When the absurd of the absurd is real.

In fact, * ^{i}*√

*i*=

*e*

^{π/2}= 4.81…

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# Category: Equations

## Imaginary Root of An Imaginary Number

## Sum of Consecutive Cubes (Visual Proof)

## Inverse Powers of Phi

## Fibonacci Right Triangle

## Visual Proof (sum of cubes)

## “Stubborn” Number 33

## Sangaku: Semicircle inscribed in a right triangle

## Math Ambigram

## “Magic” Factorials

## Prime Fractions

The sum of the first *n* cubes is the square of the *n*th triangular number:

1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3} + . . . + *n*^{3} = (1 + 2 + 3 + 4 + 5 + . . . + *n*)^{2}.

Summation of Alternating Inverse Powers of Phi…

The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number.

The sum of the sequence of the first *n* cubes equals [*n*(*n*+1)/2]² as shown below:

1³+2³+3³+…+*n*³ = (1+2+3+…+*n*)² = [*n*(*n*+1)/2]²

It is conjectured that ** n** is a sum of 3 cubes if

Find the radius* r* of the semicircle inscribed in the right triangle below:

show solution

(by Peter Rowlett)

Solve this equation for x. Then rotate 180° and solve for x again.

The equation works either ways as 𝑥, 1, 8 and the unusual 5 have rotational symmetry. The same is true of +, =, and the horizontal line in a fraction.

There are many fun facts regarding the factorials. For instance:

- 0! = 1 by convention. As weird as it may sound, this is a fact that we must remember.

- The number of zeroes at the end of
*n*! is roughly*n*/4. - 70! is the smallest factorial larger than a
.*googol* - The sum of the reciprocals of all factorials is
.*e* - Factorials can be extended to fractions, negative numbers and complex numbers by the
.*Gamma function*

It is possible to “peel” each layer off of a factorial and create a different factorial, as shown in the neat number pattern below. A prime pattern can be found when adding and subtracting factorials. Alternating adding and subtracting factorials, as shown in the picture, yields primes numbers until you get to 9! Continue reading ““Magic” Factorials”

Did you know? You can write the number 1 as a sum of 48 different fractions, where every numerator is 1 and every denominator is a product of exactly two primes.

This problem is related to the Egyptian fractions.