(A new question of the week) Extraneous roots can not only confuse the final solution to a problem; they can also make it harder to solve in the first place if you don’t deal with them early. Here is a relatively complicated rational equation, two questions about its solutions, and several ways to make it …
A Rational Equation, With and Without Extraneous Roots Read More »
(A new question of the week) Some problems can be done either by algebra or by basic arithmetic methods and some creativity; and although algebra generally makes work easier by making it routine, sometimes special-purpose thinking (once you have thought it!) can be quicker. Here we have a problem where a creative method didn’t quite …
(A new question of the week) We often see polynomials in a simplistic way, imagining that any function whose graph resembles a polynomial is a polynomial. Much as an attempt to mimic random data often lacks essential properties of genuine randomness, so what we intend to be a polynomial often is not. As we observe …
(A new question of the week) Limits of indeterminate forms like ∞ – ∞ require us first to recognize the form, and then, often, use L’Hôpital’s rule (also called L’Hospital’s rule, as we’ll be seeing it here), or some other method. Today’s question will touch on all stages of this work for three examples, but …
(A new question of the week) We’ve looked at the concept of limit of a function from several perspectives, including why they are needed, and what the definition means. Here we have a more fundamental question, which applies to both functions and sequences: What do we mean when we say a value approaches some number? …
(A new question of the week) Real life questions of probability often require information that we don’t have – they become a job for statistics instead. But sometimes just trying some plausible numbers, as in a Fermi problem, can yield interesting results. Here we consider the probability of an injury when kids play near a …
(A new question of the week) A couple recent questions involved related subtleties in probability and combinatorics. Both were about apparent conflicts between similar problems involving cards and dice.
We’ve looked at positive integer powers of complex numbers, and imaginary powers of e. Now let’s look at roots of complex numbers, which will bring us to the idea that led to this series, namely the roots of unity.
Last week we explored how the polar form of complex numbers gives multiplication a simple geometric meaning. Here we’ll go one more step, and express polar form exponentially, which makes DeMoivre’s theorem trivial, and gives us a simple notation to replace “cis”.
Having looked at the idea of complex numbers and how to perform basic operations on them, we are ready for one of the most important features for applications: their relationship to rotation. We’ll see this first in describing complex numbers by a length and an angle (polar form), then by discovering the meaning of multiplication …