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Complex Logarithms, part 2: Natural logs with a Complex Argument:
1. Numbers in Scripture, raising the dead, correct math, specific numbers with additional meaning, as in the Bible, are at paragraphs 5, 6 ,7 and 8.
IDENTICAL WORK AS FOR A NEGATIVE ARGUMENT:
2. The work done to compute the natural log of a complex argument is nearly identical to that done for a negative argument. This work was explained at hub#12.12( . . . negative argument); therefore, I will not be repeating that here. I try to keep redundancy minimized with these hubs. In this hub I did add L4, which shows the real and imaginary parts of both Eulers formula and the complex number. I explained the real and imaginary components in hub#12.12. At L4 this is similar to vectors in which the real and imaginary parts can be equated to one another. So the real part of Euler's formula is equal to the real part of the complex number, and the imaginary part of Eulers's formula is equal to the imaginary part of the complex number. This fact is written as two simultaneous equations at L5. All the work from L6 through L24 was explained at hub #12.12.
EULER'S ANGLE, AND THE COMPLEX NUMBER ANGLE:
3. Paragraph 7 of hub#12.12 tries to explain the difference between the angle in Euler's formula, and the angle inherent in the complex number, but I'll expound upon it a little more here. Euler's formula has a life of its own. It is derived with the complex analysis of the infinite series that represents e, the base of the natural logarithms. In that analysis the infinite series that represents both the cosine and sine functions both show up, and this is the foundation of Euler's formula. Euler's formula is a tool to do many things including solving equations and functions as how we are using it in these hubs on complex logarithms. The angle, in radians, that is represented in the imaginary part of Euler's formula is the angle for Euler's formula; however, the angle inherent within the complex number that Euler's formula can be viewed to represent is generally different. L4 is an equation; therefore, the right side, which is the complex number, equals the left side, which is Euler's formula, which is also the same complex number. θ in Euler's formula was calculated from L13 through L17 to be 2.1328 radians which is about 122 degrees. But the angle of the complex number was calculated at L21 through L23 to be 0.3413 radians or about 20 degrees. The A at L12 is simply a notation that I put the value of X in memory A of the calculator.
TWO POSSIBLE ANSWERS FOR THE ANGLE:
4. At L17 my calculator just happen to give me the correct answer of 2.1328 radians; however, there is another possible answer of 4.1404 radians, which is about 237.8 degrees( in the 3rd quadrant). This is because cosine of 237.8 degrees will also give a value of --0.53288. If you do problems like these, and your calculator gives you the wrong answer then you must be smarter than the calculator and know which angle to choose. Going to L5 we know the sine function of the 2nd equation will be positive. This puts the angle in the second quadrant because of the cosθ = a NEGATIVE 0.53288; therefore, the 2.1328 radians( 122.2 degrees) is the correct answer. Don't forget that those three dots that look like a triangle at L20 means, "therefore".
E1. This is an edit done on 12-4-2012. Let's take a quick look at another example. At EL1 we want to find the natural log of the imaginary number i. It is all pretty self explanatory since we follow the same procedure outlined athub#12.12, and this one. At EL3 even though the real part is zero, we still write it out because we equate excos2θ to zero at EL4. When any real base raised to a power equals 1, then that power must be zero, which is noted at the end of EL5. Our final answer is at EL7.
NUMBERS IN SCRIPTURE:
5. For a birthday present a good friend gave to me, "Numbers in Scripture" by E.W. Bullinger. This book is available through Kregel Publications.Fulfillment of prophesy( link to my hub on it) is all the evidence I need to prove the Bible is supernaturally inspired by God, but flipping through Bullinger's book gives even more evidence of that fact. There is no human ever born, who has sinned, that could have been smart enough to exquisitely lace the numbers of Scripture, with their hidden meanings, throughout the pages of Scripture, not to mention to get life itself to coincide with those numbers.
BRINGING THE DEAD TO LIFE:
6. If you like to solve puzzles then stop reading now, and see if you can figure out why I chose the particular numbers I used in this hub. If you want to cut to the chase and get to the answer then just continue reading. At hub#12.12 I said 3 times( par.6 twice, and par.14 once) that with God all things are possible. Being born again is one of those impossible things that God makes possible through Jesus Christ. God gives us a measure of faith( Romans 12:3), and draws us( John 6:44), and as we obey( proven at par. 9 of hub#12:10) we are born again. Before that, we were dead in our sins( Ephesians 2:1 and 5; Colossians 2:13). I'm no theologian, and one can argue that we are brought to life prior to obedience as the Scripture I just listed indicate; however, I think Scripture is clear that the manifestation of saving faith( the kind that gets us to heaven) is obedience. The fact is that God brings us to life, and if we obey His Word we are secured in His Book of Life.
COMPLEX NUMBERS MAKE THE IMPOSSIBLE, POSSIBLE:
7. As explained at paragraph 6 of hub#12:12, complex numbers make it possible to calculate the log of a negative or complex argument. To get an accurate answer we must follow the laws of math that God structured into the field of complex numbers. An accurate answer is the only right door, and narrow gate that will get us there. It is like the correct password with 11 letters and 1 space.
THE NUMBERS CHOSEN IN THIS HUB:
8. Using Bullinger's book as a guide, the number six is the number for man. Man has 3 qualities: spirit, soul and body. 6 raised to the third power is 216. A minus 216 represents hell, buried in the ground, the center of the earth and death. The number 7 is the number of spiritual perfection, and God is spiritually perfect. There are 3 persons in the Trinity: God, Jesus Christ and the Holy Spirit. 7 raised to the 3rd power is 343. A positive 343 represents above the earth, heaven. The 343 is in the imaginary coordinate, which is another dimension outside of the real number line, the X-axis. Imagine going from the confinement of a single line to the freedom of a plane. This is analogous of us going from 3 spacial dimensions to 4. If God's laws of complex numbers are obeyed then the dead minus 6^3 number is brought from the chains of the real number line to the field of complex numbers as the vector at L24 indicates.