math-square-journal

[math-square-journal]

[math-square-journal]

[image description: a mathematical formula on a plane white background. The formula reads |sin(x+iy) | = | sin(x) + sin(iy) | /end description]  ----------------------------------------------------------------------------------------------------------- Now, following along with the math in the proof to prove this....  I start with what I know, which is the sin addition formula and the relationship between sin and cos and complex exponentials, and also the sinh and cosh formulas. The latter would not be an obvious choice but it’s in the proof I’m following, and they do jump out as I simplify the algebraic expressions.  the sin addition formula gives: sin(a+b) = sin(a)cos(b) + cos(a)sin(b)  so sin(x+iy) = sin(x)cos(iy)+cos(x)sin(iy) given, sin(x) = (1/2i)*[e^ix - e^-ix] and cos(x) = (1/2)*[e^ix + e^-ix] and given, sinh(x) = 1/2[e^x - e^-x] and cosh = 1/2[e^x + e^-x] sin(x+y) = (1/2i)*[e^ix-e^-ix]*(1/2)*[e^i(iy)+e^-i(iy)] + stuff               = (1/4i)*[e^ix-e^-ix] [e^-y+e^y] + stuff                = sin(x)*cosh(y) + stuff                =  sin(x)*cosh(y) + cos(x)*(1/2i)*[e^i(iy)-e^-i(iy)]               =  sin(x)*cosh(y) + (1/2)[ e^ix + e^-ix]*(1/2i)*[e^-y - e^y)]               =  sin(x)*cosh(y) + (-1)(1/4i)[ e^ix + e^-ix]*[e^y - e^-y)] aside: since (-1)[u]*[w_1 - w_2] = u*[w_2-w_1]                =  sin(x)*cosh(y) + (-1/i)cos(x)*sinh(y) aside: (-1/i) = i so                =  sin(x)*cosh(y) + i*cos(x)*sinh(y) Now, the desired formula, |sin(x+iy) | = | sin(x) + sin(iy) |, involves absolute values so we take the absolute value. The absolute value of a complex number is the square root of the sum of the squares of it’s imaginary and real parts. This is intuitive and also what the proof says, so I’m going with it.  | sin(x+iy) |    = | sin(x)*cosh(y) + i*cos(x)*sinh(y) |                       = \sqrt( [sin^2(x)cosh^2(y) + cos^2(x)sinh^2(y)])  aside: we can use the classic trig identity sin^2 + cos^2 = 1                       = \sqrt(  [sin^2(x)cosh^2(y) + (1 - sin^2(x))sinh^2(y)])                      =  \sqrt( [sin^2(x)cosh^2(y) + sinh^2(y) - sin^2(x))sinh^2(y) ])                      = \sqrt( [sin^2(x)*[cosh^2(y)-sinh^2(y)] + sinh^2(y)]) aside: we can change sinh and cosh back into exponential forms to see that  cosh^2-sinh^2 = 1 so                       =  \sqrt( [sin^2(x) + sinh^2(y)]) Which is a very pretty expression but it’s not quite there. It looks like the next step is to hold that thought and then make the other side of the formula also equal to this same expression. Look at the two of them and say, done.  SO the other side of the expression is  | sin(x) + sin(iy) |  we have that sin(iy) = i*sinsh(y)  SO,

| sin(x) + sin(iy)| = | sin(x) + i*sinh*y)|  which just becomes \sqrt (sin^2(x) + sinh^2(y))  WHICH IS THE SAME EXPRESSION as before!!!! BOOM. We’re done. That’s the proof, we’ve proved the original expression.  -------------------------------------------------------------- [qed] thanks to: https://www.youtube.com/watch?v=-FSMTZkaF18 for the mathematical formula. I just paused the video and went off of the formulas I could see.  Notes: I should have noted before because it would have made the algebra easier lol: 

we are given,  sin(iz) = i*sinh(z)                 as we can see:                sin(iz) = (1/2i)[e^(iz)-e^-i(iz)]                            = (1/2i)[e^-z -e^z]                            = (-1/i)(1/2)(e^z-e^-z)                            =(i)sinh(z)  cos(iz) = cosh(z)                 as we can see:                cos(iz) = (1/2)[e^-z + e^z]                             = cosh(z) 

A cute complex analysis fun fact.

[math-square-journal]

Calendar math [image description: photos of notes on calendar related information and math. They are written minimistically to let the reader deduce the context without fully explaining. Photo 1. It begins with math sentences: 30*12 = 360 February = 28 so 31 --> 7 times Months = 1, -2, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1 Jan, Feb, Mar, April, May, June, July, Aug, Sep, Oct, Nov, Dec march april may is Mr. Apmy (as a pneumonic to help me remember the order) Photo 2. A two column chart labeled January-February and March-December.  There are a series of numbers descending in each column and to the right it is marked the that these numbers reflect the start year of 2020. The January-February column of numbers is reflective of the day of the week on which January 1st falls, while March-December reflects when March 4th falls. Each list starts with the number 4 but soon diverges. The lists are composed of numbers 1 through 7 because there are only 7 days in the week. I have recorded them here in groupings of 4 at a time for ease of my own visual comparison. The Jan-February sequence is: 4671, 2456, 7234, 5712, 3567, 1345, 6123  where forward arrows indicates the pattern is plus 2 for the year subsequent the leap year, and plus 1 otherwise moving forwards in time, and going backwards in time applies a minus 2 on years that are 1 year subsequent to a leap year and minus 1 otherwise.  The March-December sequence is:  4567, 2345, 7123, 5671, 3456, 1234, 6712 It’s not remarked upon in the chart but these two sequences are the same if the digits are rotated through by 23 digits. That is the January-February pattern is the same starting in 2020 as the March-December starting in 2043. Or by rotation by 5 since the pattern is the same starting March-Dec 2020 and Jan-Feb 2025.  Photo 3: