Algebraic Manipulation of Euler's Identity Leads to a Strange Result

In summary, the conversation discusses the use of Euler's identity and how applying properties of logs to undefined numbers can result in contradictory answers. It also mentions the multivalued nature of logs in the complex domain.
  • #1
amolv06
46
0
I was playing around with Euler's identity the other day. I came across something that seems contradictory to everything else I know, but I can't really explain it.

I started with

[tex]e^{i\pi} = -1[/tex].

I rewrote this as

[tex]ln[-1] = i\pi[/tex]

Multiplying by a constant, we have

[tex]kln[-1] = ki\pi[/tex]

and using log properties I arrived at

[tex]ln[-1^{k}] = ki\pi[/tex]

Now if I set k equal to any even number, I have

[tex]ln[1] = 0 = ki\pi[/tex]

This seems to imply that [tex]i\pi[/tex] is 0, however it is not. Furthermore, any even value of k gives the same answer. Why is this?
 
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  • #2
ln(-1) = (2/2)*ln(-1) = (1/2)*2*ln(-1) = (1/2)*ln(-1^2) = (1/2)*ln(1) = (1/2)*0 = 0

Therefore, ln(-1) = 0.

In order for properties of logs to work, you must be taking the log of a valid number. You can not take the log of a negative number, therefore, when you apply the properties of logs to something that is undefined, you get ludicrousness.
 
Last edited:
  • #4
Thanks!
 

1. What is Euler's Identity?

Euler's Identity is a mathematical equation that connects the five most fundamental numbers in mathematics: 1, 0, π, e, and i (the imaginary number). It is written as e + 1 = 0 and is often considered one of the most beautiful and elegant equations in mathematics.

2. What is algebraic manipulation?

Algebraic manipulation is the process of rearranging and manipulating mathematical expressions using basic algebraic properties such as the distributive property, associative property, and commutative property.

3. How does algebraic manipulation change Euler's Identity?

When algebraically manipulating Euler's Identity, we can rearrange the equation in a variety of ways using basic algebraic properties. This can lead to different, but equivalent, forms of the equation, which may have unexpected results.

4. What is the strange result that can arise from algebraic manipulation of Euler's Identity?

The most commonly known strange result that can arise from algebraic manipulation of Euler's Identity is e = -1, which is often referred to as "Euler's formula." This result is strange because it connects three seemingly unrelated numbers (e, i, and π) in a simple equation that equals a negative real number.

5. How is algebraic manipulation of Euler's Identity used in real life?

Algebraic manipulation of Euler's Identity is used in various areas of mathematics, physics, and engineering to solve complex problems and understand the relationships between different mathematical concepts. It also has applications in signal processing, electrical engineering, and quantum mechanics, among others.

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