Intuitive understanding of Euler's identity?

In summary, the conversation discusses gaining an intuitive understanding of Euler's identity and how raising e to the power of i rotates the real value into the imaginary plane. It is explained that this property is exclusive to e^x and that raising 2^{i*pi} can also be given a value using the definition of e^{ix}. It is also mentioned that understanding imaginary space can be useful in understanding wavefunctions expressed in exponential form. The difference between log and ln is also clarified.
  • #1
HuskyLab
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I'm trying to get a more intuitive understanding of Euler's identity, more specifically, what raising e to the power of i means and why additionally raising by an angle in radians rotates the real value into the imaginary plane. I understand you can derive Euler's formula from the cosx, sinx and ex Taylor series with the addition of i to form the identity. I understand the algebra but is this property only exclusive to e^x ? Does raising let's say 2^{i*pi} mean anything at all? The only thing we are changing is the base, after all e is just a constant. I had a quick look at the Taylor series for 2^x but by a quick comparison, there were some nasty constants, no direct relation seemed apparent. If I said anything that's wrong just me know. Thanks.
 
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  • #2
HuskyLab said:
Does raising let's say 2^{i*pi} mean anything at all?
One way to get an intuitive sense is as follows:

##e^x## is defined to be
$$\lim_{n\to\infty}\left(1+\frac xn\right)^n$$
So to give a value for ##e^{ix}## we look at the sequence ##(s_n)_{n\in\mathbb N}## such that ##s_n=\left(1+\frac {ix}n\right)^n##. We can write ##s_n = a_n+b_ni## in real and imaginary parts and then prove, by expanding each sequence element using the binomial expansion, that for real ##x##, the sequences ##(a_n)## and ##(b_n)## converge, say to values ##a## and ##b##. Then we conclude that
$$e^{ix} =
\lim_{n\to\infty}\left(1+\frac {ix}n\right)^n
=\lim_{n\to\infty}\left(a_n+ib_n\right)
=\lim_{n\to\infty}a_n+i\lim_{n\to\infty}b_n
=a+ib$$

Having done that, we can then define ##2^{ix}=\left(e^{\log 2}\right)^{ix}=e^{i(x\log 2)}## and apply the definition of ##e^{ix}## that we just made, to get a value for this.

Note that we did not need to use either Taylor series or trig functions anywhere in this.
 
  • #3
andrewkirk said:
One way to get an intuitive sense is as follows:

##e^x## is defined to be
$$\lim_{n\to\infty}\left(1+\frac xn\right)^n$$
So to give a value for ##e^{ix}## we look at the sequence ##(s_n)_{n\in\mathbb N}## such that ##s_n=\left(1+\frac {ix}n\right)^n##. We can write ##s_n = a_n+b_ni## in real and imaginary parts and then prove, by expanding each sequence element using the binomial expansion, that for real ##x##, the sequences ##(a_n)## and ##(b_n)## converge, say to values ##a## and ##b##. Then we conclude that
$$e^{ix} =
\lim_{n\to\infty}\left(1+\frac {ix}n\right)^n
=\lim_{n\to\infty}\left(a_n+ib_n\right)
=\lim_{n\to\infty}a_n+i\lim_{n\to\infty}b_n
=a+ib$$

Having done that, we can then define ##2^{ix}=\left(e^{\log 2}\right)^{ix}=e^{i(x\log 2)}## and apply the definition of ##e^{ix}## that we just made, to get a value for this.

Note that we did not need to use either Taylor series or trig functions anywhere in this.
Thanks, is that a log(base 10) in ##e^{i(x\log 2)}##? I understand the algebra you mention although I'm trying to gain an understanding of how best to understand imaginary space, particularly its application to wavefunctions expressed in exponential form. One thing I find very strange is that if you include a log constant in the first expression for ##e^{ix}## you get ##({e^{log10}})^{ix}## it doesn't seem to follow any relation in regards to ##n^{ix}=\left(e^{\log n}\right)^{ix}=e^{i(x\log n)}## for n=2,3,4,etc If the log was a natural log on the other hand I do see a pattern.
 
