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Studying Mathematics Thread
#43
RE: Studying Mathematics Thread
(March 8, 2018 at 3:49 am)robvalue Wrote: Very nice! Good work, and it shows what I was saying about how understanding helps the memory. It all follows, it's not just meaningless drivel to memorise.

Now do the formula for cubics Wink Just kidding, I didn't see that done until university, and it hurt my poor brain.

Funnily enough, I remember hearing that there is also a formula for power 4 polynomials, but that there cannot be a general formula for power 5 or more. I wonder if my memory is correct there. There's probably some restriction about the kind of formula we're talking about.

Instead, we can use iterative methods and such, for approximate solutions.

Your memory is correct. The allowed operations are addition, subtraction, multiplication, division, and extraction of roots.

The quadratic formula shows it is possible to solve a general quadratic via these operations. And yes, there is a procedure for cubic and a quartic equations

it turns out that for every polynomial there is an object called a group that describes the symmetries between the roots of that polynomial. If that group has a certain property (called solvability) , then the polynomial is solvable via radicals. It turns out that the relevant group is NOT solvable for the general equation of degree 5 or higher.

A specific quintic that cannot be solved via radicals is x^5 -5x+5.

(March 8, 2018 at 9:48 am)robvalue Wrote: Another interesting thing about complex numbers is that you sometimes get extra solutions you wouldn't normally think of.

For example, consider

x^3 = 1

What's the solution?

x = 1

Obviously. But there's also two other solutions, if we allow x to be complex. They fall on the unit circle drawn in the complex plane around the origin, at angles 120 and 240 degrees from the positive real axis. Multiplying those numbers by themselves rotates them around the origin, so 3 * 120 = 360 and 3 * 240 = 720 leaving you back on the positive real axis at 1.

The same thing happens for every polynomial equation of the form x^n =1. The result is a regular n-sided polygon in the complex plane.

(March 8, 2018 at 3:53 am)Grandizer Wrote:
(March 8, 2018 at 3:49 am)robvalue Wrote: Very nice! Good work, and it shows what I was saying about how understanding helps the memory. It all follows, it's not just meaningless drivel to memorise.

Now do the formula for cubics Wink Just kidding, I didn't see that done until university, and it hurt my poor brain.

Funnily enough, I remember hearing that there is also a formula for power 4 polynomials, but that there cannot be a general formula for power 5 or more. I wonder if my memory is correct there. There's probably some restriction about the kind of formula we're talking about.

lol, I don't even know what it's supposed to be like (the cubic formula). But I can imagine it being rather tedious to derive. Maybe a few years from now once I get into studying proofs for real.

The formula itself is nasty, but the procedure can be understood.

Given a cubic equation, a x^3 +b x^2 +c x +d=0, first divide by the leading coefficient to get

x^3 +B x^2 +C x +D=0

Then let  y=x-B/3. It turns out that when you do the algebra, this will eliminate the quadratic term, so you have
an equation of the form

y^3 + E y +F=0.

Now comes the trick. Multiplying out shows that (u+v)^3 -(3uv)(u+v) =u^3 + v^3. We try to force our equation into this form.

So, we put y=u+v, giving one more variable to play with and then require that -3uv=E, so v=-E/(3u),

Plugging in gives
0=y^3 +Ey +F=(u+v)^3 -(3uv)(u+v) +F =u^3 +v^3 +F =u^3 +(-E/(3u) )^3 +F.

Expand this and multiply through by 27u^3 gives

27 u^6 +27F u^3 -E^3 =0.

This is now a quadratic in u^3, which can be solved by the quadratic formula. This u^3, which gives u, which gives v, which gives y, which finally gives x.

In finding u from u^3, a cube root has to be taken and it is crucial to remember that there are three complex cube roots, which ultimately gives the three roots to the cubic.

