### Saturday, March 7, 2015

#### Product of numbers, each of which is the difference of two squares

Prove that if p1 = t1^2 - s1^2 and p2 = t2^2 - s2^2 then p1 * p2 = ( t1 * t2 - s1 * s2)^2 - (t1 * s2 - s1 * t2)^2 and p1 * p2 = ( t1 * t2 + s1 * s2)^2 - (t1 * s2 + s1 * t2)^2

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### Wednesday, February 25, 2015

#### My book, "Mathematical Essays"

My book, "Mathematical Essays" is now available on Amazon.com http://www.amazon.com/dp/1500735388/ http://www.amazon.co.uk/dp/1500735388/ http://www.amazon.de/dp/1500735388/ https://www.createspace.com/4935112 Table of contents Introduction to the irrational plane mathematical induction...

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### Sunday, February 22, 2015

#### Two variable equation with only a few integer solutions

Find the integer solutions to x y + 3 x + 5 y = 62

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### Sunday, February 15, 2015

#### Is zero even or odd?

Andy: How do I know whether zero is even or odd? Billie: How much is zero plus two? Andy: Zero plus two is two. Billie: How much is four minus two? Andy: Four minus two is two. Billie: If I subtract any even number from an even number, do I ever get an odd number. Andy: I see it now. Even minus even...

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### Saturday, January 3, 2015

#### Pythagorean Triples

Pythagorean Triples If x^2 + y^2 = z^2 and a^2 + b^2 = c^2, then [1] (a x – b y)^2 + (a y + b x)^2 = (c z)^2 [2] (a x + b y)^2 + (a y – b x)^2 = (c z)^2 [3] (b y)^2 + (az + cx)^2 = (cz + ax)^2 [4] (b y)^2 + (az - cx)^2 = (cz - ax)^2 [5] (a y)^2 + (b z + c x)^2 = (c z + b x)^2 [6] (a y)^2 + (b z - c...

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### Monday, December 8, 2014

#### Algebraic equivalent to the Pythagorean Theorem

If x = (a^2 - b^2), y = (2 a b) and z = (a^2 + b^2), then x^2 + y^2 = z^2. a = 2, b = 1, x = 2^2 - 1^2 = 4 - 1 = 3 y = 2 * 2 * 1 = 4, z = 2^2 + 1^2 = 5 3^2 + 4^2 = 5^2 9 + 16 = 25 a = 3, b = 2, x = 3^2 - 2^2 = 9 - 4 = 5 y = 2 * 3 * 2 = 12 z = 3^2 + 2^2 = 9 + 4 = 13 5^2 + 12^2 = 13^2 25 + 144 = 169

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### Saturday, November 8, 2014

#### Algebraic language # 5

Algebraic language #5 How do we get from the numbers 0 and 1 to all the numbers that we know? Pick a point on a horizontal line. Call it 0. Pick another point to the right of 0. Call it 1. Measure the distance from the point 0 to the point 1. The number 1 is also that distance. Find the point on the...

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### Monday, November 3, 2014

#### Algebraic language # 4

Algebraic language # 4 Is a square a rectangle? The non-mathematician might say "no". The mathematician will say "yes". Rectangle is Rect-Angle, which means "right angle". The defining characteristic of a rectangle is that it has 4 ninety-degree angles. A square has 4 ninety-degree angles. A square...

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### Saturday, November 1, 2014

#### Algebraic language # 3

Algebraic language #3 How do we prove that 2 + 2 = 4? A child proves it very easily. The child picks up two toys and puts them before you. Then the child picks up two more toys and puts them before you. Then the child counts 1,2,3,4. See, I proved that 2 + 2 = 4. A mathematician needs to work...

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### Friday, October 31, 2014

#### Algebraic language # 1

Algebraic language # 1 Any letter may be a temporary name for a number. If n is the temporary name for the number 6, then it is true that n + 5 = 11. If x + 5 = 11 is a true statement, then x must be a temporary name for the number 6.

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