Exploring Pandigital Formulas for Euler’s Number ( e ) and Simplifying Complex Expressions

Introduction to Pandigital Formulas

Mathematics often involves creativity in expressing complex numbers and constants in novel and succinct ways. One such example is the approximation of Euler's number ( e ) using a pandigital formula discovered by Richard Sabey. This formula, which was later featured in a Numberphile video and recognized by Erich Friedman's puzzle, showcases a method for constructing ( e ) using a pandigital representation involving the digits 1 through 9. Let's delve into how we can approximate and simplify such expressions.

Pandigital Formula for ( e )

A notable pandigital formula for approximating ( e ) is:

$$left(1 cdot 1.2^{9^{6times7}} right)^{5^{3^{84}}} approx e$$

This expression is approximated to an astounding 836,842,898,906,842,594,381,759,091,645,001,887,164 decimal digits. While it is not the optimal solution, it stands as a remarkable achievement in the world of mathematical puzzles.

Understanding the Structure of the Formula

The formula leverages the pandigital digits 1 to 9 to construct an expression that converges to ( e ). The key to understanding such a formula lies in recognizing the underlying mathematical principles, particularly the exponential and logarithmic properties.

Simplifying Complex Expressions

Let's consider another complex expression and simplify it:

$$left(9^{-4^{6 cdot 7}}right)^{3^{2^{85}}}$$

We can break down this expression step by step:

First, calculate the exponent in the base: $$6 cdot 7 42$$ Therefore, we have: $$9^{-4^{42}}$$

Next, rewrite the base:

The base 9 can be expressed as (3^2): $$9^{-4^{42}} 3^{2 cdot -4^{42}} 3^{-2 cdot 4^{42}} 3^{-2^{1 cdot 42}} 3^{-2^{85}}$$

This means:

$$9^{-4^{42}} 3^{-2^{85}}$$

Substituting back into the expression:

$$left(1 cdot 9^{-4^{42}}right)^{3^{2^{85}}} left(1 cdot 3^{-2^{85}}right)^{3^{2^{85}}}$$

Define:

$$x 3^{2^{85}}$$

Substitute (x):

$$left(1 cdot frac{1}{x}right)^x left(frac{1}{x}right)^x$$

Now, consider the properties of (3^{-2^{85}}):

$$3^{-2^{85}}$$

is a very small number but not quite zero. Substituting this back, we get:

$$left(frac{1}{3^{2^{85}}}right)^{3^{2^{85}}} 1$$

Thus, the value of the original expression:

$$left(9^{-4^{6 cdot 7}}right)^{3^{2^{85}}}$$

is approximately:

1

Conclusion

Through this exploration, we have seen how mathematical creativity can lead to fascinating and elegant solutions. The use of pandigital formulas and simplification techniques not only provide insights into the properties of mathematical constants but also challenge our understanding of complex expressions.

Keywords

Pandigital Formula

Simplification Techniques