Whew!
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Messages In This Thread
Studying Mathematics Thread - by Grandizer - February 28, 2018 at 11:22 am
RE: Studying Mathematics Thread - by Silver - February 28, 2018 at 11:31 am
RE: Studying Mathematics Thread - by Tiberius - February 28, 2018 at 11:36 am
RE: Studying Mathematics Thread - by Whateverist - February 28, 2018 at 3:58 pm
RE: Studying Mathematics Thread - by polymath257 - February 28, 2018 at 11:45 am
RE: Studying Mathematics Thread - by robvalue - February 28, 2018 at 1:21 pm
RE: Studying Mathematics Thread - by polymath257 - February 28, 2018 at 2:08 pm
RE: Studying Mathematics Thread - by Edwardo Piet - February 28, 2018 at 2:10 pm
RE: Studying Mathematics Thread - by polymath257 - February 28, 2018 at 2:35 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - February 28, 2018 at 8:55 pm
RE: Studying Mathematics Thread - by Grandizer - February 28, 2018 at 9:00 pm
RE: Studying Mathematics Thread - by Edwardo Piet - February 28, 2018 at 9:02 pm
RE: Studying Mathematics Thread - by Fireball - February 28, 2018 at 3:53 pm
RE: Studying Mathematics Thread - by Alexmahone - February 28, 2018 at 3:54 pm
RE: Studying Mathematics Thread - by LadyForCamus - February 28, 2018 at 10:22 pm
RE: Studying Mathematics Thread - by Grandizer - February 28, 2018 at 10:59 pm
RE: Studying Mathematics Thread - by polymath257 - March 3, 2018 at 7:12 pm
RE: Studying Mathematics Thread - by Grandizer - March 3, 2018 at 7:52 pm
RE: Studying Mathematics Thread - by polymath257 - March 3, 2018 at 9:00 pm
RE: Studying Mathematics Thread - by robvalue - March 4, 2018 at 7:12 am
RE: Studying Mathematics Thread - by Grandizer - March 6, 2018 at 7:58 am
RE: Studying Mathematics Thread - by polymath257 - March 6, 2018 at 9:21 am
RE: Studying Mathematics Thread - by robvalue - March 7, 2018 at 1:51 am
RE: Studying Mathematics Thread - by polymath257 - March 7, 2018 at 8:19 am
RE: Studying Mathematics Thread - by Grandizer - March 7, 2018 at 8:41 am
RE: Studying Mathematics Thread - by robvalue - March 7, 2018 at 8:48 am
RE: Studying Mathematics Thread - by polymath257 - March 7, 2018 at 8:49 am
RE: Studying Mathematics Thread - by robvalue - March 7, 2018 at 8:30 am
RE: Studying Mathematics Thread - by polymath257 - March 7, 2018 at 8:42 am
RE: Studying Mathematics Thread - by robvalue - March 7, 2018 at 8:53 am
RE: Studying Mathematics Thread - by polymath257 - March 7, 2018 at 9:01 am
RE: Studying Mathematics Thread - by robvalue - March 7, 2018 at 10:19 am
RE: Studying Mathematics Thread - by polymath257 - March 7, 2018 at 10:43 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - March 8, 2018 at 1:54 am
RE: Studying Mathematics Thread - by robvalue - March 8, 2018 at 2:46 am
RE: Studying Mathematics Thread - by Grandizer - March 8, 2018 at 3:39 am
RE: Studying Mathematics Thread - by robvalue - March 8, 2018 at 3:49 am
RE: Studying Mathematics Thread - by Grandizer - March 8, 2018 at 3:53 am
RE: Studying Mathematics Thread - by polymath257 - March 8, 2018 at 10:47 am
RE: Studying Mathematics Thread - by Edwardo Piet - April 20, 2018 at 10:22 am
RE: Studying Mathematics Thread - by Grandizer - April 20, 2018 at 11:10 am
RE: Studying Mathematics Thread - by Grandizer - April 20, 2018 at 10:39 pm
RE: Studying Mathematics Thread - by polymath257 - April 21, 2018 at 10:06 am
