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 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.