  

         We must prove, for this same k, the formula 1 + 2 + ... + k + (k+1) = (k+1)(k+2)/2.

        1 + 2 + ... + k + (k+1) = k(k+1)/2 + (k+1) = (k(k+1) + 2 (k+1))/2 = (k+1)(k+2)/2.

2.

Write the first five terms of the sequence

a0=1 an=a(n-1)+2

a0=1

a1=3

a2=5

a3=7

Determinants of a matrix

Determinant: the associated algebraic number associated with a matrix

Matrix: an array of numbers set inside of a two brackets

Notations of determinants differ. There are 2 ways to do a determinant. 

How to do a 2x2 matrix determinant

This is quite a simple application of timesing the top left and bottom right numbers then subtracting the multiplication of the bottom left and top right numbers. 

Determinant of 3x3 matrices. 

These are hard yet simple. The easiest way is to take the first 2 columns, copy them, and move them to the end of the matrix. After that take the first 3 diagonals starting at the top left and going down and multiply those numbers in their individual columns. Then add the other 2 columns to the first. after that take the columns going diagonally up from the bottom right and across and multiply those together. Then subtract all three of those columns from the columns of the first three. This will give a person their matrix.