RE: Studying Mathematics Thread - by robvalue - March 8, 2018 at 4:00 am
RE: Studying Mathematics Thread - by Grandizer - March 8, 2018 at 4:11 am
RE: Studying Mathematics Thread - by Grandizer - March 8, 2018 at 8:43 am
RE: Studying Mathematics Thread - by robvalue - March 8, 2018 at 9:48 am
RE: Studying Mathematics Thread - by Grandizer - March 15, 2018 at 1:49 am
RE: Studying Mathematics Thread - by polymath257 - March 18, 2018 at 10:43 pm
RE: Studying Mathematics Thread - by Grandizer - March 19, 2018 at 9:11 am
RE: Studying Mathematics Thread - by polymath257 - March 19, 2018 at 9:32 pm
RE: Studying Mathematics Thread - by polymath257 - March 22, 2018 at 9:34 am
RE: Studying Mathematics Thread - by Grandizer - March 22, 2018 at 10:06 am
RE: Studying Mathematics Thread - by Grandizer - March 22, 2018 at 9:58 pm
RE: Studying Mathematics Thread - by polymath257 - March 23, 2018 at 7:44 am
RE: Studying Mathematics Thread - by Grandizer - March 24, 2018 at 8:45 am
RE: Studying Mathematics Thread - by polymath257 - March 24, 2018 at 9:52 am
RE: Studying Mathematics Thread - by Grandizer - April 4, 2018 at 10:35 pm
RE: Studying Mathematics Thread - by polymath257 - April 5, 2018 at 7:41 am
RE: Studying Mathematics Thread - by robvalue - April 5, 2018 at 8:48 am
RE: Studying Mathematics Thread - by polymath257 - April 5, 2018 at 9:01 am
RE: Studying Mathematics Thread - by Grandizer - April 5, 2018 at 9:03 am
RE: Studying Mathematics Thread - by Grandizer - April 5, 2018 at 8:25 pm
RE: Studying Mathematics Thread - by robvalue - April 6, 2018 at 2:18 am
RE: Studying Mathematics Thread - by Grandizer - April 6, 2018 at 2:58 am
RE: Studying Mathematics Thread - by robvalue - April 6, 2018 at 4:16 am
RE: Studying Mathematics Thread - by Grandizer - April 6, 2018 at 4:22 am
RE: Studying Mathematics Thread - by polymath257 - April 6, 2018 at 7:53 am
RE: Studying Mathematics Thread - by robvalue - April 9, 2018 at 12:58 pm
RE: Studying Mathematics Thread - by Grandizer - April 9, 2018 at 11:46 am
RE: Studying Mathematics Thread - by Silver - April 9, 2018 at 1:01 pm
RE: Studying Mathematics Thread - by robvalue - April 9, 2018 at 1:09 pm
RE: Studying Mathematics Thread - by polymath257 - April 9, 2018 at 1:20 pm
RE: Studying Mathematics Thread - by robvalue - April 9, 2018 at 1:22 pm
RE: Studying Mathematics Thread - by Grandizer - April 14, 2018 at 2:07 am
RE: Studying Mathematics Thread - by polymath257 - April 14, 2018 at 10:24 am
RE: Studying Mathematics Thread - by robvalue - April 14, 2018 at 10:41 am
RE: Studying Mathematics Thread - by Grandizer - April 15, 2018 at 5:18 am
RE: Studying Mathematics Thread - by Jarrey - April 15, 2018 at 7:12 am
RE: Studying Mathematics Thread - by robvalue - April 15, 2018 at 7:53 am
RE: Studying Mathematics Thread - by Grandizer - April 15, 2018 at 9:21 am
RE: Studying Mathematics Thread - by polymath257 - April 15, 2018 at 10:26 am
RE: Studying Mathematics Thread - by Grandizer - April 19, 2018 at 10:39 pm
RE: Studying Mathematics Thread - by robvalue - April 21, 2018 at 3:44 am
RE: Studying Mathematics Thread - by Edwardo Piet - April 21, 2018 at 10:09 am
RE: Studying Mathematics Thread - by Grandizer - April 23, 2018 at 5:51 pm
RE: Studying Mathematics Thread - by polymath257 - April 23, 2018 at 10:09 pm
RE: Studying Mathematics Thread - by Divinity - April 23, 2018 at 7:19 pm
RE: Studying Mathematics Thread - by Grandizer - April 23, 2018 at 8:32 pm
