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
More about Fibonacci Numbers
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.
Monday, May 28, 2012
Arithmetic Pattern
12*12 = 144 144 - 1 = 143 = 11 * 13 143 - 3 = 140 = 10 * 14 140 - 5 = 135 = 9 * 15 135 - 7 = 128 = 8 * 16 128 - 9 = 119 = 7 * 17 119 - 11 = 108 = 6 * 18 108 - 13 = 95 = 5 * 19 95 - 15 = 80 = 4 * 20 80 - 17 = 63 = 3 * 21 63 - 19 = 44 = 2 * 22 44 - 21 = 23 = 1 * 23 23 - 23 = 0 = 0 * 24
Sign in to see full entry.
Friday, May 18, 2012
The Golden Rectangle
The ancient Greeks expressed the opinion that the rectangle with adjacent sides in a particular ratio was the most beautiful of the possible rectangles. To construct this rectangle, they started with a rectangle whose length was twice its width. They drew in the diagonal of this 1 by 2 rectangle....
Sign in to see full entry.
Tuesday, May 8, 2012
Arithmetic Exercise
Ask a friend to: Think of a positive number. Example: 2 Add 3. 2+3 = 5 Multiply the result by itself. 5 * 5 = 25 subtract 9. 25 - 9 = 16. divide by original number. 16 / 2 = 8 subtract 6. 8 - 6 = 2. Then you can tell your friend, "If you did all the arithmetic correctly, the result is your original...
Sign in to see full entry.
Sunday, April 15, 2012
puzzle of sum to composite
Prove that every positive composite integer is the sum of 4 positive integers such that the product of the four integers is a square integer.
Sign in to see full entry.
