Imagine a veeerrryyyy long rocket travelling through space at “ludicrous” speed. From our stationary perspective the length of the rocket would appear shorter (length contraction), clocks on the rocket would be ticking more slowly (time dilation) and clocks spread throughout the rocket would show different times (asynchronous time).
Twelve tone music. Wow. Found this extraordinary creative, explanatory composition by ViHartViHart. I’m hooked! http://www.youtube.com/watch?v=4niz8TfY794
Complex roots can be used to calculate the exact trigonometric value for unusual angles; for example cos(2π/5) and sin(2π/5). Choosing z5=1 will produce five complex roots aligned as shown in the first diagram. All five roots must be equally spaced over 2π, so their radial spacing will be 2π/5.
A right triangle can be superimposed on z2 as shown in diagram 2 with the angle at the origin being 2π/5.
For z5=1 the modulus for all the roots =1, so z1 … z5 are all on a unit circle (diagram 3). Thus from trigonometry, the x-coordinate of z2= cos(2π/5) and the y-coordinate of z2 = sin(2π/5).
Calculating their value requires some clever maths.
In Cartesian form let z5=(x+yi)5. If z5=1+0i, then (x+yi)5=1+0i. So the first step is to expand (x+yi)5 collect Re() and Im() parts and equate the Re() part to 1. (Diagram 4 lines 1 and 2)
This real part contains both x and y. In order to solve for x we first eliminate y by returning to the fact that z2 is on a unit circle so x2+y2=1. Rearranging this gives y2=1-x2 which can be used as a substitute for y2 and y4 since y4=(y2)2. (Diagram 4 lines 3, 4 and 5).
Now we have an equation in x only and resorting to a CAS calculator to solve it yields three exact values for x (Diagram 5 lines 1 and 2).
As z2 is in the first quadrant we choose the positive value of x as cos(2π/5) and use this value to calculate y=sin(2π/5) (diagram 5 lines 3 and 4).
Found this fascinating link to the historically famous (1748) “Witch of Agnesi” curve which I read about in Jennifer Ouolette’s very readable book The Calculus Diaries, courtesy of Mildura library. You will need java to activate the applet. Persevere; my browser initially blocked the javascript.
http://dynamicgeometry.com/JavaSketchpad/Gallery/Trigonometry_and_Analytic_Geometry/The_Witch_of_Agnesi.html
See explanation in frame below.
Referred to by Emma Ayres on ABC Classic Fm this morning. Pythagoras would be pleased http://www.bbc.co.uk/news/science-environment-26151062
Found in Mildura library: The Calculus Diaries by Jennifer Ouellette, 2010. Chapter one recounts how Archimedes thwarted the initial invasion of Syracuse by the Romans in 213 BC by building a giant, curved array of mirrors which he used to focus the Sun’s rays on to Roman ships at anchor, causing them to catch fire. Myth Busters, however, have debunked the idea: http://www.discovery.com/tv-shows/mythbusters/videos/death-ray-minimyth.htm .
Enjoying reading Einstein’s Heroes by Monash University professor Robyn Arianrhod. In this very easy to read treatise on famous scientific mathematicians, Arianhrod pays tribute to Galileo, Newton, Faraday and especially James Clerk Maxwell who drew together and formalised the four field equations that completely describe electromagnetism. Indeed, Arianhrod points out that Maxwell’s equations enabled him to predict the existence of radio waves and conjecture that light was part of an extensive spectrum of electromagnetic radiation.
Apparently Arianhrod has written a second book entitled Seduced by Logic about two women of mathematics. I wonder if any of my readers can recommend it?
Excited to be starting my aussie maths tutoring page. Available in person for students in Mildura or via Skype.
aussiemathstutor 