RE: Studying Mathematics Thread - by Divinity - April 23, 2018 at 10:45 pm
RE: Studying Mathematics Thread - by Grandizer - April 24, 2018 at 2:18 am
RE: Studying Mathematics Thread - by polymath257 - April 24, 2018 at 8:13 am
RE: Studying Mathematics Thread - by Fireball - April 23, 2018 at 8:20 pm
RE: Studying Mathematics Thread - by Grandizer - April 24, 2018 at 4:43 am
RE: Studying Mathematics Thread - by polymath257 - April 24, 2018 at 8:24 am
RE: Studying Mathematics Thread - by robvalue - April 24, 2018 at 5:09 am
RE: Studying Mathematics Thread - by Grandizer - April 24, 2018 at 8:18 am
RE: Studying Mathematics Thread - by polymath257 - April 24, 2018 at 11:24 am
RE: Studying Mathematics Thread - by polymath257 - April 26, 2018 at 7:48 am
RE: Studying Mathematics Thread - by Grandizer - April 27, 2018 at 7:19 pm
RE: Studying Mathematics Thread - by polymath257 - April 27, 2018 at 8:53 pm
RE: Studying Mathematics Thread - by Grandizer - April 28, 2018 at 1:41 am
RE: Studying Mathematics Thread - by polymath257 - April 28, 2018 at 9:51 am
RE: Studying Mathematics Thread - by Grandizer - April 28, 2018 at 11:35 pm
RE: Studying Mathematics Thread - by polymath257 - April 29, 2018 at 9:00 am
RE: Studying Mathematics Thread - by Grandizer - April 30, 2018 at 9:48 am
RE: Studying Mathematics Thread - by polymath257 - April 30, 2018 at 1:15 pm
RE: Studying Mathematics Thread - by Grandizer - April 30, 2018 at 7:42 pm
RE: Studying Mathematics Thread - by polymath257 - April 30, 2018 at 8:09 pm
RE: Studying Mathematics Thread - by Grandizer - May 1, 2018 at 2:19 am
RE: Studying Mathematics Thread - by Grandizer - April 28, 2018 at 9:29 am
RE: Studying Mathematics Thread - by polymath257 - May 1, 2018 at 7:55 am
RE: Studying Mathematics Thread - by Grandizer - May 1, 2018 at 7:59 am
RE: Studying Mathematics Thread - by Fireball - May 1, 2018 at 11:43 pm
RE: Studying Mathematics Thread - by Grandizer - May 2, 2018 at 6:35 am
RE: Studying Mathematics Thread - by Fireball - May 2, 2018 at 11:43 am
RE: Studying Mathematics Thread - by polymath257 - May 4, 2018 at 8:14 am
RE: Studying Mathematics Thread - by Fireball - June 15, 2018 at 12:35 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - June 15, 2018 at 11:45 am
RE: Studying Mathematics Thread - by polymath257 - June 15, 2018 at 3:46 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - July 6, 2018 at 3:23 am
RE: Studying Mathematics Thread - by polymath257 - July 6, 2018 at 8:42 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - July 17, 2018 at 2:02 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 27, 2018 at 1:59 am
RE: Studying Mathematics Thread - by The Valkyrie - September 27, 2018 at 2:01 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 27, 2018 at 2:10 am
RE: Studying Mathematics Thread - by polymath257 - September 27, 2018 at 8:07 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 27, 2018 at 7:13 pm
RE: Studying Mathematics Thread - by Grandizer - September 27, 2018 at 7:20 pm
RE: Studying Mathematics Thread - by polymath257 - September 27, 2018 at 9:06 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 28, 2018 at 5:34 pm
RE: Studying Mathematics Thread - by Aliza - September 28, 2018 at 6:57 pm
RE: Studying Mathematics Thread - by polymath257 - September 29, 2018 at 9:16 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 29, 2018 at 10:50 am
