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A student was exploring different features of the graphing calculator and got the result
Ln(–1) = 3.14159265 i. Before you say that it is impossible … we cannot take logarithms of negative numbers … let us first admit that mathematics is a really big topic. It is difficult to be precise all the time in math. For example, in your first year in algebra, you probably saw the following problem:
Your first reaction was that that equation had no solutions. Nothing squared comes out to a negative one. However, later you learned that there are numbers called "complex numbers." These are written in the form a + bi where
.Thus, to solve the above equation, we do the following:
Thus, when you said x 2 + 1 = 0 has no solutions, what you meant was that it had no solutions in the real number system.
Thus, can we solve Ln(–1) = x using complex numbers?
First, let's check for ourselves the claim Ln(–1) = 3.14159265 i. If we put it into the calculator, the output does not immediately match up.
What gives? Did you know your calculator is set by default to work in the Real Number system? Did you know it could be reset to work in the Complex Number realm? Go to Mode and reset as shown in the screen below. Trying the Ln(–1) again reproduces the student's claim.
Hopefully you recognize our friend, the number p. Thus, the claim is that Ln(–1) = p i. Next, recalling that any logarithm is really just an exponent, we are allowed to rewrite the equation in exponential form, as follows.
Try that on you calculator … in both complex mode and real mode. Is it true, according to the machine?
There are times the calculator produces errors (often due to rounding). This is not the case. It is a mathematical fact that Ln(–1) = p i and e p i = –1. The reason is somewhat advanced. Rather than my trying to stumble through it, I suggest you check out the reason at the World of Mathematics web site.
It has been said that the identity e p i + 1 = 0, an equation connecting the fundamental numbers i, p, e, 1, and 0 (zero), is one of the most beautiful of all equations. The majesty of this equation is that the five fundamental numbers are joined in such a simple way, using addition and the raising to an exponent power (thus multiplication), which are the two most basic operations in mathematics. Thus in one simple equation, we see, arguable, the five most important numbers in math joined by the two most basic operations. The idea also extends to summation, trigonometry, and the field of analysis ... again … topics of fundamental importance. It is staggering that such fundamentally important ideas are linked so elegantly.
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