https://www.youtube.com/watch?v=iwS6y1POULQ It is a standard convention that the degree of the zero polynomial is negative infinity. That is, $$deg(0)=-infty.$$ We explore the reasons for this strange definition, justify it, and look at some of its implications in this video.
Why is $\deg(0)=-\infty$? Read More »
https://www.youtube.com/watch?v=d0avql3E6ik We show how to explicity compute the square roots of a complex number that is given in rectangular form. We perform the derivation in a motivated way, instead of simply mechanically verifying the formula.
Square Root of a Complex Number Read More »
https://www.youtube.com/watch?v=t9RuFHcyQRc&t=60s The ordinary way of proving the quadratic formula $$x=frac{-bpmsqrt{b^2-4ac}}{2a}$$ is through a process called “completing the square.” But there is one step at the end which requires casework that almost everyone glosses over. In this video, we show a little-known modificiation of completing the square that avoids the need for any casework.
The Quadratic Formula Read More »
https://www.youtube.com/watch?v=V7JK9yqP7Q0 How can we find all of the complex $n^{text{th}}$ roots of a complex number, if the complex number is given in polar or trigonometric form? We show how to achieve this goal in our video.
$n^{\text{th}}$ Roots of a Complex Number Read More »
https://www.youtube.com/watch?v=k2gO530rwmU De Moivre’s formula allows us to conveniently compute the powers of complex numbers that are written in polar or trigonometric form. We prove de Moivre in this video, which states that $$(re^{itheta})^n = r^n e^{i(ntheta)}.$$
De Moivre’s Formula Read More »
https://www.youtube.com/watch?v=b5WdpOp4r8A Similar to the real number triangle inequality that uses the absolute value, there is a triangle inequality for complex numbers that uses the complex modulus $$|z+w|le |z|+|w|.$$ We show how to prove this inequality here.
Complex Triangle Inequality Read More »
https://www.youtube.com/watch?v=C4yq-GSNO6k Many students know the double angle trigonometric identities, but what about triple angles? We derive the triple angle identities here using the addition and double angle identities.
Triple Angle Trig Identities Read More »
https://www.youtube.com/watch?v=PbrJHqBrSTk For sine and cosine, there are double and half angle trigonometric identities which we explore here. In particular, the cosine double angle identity has three forms, from among which we can choose one strategically depending on the situation. We prove these identities using the addition identities and the main Pythagorean identity.
Double & Half Angle Trig Identities Read More »
https://www.youtube.com/watch?v=QfIvr-6xM-o The Pythagorean theorem is among the most famous theorems in math, with almost all school students studying it at some point. But how can we prove it? The Indian mathematician Bhaskara had a method that we modify and simpify here to prove the Pythagorean theorem and its converse.
Pythagorean Theorem Read More »
https://www.youtube.com/watch?v=O_tg4X88HGE The angle bisector theorem gives us a remarkably simple relationship between the line segments that are produced by a cevian that is an angle bisector and the sides of a triangle. We discuss and prove this equation here.
Angle Bisector Theorem Read More »