RE: Studying Mathematics Thread - by polymath257 - September 29, 2018 at 6:02 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 29, 2018 at 10:01 pm
RE: Studying Mathematics Thread - by Aliza - September 30, 2018 at 2:40 pm
RE: Studying Mathematics Thread - by Aliza - September 29, 2018 at 6:21 pm
RE: Studying Mathematics Thread - by Tiberius - September 27, 2018 at 2:11 am
RE: Studying Mathematics Thread - by Aliza - September 27, 2018 at 2:23 am
RE: Studying Mathematics Thread - by robvalue - September 28, 2018 at 1:35 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 28, 2018 at 8:02 pm
RE: Studying Mathematics Thread - by Aliza - September 28, 2018 at 8:21 pm
RE: Studying Mathematics Thread - by Fireball - September 29, 2018 at 7:56 pm
RE: Studying Mathematics Thread - by Aliza - September 29, 2018 at 8:05 pm
RE: Studying Mathematics Thread - by Fireball - September 29, 2018 at 9:08 pm
RE: Studying Mathematics Thread - by Aliza - September 30, 2018 at 12:12 am
RE: Studying Mathematics Thread - by Fireball - September 30, 2018 at 7:56 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 30, 2018 at 10:11 pm
RE: Studying Mathematics Thread - by Aliza - September 30, 2018 at 10:18 pm
RE: Studying Mathematics Thread - by Fireball - September 30, 2018 at 10:44 pm
RE: Studying Mathematics Thread - by polymath257 - October 1, 2018 at 8:06 am
RE: Studying Mathematics Thread - by LastPoet - October 2, 2018 at 7:40 am
RE: Studying Mathematics Thread - by polymath257 - October 2, 2018 at 7:58 am
RE: Studying Mathematics Thread - by Reltzik - October 2, 2018 at 10:49 am
RE: Studying Mathematics Thread - by Reltzik - October 2, 2018 at 10:58 am
RE: Studying Mathematics Thread - by polymath257 - October 2, 2018 at 11:34 am
RE: Studying Mathematics Thread - by Grandizer - October 2, 2018 at 11:05 am
RE: Studying Mathematics Thread - by Aliza - October 2, 2018 at 11:29 am
RE: Studying Mathematics Thread - by Grandizer - October 2, 2018 at 11:35 am
RE: Studying Mathematics Thread - by polymath257 - October 2, 2018 at 12:03 pm
RE: Studying Mathematics Thread - by Aliza - October 2, 2018 at 12:50 pm
RE: Studying Mathematics Thread - by Grandizer - October 2, 2018 at 1:23 pm
RE: Studying Mathematics Thread - by polymath257 - October 2, 2018 at 1:29 pm
RE: Studying Mathematics Thread - by Aliza - October 3, 2018 at 1:53 am
RE: Studying Mathematics Thread - by polymath257 - October 3, 2018 at 8:30 am
RE: Studying Mathematics Thread - by Fireball - October 3, 2018 at 11:30 am
RE: Studying Mathematics Thread - by polymath257 - October 3, 2018 at 12:25 pm
RE: Studying Mathematics Thread - by Fireball - October 3, 2018 at 1:03 pm
RE: Studying Mathematics Thread - by robvalue - October 1, 2018 at 1:54 am
RE: Studying Mathematics Thread - by Reltzik - October 2, 2018 at 2:51 am
RE: Studying Mathematics Thread - by Grandizer - October 5, 2018 at 1:19 pm
RE: Studying Mathematics Thread - by Fireball - October 9, 2018 at 8:03 pm
RE: Studying Mathematics Thread - by Grandizer - October 10, 2018 at 12:13 am
RE: Studying Mathematics Thread - by Aliza - October 10, 2018 at 3:10 pm
RE: Studying Mathematics Thread - by Grandizer - October 10, 2018 at 11:34 pm
RE: Studying Mathematics Thread - by Reltzik - October 11, 2018 at 12:20 am
RE: Studying Mathematics Thread - by Aliza - October 11, 2018 at 12:30 am
