On the Liars Paradox

I recently came across Arthur Priors argument, his so-called resolution of the Liar’s Paradox.  I adamantly disagree with this mans entire argument and resolution of the paradox.  This post aims to discredit the argument.

Trivial Series Convergence Tests

It astounds me how frequently these trivial convergence/divergence tests are overlooked.  It astounds me how some of this causes confusion and stumps people.

Divisibility Rules: 11-16, and Some

A continuation of divisibility rules.  For rules for the digits 0 through 10, refer to the post titled Divisibility Rules: 0-10.

A Technique for Finding Complex Roots

In regards to finding the roots of a polynomial.

This method I have found useful and effective on multiple instances.  It isnt always effective, however – it depends heavily on the polynomial itself and you might waste some time applying it ineffectually.  But if it does works… you simplify the problem and save yourself a lot of work. Its one of those tricks that you might use when you run out of options otherwise.

Divisibility Rules: 0-10

A list of divisibility tests for factoring integers.

Upper and Lower Bounds Tests

Restrict the domain of possible real roots with a simple theorem. Applying these tests are not out of your way in the slightest – you will inevitably be doing these operations anyway when you are checking for a root.  Being aware of this theorem will help a lot.

Whenever you check a root by polynomial long division or synthetic division, these tests apply and you’ve already done all the work!

Conjugate Pairing

The concept of conjugate pairing, as it pertains to finding the roots of a polynomial.

I have noted that some people are confused about the existence of irrational roots that dont have conjugate pairs.

Rational Roots Theorem

This is a discussion on the method of the Rational Roots Theorem.  This method allows you to find all of the rational roots of any polynomial with rational (integer) coefficients.

Descartes’ Rule of Signs

This is a discussion on Descartes’ Rule of Signs.  The rule allows the mathematician to determine how many positive real, negative real, and non-real complex roots a polynomial can have.  This significantly reduces your efforts in searching for roots, or zeros, of a polynomial.

Finding Fibonacci Numbers and Their Indices

The nth Fibonacci Number, F(n), where φ is the Golden Ratio.