Open questions about irrational numbers
Is π+e irrational? Is sqrt(2)+sqrt(3)? One of the open questions (they don't know either way) dealing with irrational numbers is: is the sum of of two irrational numbers irrational? To address this, let's think about the crucial property of irrationals: the unique-infinite-sequence (or uis) following the decimal/binary point representing the unique fractional part of an irrational number (as explained in the previous article). So in decimal format, 3-e and 4-π are what I call the uis-complements of e and π respectively. The reason I choose this term is because under addition, they produce whole numbers. It's not hard to visualize this critical feature in binary format; please try it.
When we fully understand 3-e, 4-π, 2-sqrt(2),.. are unique irrational numbers themselves, this so-called open question becomes almost trivial. From a probability standpoint anyways.. If we randomly choose a number from the real line, chances are it will be irrational is 100%! This is the astounding result from real analysis. What are the chances we choose a particular irrational number? 0%. So.. The chances π+e or π-e is rational is zero. When we think about the uniqueness of irrational complements, we're Confident about this claim.
Now we're ready to state another conjecture. Here, we introduce another term: distinct. For purposes of this discussion, I define distinct analogously to linear-independence. Two irrational numbers are distinct if you cannot make a liner combination of one out of the other using rational numbers. More explicitly, two irrational numbers are distinct, x and y, if you cannot find any non-zero rational numbers, a and b, such that y = ax + b is true. Conjecture 1:
The sum of two distinct irrational numbers is irrational.
The product or quotient of two distinct irrational numbers is irrational.
The exponentiation of two distinct irrational numbers is irrational.
One and two together equate with conjecture 4:
The set of distinct irrational numbers is a field.
Why is this important? In a sense, we don't 'need' rational numbers in math! (Remember, the rationals include the integers – the 'counting numbers'.) Any rational number can be 'approximated' with a nearby irrational. In a very real sense, we can dispense with all other numbers! ..Of course, these statements are facetious computationally but illustrate the overwhelming density of the irrationals.
Convince yourself: if you doubt above, try the operations above on your calculator. For instance, try using sqrt(2) and sqrt(3) as components and look at the results you get. Use any two distinct irrational numbers.. If you find two that violate a statement above, let me know! We all can make mistakes!