RE: Studying Mathematics Thread - by Reltzik - October 11, 2018 at 12:35 am
RE: Studying Mathematics Thread - by Aliza - October 11, 2018 at 12:53 am
RE: Studying Mathematics Thread - by Aliza - October 11, 2018 at 12:41 pm
RE: Studying Mathematics Thread - by Reltzik - October 11, 2018 at 2:47 pm
RE: Studying Mathematics Thread - by DodosAreDead - October 11, 2018 at 12:21 am
RE: Studying Mathematics Thread - by EgoDeath - October 9, 2018 at 4:20 pm
RE: Studying Mathematics Thread - by Reltzik - October 9, 2018 at 7:55 pm
RE: Studying Mathematics Thread - by EgoDeath - October 10, 2018 at 9:46 am
RE: Studying Mathematics Thread - by Reltzik - October 10, 2018 at 3:37 pm
RE: Studying Mathematics Thread - by EgoDeath - October 10, 2018 at 11:29 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - October 10, 2018 at 12:07 am
RE: Studying Mathematics Thread - by Fireball - October 11, 2018 at 2:28 am
RE: Studying Mathematics Thread - by Aliza - October 11, 2018 at 10:08 am
RE: Studying Mathematics Thread - by Grandizer - October 11, 2018 at 7:31 pm
RE: Studying Mathematics Thread - by Fireball - October 11, 2018 at 8:34 pm
RE: Studying Mathematics Thread - by Reltzik - October 12, 2018 at 11:34 pm
RE: Studying Mathematics Thread - by Aliza - October 13, 2018 at 2:07 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - October 13, 2018 at 6:40 pm
RE: Studying Mathematics Thread - by Fireball - October 13, 2018 at 8:15 pm
RE: Studying Mathematics Thread - by polymath257 - October 14, 2018 at 11:33 pm
RE: Studying Mathematics Thread - by Grandizer - October 27, 2018 at 11:39 am
RE: Studying Mathematics Thread - by Fireball - October 27, 2018 at 11:54 am
RE: Studying Mathematics Thread - by Aliza - October 28, 2018 at 12:23 am
RE: Studying Mathematics Thread - by Grandizer - October 28, 2018 at 7:42 pm
RE: Studying Mathematics Thread - by Silver - October 30, 2018 at 6:03 am
RE: Studying Mathematics Thread - by Aliza - October 30, 2018 at 10:55 am
RE: Studying Mathematics Thread - by Grandizer - October 31, 2018 at 10:14 am
RE: Studying Mathematics Thread - by Grandizer - November 3, 2018 at 1:15 am
RE: Studying Mathematics Thread - by Grandizer - November 3, 2018 at 6:30 am
RE: Studying Mathematics Thread - by Fireball - October 30, 2018 at 6:44 am
RE: Studying Mathematics Thread - by Jehanne - November 2, 2018 at 9:02 am
RE: Studying Mathematics Thread - by Macoleco - November 3, 2018 at 11:15 pm
RE: Studying Mathematics Thread - by polymath257 - November 4, 2018 at 9:26 am
RE: Studying Mathematics Thread - by Jehanne - November 5, 2018 at 9:55 am
RE: Studying Mathematics Thread - by polymath257 - November 5, 2018 at 1:03 pm
RE: Studying Mathematics Thread - by Jehanne - November 6, 2018 at 8:37 am
RE: Studying Mathematics Thread - by polymath257 - November 7, 2018 at 1:31 pm
RE: Studying Mathematics Thread - by Jehanne - November 7, 2018 at 3:51 pm
RE: Studying Mathematics Thread - by polymath257 - November 7, 2018 at 5:30 pm
RE: Studying Mathematics Thread - by Jehanne - November 8, 2018 at 11:34 pm
RE: Studying Mathematics Thread - by polymath257 - November 9, 2018 at 10:20 am
RE: Studying Mathematics Thread - by Silver - November 8, 2018 at 10:22 am
RE: Studying Mathematics Thread - by bennyboy - November 19, 2018 at 12:48 am
RE: Studying Mathematics Thread - by Fireball - November 19, 2018 at 1:28 am

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