Algebra

Harmonic Series Divergence

Fractions of the form $frac{1}{n}$ go to zero as $n$ goes to infinity, but what if we add them all up? A surprising classic result is that this sum can get arbitrarily large. We prove the divergence of the harmonic series here.

Logarithmic Change of Base Identity

https://www.youtube.com/watch?v=Of0LRRMoO78 The change of base identity for logarithms is a highly useful technique for dealing with logarithm problems. We prove this identity $$log_b x = frac{log_c x}{log_c b}$$ and then use it to derive a few other identities as corollaries.

Rationalizing a Denominator

When we have square roots in the denominator of a fraction, it can be desirable to transform the fraction into the same number but with a different presentation where the denominator is rational. We show a method for achieving this goal, with the main technique being the difference of squares factorization $$a^2 – b^2 = …

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Inverses in a Binary Operation are Unique

Adding a number to its negative produces $0$. Multiplying a non-zero number by its reciprocal produces $1$. We show that, given a fixed element that has an inverse in an associative binary operation (like addition or multiplication), the inverse is unique. So a number cannot have more than one negative or more than one reciprocal.

Identity Element of a Binary Operation is Unique

Given a binary operation, an identity element is like $0$ for addition or $1$ for multiplication: operating on a different number with it does not change that number. We show that, if an identity element exists, it is unique. So there cannot be two identity elements.