# Mathematics

### Thursday, September 19, 2013

#### Linear transformation of Pythagorean triples into Pythagorean triples

For any integer q, If x^2 + y^2 = z^2 Then ((2q^2 -1)x -(2q)y + (2 q^2 )z)^2 +((2q )x -(1 )y + (2q )z)^2 =((2q^2 )x -(2q)y + (2q^2+1)z)^2 Proof: ((2q^2 -1)x -(2q)y + (2 q^2 )z)^2 +((2q )x -(1 )y + (2q )z)^2 -((2q^2 )x -(2q)y + (2q^2+1)z)^2 = ((2q^2-1)^2 + (2q)^2 - (2q^2)^2) x^2 +((2q)^2 + 1 -... Sign in to see full entry.

### Tuesday, June 11, 2013

You have all seen the derivation of the quadratic formula solution to A x ^ 2 + B x + C = 0. A x y + B x + C y = D is the two variable generalization of the quadratic equation. Divide through by A x ^ 2 + (B/A) x + (C/A) = 0 x ^ 2 + (B/A) x = - (C/A) x y + (B/A) x + (C/A) y = (D/A) Complete the... Sign in to see full entry.

### Friday, May 31, 2013

#### Newtonian square root algorithm

The Newtonian algorithm for calculating square root: We easily see that the square root of 49 is 7. But what about the square root of 50. We know that the square root of 50 must be slighly more than 7. 7 50/7 > sqrt(50) (7 + 50/7)/2 is closer to the square root of 50 than either 7 or 50/7. (7 +... Sign in to see full entry.

### Thursday, May 30, 2013

#### sum of like powers

Sum Like Powers You have probably seen the formula for the sum of the first n positive integers, and maybe even the formula for the sum of the first n square integers and first n cube integers. There is a simple rule for progressing from a formula for the sum of the first n integers raised to a... Sign in to see full entry.

### Monday, October 29, 2012

Pick any odd positive integer. It is the difference of two squares. For example, 55 = 8*8 - 3*3 Then we can make a number series by adding up and down as follows: 8 * 8 - 3 * 3 = 55 7 * 9 - 2 * 4 = 55 6 * 10 - 1 * 5 = 55 5 * 11 - 0 * 6 = 55 4 * 12 + 1 * 7 = 55 3 * 13 + 2 * 8 = 55 2 * 14 + 3 * 9 = 55... Sign in to see full entry.

### Saturday, July 28, 2012

#### Seventy Two

Seventy Two 2*2*2*3*3 = 72 2 * (2*2*3*3) = 72 2 * 36 = 72 3 * (2*2*2*3) = 72 3 * 24 = 72 (2*2) * (2*3*3) = 72 4 * 18 = 72 (2*3) * (2*2*3) = 72 6 * 12 = 72 (2*2*2) * (3*3) = 72 8 * 9 = 72 Sign in to see full entry.

### Sunday, July 15, 2012

Consider a generalized Fibonacci sequence, G0, G1, G2, G3, G4,..... Suppose we imagine that there is a number x such that G0 = 1 G1 = x G2 = x^2 G3 = x^3 G4 = x^4 etc Is such an x possible? The only requirement is that G2 = x^2 G2 = G0 + G1 x^2 = 1 + x Multiply through by x. x^3 = x + x^2 G3 = G1 +... Sign in to see full entry.

### Saturday, July 14, 2012

#### Fibonacci Numbers

Fibonacci Numbers Fibonacci numbers are numbers in the Fibonacci sequence. The Fibonacci starts out as follows: 1,1,2,3,5,8,13,21,34,... After the second term, each term is the sum of the two preceeding it. Use the following convention to name the Fibonacci numbers. The Fibonacci sequence is... Sign in to see full entry.

### Monday, July 2, 2012

#### Square root of 2 is not the ratio of integers. It is irrational.

Rational numbers are the RATIO of integers. That is why they are called rational numbers. Irrational numbers are not the ratio of integers. That is why they are called irrational numbers. Since sqrt(2) was proven to be not rational, it was proven to not be the ratio of integers. (a+b)^2 + (a-b)^2 =... Sign in to see full entry.

### Sunday, July 1, 2012

#### 0! and 0^0

1! = 1 2! = 1 * 2 3! = 1 * 2 * 3 etc So how would we decide what 0! should be? Reverse the process. 3! = 6 2! = 3!/3 = 6/3 = 2 1! = 2!/2 = 2/1 = 1 0! = 1!/1 = 1/1 = 1 (-1)! = 0!/0 = 1/0. oops. (-1)! does not exist. thus (-2)! does not exist. etc We can find a factorial for all numbers except for the... Sign in to see full entry.

Copy (or write down) this entry's web address (URL), which is:

Next, go to the email or web page where you want to link to this entry, and paste (or type) the web address.

Page:     1  2  3  4  5  Next > Last >>