No.23

This week, we learned an interesting math topic, the Pascal's triangle, which was named after the French mathematician Blaise Pascal. However, Chinese and Indian mathematicians also found this rule decades before him. The Chinese mathematician Yang Hui was a famous guy in Song dynasty. He worked on magic squares, magic circles and binomial theorems. He was well known as the founder of the "Yang Hui's Triangle".(same as Pascal's.)

No.24

Every nonempty set of nonnegative integers has at least element 

• Let C = {n: P(n) is false} (the set of “counterexamples”)
• •Assume C is nonempty in order to derive a contradiction
• •Let m be the smallest element of C
• •Derive a contradiction (perhaps by finding a smaller member of C)
• 
  

No.21

Mathematical induction is a form of mathematical proof. There are two steps of mathematical induction:
1. Prove the statement is true at the starting point.
2. Assume the statement is true for n. Prove the statement is true for n+1.
Example: 1+3+5+7+...+(2n-1)=n^2
Step1 n=1
    LHS=1
    RHS=1^2=1
Since LHS=RHS is true for n=1
Step2 assume true for n. Show true for n+1
1+3+5+...+(2n+1)=(n+1)^2
    LHS=1+3+5+...+(2n-1)+(2n+1)
           = n^2+ 2n+1
    RHS=(n+1)^2
Since LHS=RHS is true for n+1
Hence the statement is true for all n element of natural number.

No.20

In today's lesson, we learned how to find the common difference of arithmetic sequences, which should be a review lesson for most of use. Arithmetic sequence is defined as consecutive terms have a common difference. For example, 2,4,6,8,10....
the common difference for this should be 2. Since the term is always 2 more than the previous one.

No.19

Today we reviewed the lesson of sequences in Algebra 2. A sequence is a function whose domain is natural numbers. Rather than using function notations, they are usually written in  a_n  form.  Here is a youtube clip about intro to sequences.  

No.18

Why does 0! = 1 ?

Usually n factorial is defined in the following way:

n! = 1*2*3*...*n

But this definition does not give a value for 0 factorial, so a natural question is: what is the value here of 0! ?

A first way to see that 0! = 1 is by working backward. We know that:

                1! = 1
     2! = 1!*2 
                2! = 2
     3! = 2!*3
                3! = 6
     4! = 3!*4
                4! = 24
             

We can turn this around:

                4! = 24
     3! = 4!/4
                3! = 6
     2! = 3!/3
                2! = 2
     1! = 2!/2
                1! = 1
     0! = 1!/1
                0! = 1

In this way a reasonable value for 0! can be found.

How can we fit 0! = 1 into a definition for n! ? Let's rewrite the usual definition with recurrence:

      1! = 1
      n! = n*(n-1)! for n > 1
      

Now it is simple to change the definition to include 0! :

      0! = 1
      n! = n*(n-1)! for n > 0
      

Why is it important to compute 0! ?

An important application of factorials is the computation of number combinations:

               n!
   C(n,k) = --------
            k!(n-k)!
         

C(n,k) is the number of combinations you can make of k objects out of a given set of n objects. We see that C(n,0) and C(n,n) should be equal to 1, but they require that 0! be used.

                      n!
   C(n,0) = C(n,n) = ----
                     n!0!

So 0! = 1 neatly fits what we expect C(n,0) and C(n,n) to be.

Can factorials also be computed for non-integer numbers? Yes, there is a famous function, the gamma function G(z), which extends factorials to real and even complex numbers. The definition of this function, however, is not simple:

          inf.
   G(z) = INT x^(z-1) e^(-x) dx
           0

Note that the extension of n! by G(z) is not what you might think: when n is a natural number, then G(n) = (n-1)!

The gamma function is undefined for zero and negative integers, from which we can conclude that factorials of negative integers do not exist.

(from Math Forum)

No.17

http://voicethread.com/?#q.b5433149.i27621334

No.16

https://sites.google.com/site/chapter8reviewgroup3/home

No.15

http://www.educreations.com/lesson/view/determinant-of-a-matrix/16850106/?s=ciem9G&ref=app

No.14

Example 6 explains a system with no solution. Because the third row of the matrix [0 0 0] means the linear equation of it will be inconsistent. 0=-2 will never work. So this is no solution.

No.13

When finding the code
1. Find the inverse of the encryption
2. Put the number by three and multiply by the encryption
3. Find code. 0 is space, 1 is A.....

No.12

The first problem is not invert arable. Because the third equation is all zeros. Singular