Complex number fundamentals | Ep. 3 Lockdown live math

What people said

Recommends Smriti Tiwari · 2017-10 · Mathematics Stack Exchange · on “Is $\{(a, b, c) \in \mathbb{C}^3 : a^3 = b^3\}$ a subspace of $\mathbb{C}^3$”

As somebody else commented, the difference in $\mathbb{C}$ is that $a^3 = b^3$ does not imply $a = b$ . Essentially, complex numbers have multiple different cube roots, just like real numbers have two different square roots (e.g. $\sqrt{1}$ can be $-1$ or $1$ ). To disprove that the set in question is a subspace of $\mathbb{C}$ , we can provide a counterexample of two complex vectors that are members of the set, while their sum is not: Let $x = (e^{i\frac{π}{3}}, -1, 0)$ and $ y = (1,1,0)$ . (Note: $e^{i\frac{π}{3}}$ is a shorthand for $cos(\frac{π}{3}) + sin(\frac{π}{3})i$ , which is a point on the unit circle drawn in the complex plane, see this wonderful video [this video] for an in-depth expl

Recommends Rob Dukes · 2021-10 · Mathematics Stack Exchange · on “What does it mean to divide imaginary numbers exactly?.”

The other answers are correct, but there is a geometric meaning to complex numbers that adds a lot to our understanding. Historically, they were thought to be little more than a curiosity until Caspar Wessel gave them a geometric meaning. Then Gauss and Euler expanded on that, of course. Here’s the visual: The real number line is horizontal. Picture a second real number line at a right angle to it. So now you have the Cartesian plane. Any complex number a + bi is placed at the point (a,b). i itself is thus 0 + 1i, so it is placed one unit above the origin. Now, with a little trigonometry it’s possible to redefine the point with the quantities r and theta, where r is the point’s distance from

Recommends Thomas · 2022-04 · Mathematics Stack Exchange · on “Is $\{(a, b, c) \in \mathbb{C}^3 : a^3 = b^3\}$ a subspace of $\mathbb{C}^3$”

As somebody else commented, the difference in $\mathbb{C}$ is that $a^3 = b^3$ does not imply $a = b$ . Essentially, complex numbers have multiple different cube roots, just like real numbers have two different square roots (e.g. $\sqrt{1}$ can be $-1$ or $1$ ). To disprove that the set in question is a subspace of $\mathbb{C}$ , we can provide a counterexample of two complex vectors that are members of the set, while their sum is not. Let $x = (e^{i\frac{π}{3}}, -1, 0)$ and $ y = (1,1,0)$ . Its sum won't satisfy the property $a^{3}=b^{3}$ (Note: $e^{i\frac{π}{3}}$ is a shorthand for $cos(\frac{π}{3}) + sin(\frac{π}{3})i$ , which is a point on the unit circle drawn in the complex plane, see

Recommends PrincessEev · 2023-07 · Mathematics Stack Exchange · on “Why do complex numbers lend themselves to rotation?”

Take two complex numbers in their polar forms, $$ z := \alpha e^{i \theta} \qquad w := \beta e^{i \varphi} $$ (Here, $\alpha,\beta$ are their magnitudes, and $\theta,\varphi$ the angle they make with the positive real axis up to a multiple of $2\pi$ .) Then $$ zw = \alpha \beta e^{i(\theta+\varphi)} $$ Think about what this means for $z$ and $w$ . Multiplying $z$ by $w$ (when one thinks of $z$ as a vector, i.e. an arrow pointing from the origin to $z$ in the plane) stretches $z$ by a factor of $\beta$ , and then rotates it by an angle of $\varphi$ radians about the origin. That is, multiplication is naturally an operation about stretching and rotating. 3Blue1Brown discusses this in more deta

Recommends Benjamin Wang · 2023-10 · Mathematics Stack Exchange · on “Numbers Not Expressed with 0-9”

Here's a intro [this video] to complex numbers presented in a conversation-like way by 3Blue1Brown, and it sounds like you might enjoy it.

Recommends RajS · 2023-11 · Mathematics Stack Exchange · on “How division rotates complex number in direction opposite to multiplication?”

At 1:06 [this video] timestamp of 3b1b Complex numbers fundamental video, Grant says $$\text{cis}(\alpha)=\frac{1}{\text{cis}(-\alpha)}$$ where $$\text{cis}(\alpha)=\cos(\alpha)+i\sin(\alpha)$$ He seem to give the fact that multiplying vector by constant $>1$ is equivalent to stretching the vector while by constant $<1$ is equivalent to squishing the vector However, I dont get how vectors gets flipped vertically when taken inverse, that is I dont get how $\text{cis}(\alpha)=\frac{1}{\text{cis}(-\alpha)}$ I tried to visualize it: [LINK] I confirmed this fact by quickly writing a python code. Also tried to prove this by pen pensil for $\alpha=45^o$ and then algebraically proving $\text{cis}(\alpha)

Cites prarabdh shivhare · 2020-04 · Mathematics Stack Exchange · on “3blue1brown's complex numbers fundamental video: rotating a point by 30 degrees”

Grant says that multiplying any complex number represented by a point on the Argand plane by $cis(30^{\circ})$ i.e. $\cos30^{\circ} +i\sin(30^{\circ})$ rotates that point by $30$ degrees counterclockwise on the Argand plane. Maybe I did not pick on what he was trying to say in the video, but can someone elaborate on why this happens? Even people who haven't watched the video........ Here's a link by the way: [this video] . Start watching after 32:10. I get why multiplying $cis(30^{\circ})$ to $1$ , rotates $1$ by $30$ degrees counterclockwise, but can someone algebraically prove it for any general complex number (z) ? It's easy to see with the Euler notation... but I am not well-worsed with it (

Cites hyperpallium · 2020-05 · Mathematics Stack Exchange · on “Denesting radicals e.g. $\sqrt{ 8 - 4\sqrt{3}}$ to $\sqrt{6} - \sqrt{2}$”

How to find the square root of an irrational expression, to simplify that root. e.g.: $$ \sqrt{ 8 - 4\sqrt{3} } = \sqrt{6} - \sqrt{2} $$ Easy to verify: \begin{align} (\sqrt{6} - \sqrt{2})^2 = 6 - 2\sqrt{12} +2 = 8 - 4 \sqrt{3} \end{align} But how to work it out in the first place? I feel there's a standard technique (Completing-the-square? Quadratic formula?), but don't recall it or what it's called... BTW: this came up in verifying equivalence of different calculations of $\cos{75°}$ (the above divided by $4$ ), as $\cos{\frac{90°+60°}{2}}$ vs $\cos{(45°+30°)}$ , from 3Blue1Brown's lockdown video on complex numbers and trigonometry [this video] .