Proving the uncountability of the number of irrational powers of
... Considering the equation xx = y for any real given y, Theorem 3.1 estimates the solution x. Theorem 3.1 Let us consider the equation xx = y, for a given real y. Then, the following estimations of solution x hold: (a) if y > ee , then x < ln(y). (b) if y > e, then x > ln(ln(y)). Proof (a) Since y = x ...
... Considering the equation xx = y for any real given y, Theorem 3.1 estimates the solution x. Theorem 3.1 Let us consider the equation xx = y, for a given real y. Then, the following estimations of solution x hold: (a) if y > ee , then x < ln(y). (b) if y > e, then x > ln(ln(y)). Proof (a) Since y = x ...
A System of Equations
... any pair of coordinates that makes the equation true. • If you plot all of the solutions to a linear equation on a coordinate plane, they form the line. All the points on the line are solutions to the equation- they make the equation true. • A linear equation as an infinite amount of solutions- ther ...
... any pair of coordinates that makes the equation true. • If you plot all of the solutions to a linear equation on a coordinate plane, they form the line. All the points on the line are solutions to the equation- they make the equation true. • A linear equation as an infinite amount of solutions- ther ...
Propositional Statements Direct Proof
... Proof by Contradiction Given p → q, suppose that q is not true and p is true to deduce that this is impossible. In other words, we want to show that it is impossible for our hypothesis to occur but the result to not occur. We always begin a proof by contradiction by supposing that q is not true (¬q) ...
... Proof by Contradiction Given p → q, suppose that q is not true and p is true to deduce that this is impossible. In other words, we want to show that it is impossible for our hypothesis to occur but the result to not occur. We always begin a proof by contradiction by supposing that q is not true (¬q) ...
Notes for week 11.
... Prove that 2 | a(a+1), for all a ∈ ℕ. An informal proof of this result could be the observation that either a or (a+1) must be divisible by 2, and therefore the product a(a+1) must also be divisible by 2. However, in our studies we saw a very similar example that provides a “template” for proving th ...
... Prove that 2 | a(a+1), for all a ∈ ℕ. An informal proof of this result could be the observation that either a or (a+1) must be divisible by 2, and therefore the product a(a+1) must also be divisible by 2. However, in our studies we saw a very similar example that provides a “template” for proving th ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".