>> The next topic in mathematics that I want to cover is Algebra and Linear Algebra. First of all, we'll look at complex numbers and logarithms and then matrices and determinants. First of all, in complex numbers, the definitions of complex numbers are given here. And the definition is Z, a complex number, is equal to a+jb where in that equation A is the real component and B is the imaginary component and J is the square root of -1 although many disciplines and more commonly I is used instead of j, but here we'll use j just to follow the notation of the handbook. So a complex number can be displayed in terms of the diagram which is shown here, which is sometimes called an Argand diagram, where the horizontal axis is the real axis or the value of the real component or the real part of the imaginary number and the vertical axis is the imaginary axis which shows the magnitude of the imaginary part. So, the point here Z, is A plus JB and this can also be expressed in polar form where the radius here is C and the magnitude of the radius is just equal to the square root of a squared plus b squared and the angle theta. And manipulations on complex numbers can be performed fairly easily. For example, the summation of two complex numbers, Z1 is A + JB1. Z2 is A2 + JB2, then to sum those equations we simply add up the real components and add up the imaginary components as shown here. In the Polar Coordinates System, we can express it like this where r is the radius, or the magnitude of the vector, or the imaginary number. And, in this case, we have r is equal to cosign theta and Y, X is equal to r cosign theta and Y component is equal to sign theta, etcetera. And the rest of the equations as given here. Another convenient form to express this in terms of Euler's identity, exponential E to the J theta is equal to cosign theta plus J sign theta, is another way to express the complex number. So let's do an example on that. The polar coordinates of a complex number are magnitude four and angle 56 degrees. It's rectangular components or coordinates are most nearly which of these. So here we simply use the definitions that x, the real component, is equal to r cosign theta is equal to four cosign 56 degrees is 2.24. Y, the imaginary component, is r sign theta which is equal to 4 sign 56 degrees which is equal to 3.32. So the answer is C and you can also always confirm for yourself you want that indeed the magnitude or the radius is equal to the square root of the sum of the squares of those two components. Next we look at logarithms and the basic definition of the logarithm of a number say x to the base b is written like this. Log of b of x is equal to c where b to the raised to the power c is equal to x, is the definition of a logarithm of base b. And in particular most commonly we use so called natural logarithms written as LN of X which is a logarithm raised to the base E. And also, logarithms to the base 10 usually written as LNG without the subscript 10 is written like that, are the most common ones. To convert the basis of two logarithms, we have this equation here. Log B of X is equal to log A of X divided by log A of B. So, for an example, we given that log of six is equal .778. And because it doesn't show the base there, implicitly this is log to the base 10 and similarly the Logarithm 8 is .9 over 3. The question is what is the value of log to the base six of eight of these alternatives? So we use our general equation for converting between bases. Log b of x is log a of x divided by log a of b. So in this case, we have logarithm to the base b of x is equal to log 10 of x divided by log 10 of b where x and b are the two values which were given so therefore the logarithm to the base 6 of 8 is log to the base 10 of 6 divided by log to the base 10 of 8. And these values we were given as .778 and point-nine-oh-three. And the ration when we compute that is equal to point-eight-six-two, so the answer is B. And this concludes our discussion of logarithms.
