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One of the main complaints that students have when struggling with their math homework is that they don't understand why they need to learn this in the first place.  After all, how often do we actually use calculus or trigonometry in our daily lives?   I always make an effort to correct this false assumption in my students.  Everything that we learn in math connects to reality in often unexpected ways.  For this reason, I like to find out what it is that interests my student, or what their career goals are, so that I may show them how the math connects.   Take the example of logarithms.  For the student with an ear for music, I can explain how logarithmic scales describe the relationships between musical tones, and true understanding of musical theory requires an understanding of this field of math.  For the student who plans to go into the medical field, logarithms can be used to help model the levels of medications in a patient's... read more

Hint 1; Complex numbers are not more complicated than any other numbers, just different. Complex number = real number + imaginary number Hint 2; imaginary numbers are not more ethereal than real numbers, just different. Imaginary numbers were invented to solve problems involving square roots of negative numbers. Hint 3; we lied to you when we told you that you cannot take the square root of a negative number. We just waited until now to tell you how to do it. i = √(-1) The letter i is used to signify the square root of -1. Any number multiplied by i is an imaginary number. Thus the square root of any negative number equals the square root of its positive value multiplied by i. Yes, it really is that easy. Examples √(-25) = 5i √(-7) =  i√(7) √(-12) = i√(12) = 2i√(3) The powers of i follow a repeating pattern that is illustrated below. i0 = 1 because any number (even imaginary ones) raised to the power of zero equals... read more

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