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6 votes

Bridging the gap between students' intuitive problem-solving abilities and expressing ideas through formal writing

I've worked with some of these types of students in the past. One trend I noticed was that these students often have experience with coding, which they tend to enjoy and excel at (since the computer ...
Justin Skycak's user avatar
0 votes

Importance of complex numbers knowledge in real roots

My favorite goes approximately like this: The geometric series $\sum\limits_{i=0}^{\infty}\frac{1}{x^n}$ converges to a function $f =\frac{1}{1-x}$ only on $-1\lt x \lt 1$. On the other hand, $f$ is ...
user58697's user avatar
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0 votes

Importance of complex numbers knowledge in real roots

Complex numbers don't have an ordering like real numbers, i.e. $a + i b\ngtr c + id$. This is both subtle and profound. To explain, assign north as imaginary and east as real and then talk about ...
user121330's user avatar
1 vote

Importance of complex numbers knowledge in real roots

The complex numbers are really just the scale-rotations in 2D geometry. So you can get lots of examples from geometry. We start with two particular scale-rotations that form a basis for building up ...
Nullius in Verba's user avatar
1 vote

Importance of complex numbers knowledge in real roots

Students who are interested in geometry might find this proof of Ptolemy's theorem to be a compelling application of complex numbers. $$ AC\cdot BD = AB\cdot CD+BC\cdot AD $$
Steven Gubkin's user avatar
6 votes

Importance of complex numbers knowledge in real roots

Consider the real initial value problem $$ \begin{cases} y'' + 2y' + 10y = 0\\ y(0) = 0\\ y'(0) = 1 \end{cases} $$ One nice way to do it is to first finding generic solutions with real inputs but ...
Steven Gubkin's user avatar
4 votes

Importance of complex numbers knowledge in real roots

In the setting of single-variable calculus, here is one of the most striking uses of complex numbers: the radius of convergence of the power series at $x = 0$ of a rational function $p(x)/q(x)$, ...
KCd's user avatar
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0 votes

Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?

A guess: Could it be a matter of time constraints? Thoroughly covering all potential outcomes and their implications is a time-intensive endeavor.
Humberto José Bortolossi's user avatar
2 votes

Importance of complex numbers knowledge in real roots

AThe book 'Visual Complex Analysis' by Tristan Needham highlights the significance of complex numbers, emphasizing their historical importance in mathematics. It details how the real solutions of ...
Humberto José Bortolossi's user avatar
6 votes

Importance of complex numbers knowledge in real roots

If series and radius of convergence are within the reach of your syllabus, then a standard example of "needing to go through the complex numbers" is the radius of convergence of the series ...
Pablo H's user avatar
  • 191
1 vote

Importance of complex numbers knowledge in real roots

This might be the oldest question in mathematics education! But you are probably not going to like Euclid’s answer. A youth who had begun to read geometry with Euclid, when he had learnt the first ...
user52817's user avatar
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6 votes

Importance of complex numbers knowledge in real roots

we need knowledge of complex numbers to work with real numbers We need to be honest that at the level of your students, there are no examples of this: in all sufficiently elementary uses of complex ...
Kostya_I's user avatar
  • 1,164
8 votes

Importance of complex numbers knowledge in real roots

Another topic that uses complex numbers in service of real numbers, and is immediately accessible to a high schooler who has learned or is learning calculus, is integration using Euler's formula. Here'...
Justin Skycak's user avatar

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