The Longest Equation Known to Man

This is the longest equation known to man. Crazy right? Known as The Standard Model, it is a theory that currently governs all existing interactions. This includes basically everything that has EVER been measured by humans, and is considered the most accurate theory to date. It has been fantastically successful in predicting and explaining data, as there are no obvious or overt violations in this theory. According to its original typist, this equation took about 4 hours to write!

Week 8, Day 2

Today we learned about Binomial Expansion. When you write out the coefficients for a binomial that is raised to a power, you are expanding a binomial. There are two methods of expansion, both of which provide binomial coefficients. The first is Binomial Theorem, which states that in the expansion of 

(x + y)^n: (x + y) ^n = x^n + (nx^n-1)y +...+ (n)C(r) (x^n-r)y^r +...+ nxy^n-1 + y^n.The coefficient of (x^n-r)y^r is given by a procedure known as combination: (n)C(r) = n! / (n-r)!r!. The second is Pascal's Triangle, in which the first andlast number in each row is 1. Every other number in each row is formed by adding the two numbers immediately above the number. A basic illustration of Pascal's triangle is:

To expand a binomial, it is easiest to use Pascal's Triangle as shown below:

Because the exponent of the binomial is 6, you would use the 6th row of the triangle. Using the given coefficients, write each appropriate term, and simplify. 

Week 8, Day 1

Today we discussed the process of mathematical induction. 

The following is an example problem demonstrating how this process is executed: 

Step 1: Prove the statement is true at the starting point (usually n = 1)

Step 2: Assume the statement is true for n. Prove the statement is true for n + 1. 

The Well-Ordering Principle is a concept equivalent to mathematical induction. It states that every non-empty set of positive integers contains a least element. The phrase "well-ordering principle" is synonymous to "well-ordering theorem." On other occasions, it is understood to bethe proposition that the set of intergers (..., -2, -1, 0, 1, 2, 3,...) comtains a well-ordered subset, called the natural numbers, in which every nonempty subset contains a least element. Depending on the framework in which the natural numbers are introduced, this (second order) property of the set of natural numbers is either an axiom or a provable theorem. In fact, there is a proof that shows how mathematical induction implies the validity of the Well-Ordering Principle. 

Ultimately, as demonstrated by this proof, mathematical induction can be correlated to the Well-Ordering Principle, and this principle, in turn, can support induction. 

What Is Zero-Factorial?

Why is 0! defined to equal 1? This question arose the other day while we were discussing factorials, and I decided to find an answer by providing a very simple proof often used to support this statement. 

If n! is defined as the product of all positive intergers from 1 to n, then:

1! = 1 x 1 = 1

2! = 1 x 2 = 2 

3! = 1 x 2 x 3 = 6

4! = 1 x 2 x 3 x 4 = 24

n! = 1 x 2 x 3 x 4... x (n-2) x (n-1) x n

And so forth.

Logically, n! can also be expressed n x (n-1)!

Therefore, at n = 1, using n! = n x (n-1)! 

1! = 1 x (1-1)!

1! = 1 x 0!

This simplifies to 1 = 0!

And there you go, a mathematical explanation for 0! = 1.

Week 7, Day 2

Today we learned how to interpret arithmetic sequences and reclusive formulas.

Arithmetic Sequences:

2, 4, 6, 8 --> n + 2 (this is referred to as a common difference or d)

The common difference can be determined by subtracting a(2) from a(1).

The Formula of Arithmetic Sequences: a(n) = a(1) + (n - 1)d (explicit formula)

2, 4, 6, 8: a(n) = 2 + (n - 1)2 

a(n) = 2 + 2n - 2 

a(n) = 2n

The Formula of the Sum of Arithmetic Sequences: S(n) = n/2 (a(1) + a(2))

S(100) = 100/2 (2 + 2(100))

S(100) = 50 (202)

S(100) = 10,100

Reclusive Formulas:

A reclusive formula must always include 2 parts: 

The first part indicates where the sequence starts and the second part indicates the direction of the sequence, for instance, whether you add or subtract. 

a(k+1) = a(k) + 1 OR a(k) = a(k-1) + 1

a(1) = 3

a(2) = 3 + 1 = 4

a(3) = 4 + 1 = 5

a(4) = 5 + 1 = 6

According to the formula provided, you would add 1 to the preceding term every time you wish to find the next term. For instance, to find the second term, you would add 1 to the first term, and so forth. 

Week 7, Day 1

Today we learned about sequences and summation notation.

There are two types of sequences: infinite and finite.

To express an infinite sequence write: a(1), a(2), a(3)... a(n).

These are referred to as the terms of the sequence. 

To express a finite sequence write: a(1), a(2), a(3)... a(n).

If the domain of the function consists of the first "n" positive intergers only, the sequence is finite.

Finding the Terms of a Sequence:

When given a sequence, otherwise known as an explicit formula, such as:

a(n) = 3n - 2 where n = 1, simply plug and chug:

a(1) = 3(1) - 2 = 1

a(2) = 3(2) - 2 = 4

a(3) = 3(3) - 2 = 7

The first three terms of this sequence are 1, 4, and 7.

Finding the nth Term of a Sequence:

When given a set of terms such as:

1, 3, 5, 7, it is most effective to first multiply a number by "n", then add or subtract as necessary:

a(n) = 2n - 1

a(1) = 2(1) - 1 = 1

a(2) = 2(2) - 1 = 3

a(3) = 2(3) - 1 = 5

a(4) = 2(4) - 1 = 7 

A Sequence That Is Defined Reclusively:

The Fibonnaci Sequence is defined reclusively, as follows:

a(0) = 1, a(1) = 1, a(k) = a(k-2) + a(k-1) where k >(=) 2

* i, n, and k are synonymous.

To find the first six terms, simply plug and chug using the sequence provided:

a(0) = 1

a(1) = 1

a(2) = a(0) + a(1) = 1 + 1 = 2

a(3) = a(1) + a(2) = 1 + 2 = 3

a(4) = a(2) + a(3) = 2 + 3 = 5

a(5) = a(3) + a(4) = 3 + 5 = 8

Factorials!:

n! = 1 x 2 x 3 x 4... (n - 1)n 

* 0! = 1

To find 6 factorial, for instance, simply multiply 1, 2, 3, 4, 5, and 6:

6! = 1 x 2 x 3 x 4 x 5 x 6 = 720

To perform factorial operations, for instance 8! / 2!6!, follow the same process:

(1 x 2 x 3 x 4 x 5 x 6 x 7 x 8) / (1 x 2)(1 x 2 x 3 x 4 x 5 x 6) = 28

Summation Notation:

Week 6, Day 3

Attached is a link to a google site we created to help you better prepare for the chapter 8 test! I hope you find this helpful! :) 

https://sites.google.com/a/maranathastudents.org/chapter-8-group-1/