Many concepts in fundamental physics and engineering depend on the existence of what we call “imaginary numbers.” Imaginary numbers have important practical consequences for our everyday lives, and they are key to understanding the philosophical implications of Quantum Mechanics. Imaginary numbers are used in Schrodinger’s Equation, and they are used in fields such as stability analysis, AC current analysis, and many others. Yet, these are numbers that are impossible to see or touch. The only numbers that we can physically see and touch are positive real numbers, which are represented by this number line. But, there also exists another very useful set of numbers called negative real numbers, which are represented by this other number line. In addition to positive and negative numbers, there exists yet another set of mysterious numbers which are just as real, even though we have chosen not to refer to them as real. Instead, we have decided to refer to these numbers as imaginary, and these are represented by these two new number lines. A number can appear anywhere along any of these four number lines. A number can also appear anywhere in the plane formed by these number lines, in which case the number is the sum of a real number and an imaginary number. The set of all the numbers that can appear in this plane are what we refer to as “complex numbers.” We can represent each complex number as an arrow, as shown. When two complex numbers are added together, their arrows add together like vectors to produce the result. This means that their real portions will add together, and their imaginary portions will add together. A complex number can be represented by its real component and its imaginary component. Or, the complex number can instead be represented by the length of its arrow, and the angle that this arrow makes with the positive real axis, in the counterclockwise direction. Suppose we have two complex numbers, represented by these two arrows, and we multiply them together. Their product will be represented by a new arrow. The length of the new arrow is the product of the length of the two original arrows. And the angle of this new arrow is the sum of the angles of the two original arrows. The length of the new arrow is the product of the length of the two original arrows. And the angle of this new arrow is the sum of the angles of the two original arrows. Suppose that we have a number that is represented by an arrow with a length of one, and an angle of 90 degrees. We will refer to this number as “i”. If we multiply “i” by itself, the product will be represented by an arrow. This arrow will have a length of 1 multiplied by 1. And it will have an angle of 90 degrees plus 90 degrees. Therefore, the new arrow will have a length of one, and an angle of 180 degrees. This number is negative 1. Therefore “i” multiplied by “i” is exactly equal to negative one. We can represent this by saying that i squared is equal to negative one. Or, we can represent this by saying that the square root of negative 1 is equal to i. If we have a function with only real numbers as inputs and outputs, then we can represent it like this, with one axis for the input, and one axis for the output. If we have a function with complex numbers as inputs, then we need two axes just to represent the input. We need one axis represent the real part of the input, and the other axis to represent the imaginary part of the input. The output of the function is also a complex number which would need two additional axes to be represented. The output of the function is a complex number that can be thought of as an arrow with a length and an angle, as was shown before. Let us call the length of the arrow the “magnitude” of the output. And let us call the angle of the arrow the “phase” of the output. But, since we already have two axes for the input, and we are limited to only a total of three spatial dimensions, let us represent the output with just a single axis. This new axis will only signify the “magnitude” of the output, using a logarithmic scale. The phase of the output will be signified by the color. This is the graph for the function where the output is exactly equal to the input. The magnitude of this function at the center of the graph is zero. When the magnitude of this function is zero, it is represented on a logarithmic scale by the center of a bottomless funnel, due to the fact that zero on a logarithmic scale is represented by a number that approaches negative infinity on the axis. Now let us consider this other function. Here, the magnitude of the function at the center of the graph approaches positive infinity, due to the fact that we are trying to divide by a number that approaches zero at the center of the graph. Also, the colors depicting the angle of the complex function are now the mirror image of what they were before. This is because when we divide by a complex number, its angle is subtracted from the angle of the result. Now let us consider this function. This function has two places where the magnitude of the output approaches infinity, and one place where the magnitude of the output approaches zero. The one place where the output approaches zero is the one place where the function’s numerator approaches zero. The two places where the denominator approaches zero are the two places where the output approaches infinity. K is a constant. If we change the value of this constant, then we change the locations where the magnitude of the function approaches infinity. Let us call the places where the magnitude becomes infinity the “poles” of the function. And let us call the place where the magnitude becomes zero the “zero” of the function. As K becomes very large, one of the poles moves far away, while the other pole approaches the location of the zero. When K is exactly equal to 4, the two poles are at the same location. The two poles are also at the same location when K is exactly equal to negative 4. And when K becomes a very large negative number, one of the poles again moves far away, while the other pole again approaches the location of the zero. Now let us consider this function. The exponential of a complex number looks like the following. If the imaginary part of the complex number is zero, then this formula becomes a simple exponential, and an exponential function plotted on a logarithmic scale looks like a straight line. On the other hand, if the real part of the complex number is zero, then we get this result, where the magnitude of the function is always exactly equal to one, and it is only the phase of the output that changes. As it turns out, “e” raised to the power of a complex number has the following formula. A complex number with a magnitude of one can be written as follows. By rearranging the terms, and using the properties of trigonometry, we can get the following formula for the cosine of an imaginary number And we can get the following formula for the sine of an imaginary number Although the sine and cosine of any real number always has a magnitude less than or equal to one, the magnitude of the sine and cosine of an imaginary number grows exponentially. The plot for the sine of the inverse of an imaginary number looks like this. Here are some other examples of functions of complex variables. The existence of complex numbers opens up the calculations in physics, mathematics, and engineering to an entire new world of possibilities, with extremely important practical consequences. 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