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  • #4
HuskyLab said:
Thanks, is that a log(base 10) in ##e^{i(x\log 2)}##? I understand the algebra you mention although I'm trying to gain an understanding of how best to understand imaginary space, particularly its application to wavefunctions expressed in exponential form. One thing I find very strange is that if you include a log constant in the first expression for ##e^{ix}## you get ##({e^{log10}})^{ix}## it doesn't seem to follow any relation in regards to ##n^{ix}=\left(e^{\log n}\right)^{ix}=e^{i(x\log n)}## for n=2,3,4,etc If the log was a natural log on the other hand I do see a pattern.
I'm reasonably certain that andrewkirk's intention is that log 2 means the natural log of 2. My preference is that if you mean "natural log," you should write ln.

Also, ##e^{ix} \ne (e^{\log 10})^{ix}## unless "log" here means "log10", which is not what andrewkirk was intending.
 
  • #5
Mark44 said:
I'm reasonably certain that andrewkirk's intention is that log 2 means the natural log of 2. My preference is that if you mean "natural log," you should write ln.

Also, ##e^{ix} \ne (e^{\log 10})^{ix}## unless "log" here means "log10", which is not what andrewkirk was intending.
Thanks for the reply. Ok, yeah, it's just that it wasn't written as ln in the equation, so one would assume it's ##log_{10}##. I have always known log with no subscript to mean ##log_{10}## (because it's the most common), not sure about other people.
 
  • #6
HuskyLab said:
Thanks for the reply. Ok, yeah, it's just that it wasn't written as ln in the equation, so one would assume it's ##log_{10}##. I have always known log with no subscript to mean ##log_{10}## (because it's the most common), not sure about other people.
It really depends on the context. In more advanced math books, ##\log## can mean ##\ln##, the natural logarithm. In some computer science textbooks, ##\log## can mean ##\log_2##. When I first learned about logarithms, many years ago, ##\log## meant the common log, ##\log_{10}##. I'm told that at least in some cases, this isn't true any longer.
 
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  • #7
HuskyLab said:
I'm trying to get a more intuitive understanding of Euler's identity, more specifically, what raising e to the power of i means and why additionally raising by an angle in radians rotates the real value into the imaginary plane.
It's really the difference between the exponential function of a real argument versus the exponential function extended to the complex numbers.

As you are aware, the Maclaurin series of ##e^x## is ##1 + x + \frac {x^2}{2!} + \dots##. The series for ##e^{x + iy}## can be obtained simply by substitution into the series I showed, and can be manipulated to obtain ##e^{x + iy} = \cos(x) + i\sin(y)##. If we let x = 0 and y = ##\pi##, it's easy to see that ##e^{i\pi} = \cos(0) + i\sin(\pi) = 1##, a very elegant equation involving some of the most fundamental constants in mathematics.

It's also easy to see that ##e^{i\theta}## represents a point on the unit circle identified by letting a ray from the origin to (1, 0) sweep around CCW through an angle of ##\theta## radians.

BTW, the usual description is "complex plane," not "imaginary plane."
 

Related to Intuitive understanding of Euler's identity?

1. What is Euler's identity?

Euler's identity is a mathematical equation that relates five fundamental mathematical constants: 0, 1, π, e, and i. It is written as e^(iπ) + 1 = 0, where e is the base of the natural logarithm, i is the imaginary unit (√-1), and π is the ratio of a circle's circumference to its diameter.

2. What is the intuitive understanding of Euler's identity?

The intuitive understanding of Euler's identity is that it connects seemingly unrelated mathematical constants in a single, elegant equation. It also demonstrates the concept of complex numbers, which have both real and imaginary components, and how they can be expressed using exponential notation.

3. How is Euler's identity used in mathematics?

Euler's identity is used in various branches of mathematics, including calculus, complex analysis, and number theory. It is also a fundamental part of many mathematical proofs and is often used to simplify complex equations.

4. Why is Euler's identity important?

Euler's identity is important because it provides a deeper understanding of the relationships between different mathematical concepts. It has also been called "the most beautiful equation in mathematics" due to its simplicity and elegance.

5. Is there a real-world application for Euler's identity?

While Euler's identity may not have a direct real-world application, it has been used to develop various mathematical models and theories that have practical applications. For example, complex numbers, which are essential in Euler's identity, are used in engineering, physics, and signal processing.

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