Raising things to i power, I came across 4.8104773809655

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In summary, the conversation discusses the meaning of taking something to the i power and the discovery of a number, approximately 4.8104773809655, which when raised to the i power equals i. It is also mentioned that the only solution for i^x = 0 is when x is a complex number and a=0. The conversation also looks at the equation a^i = e^(i ln(a)) to determine when it approaches zero. Finally, it is concluded that e^(ix) approaches zero as x goes to infinity.
  • #1
SteveRives
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I was working on the general meaning of taking things to the i power -- I was pondering the meaning of

e^pi*i = -1

The proof for this thing is well established, I was musing about meaning. An obvious question was:

What to the i power goes to zero?

As I was hunting for that number (on my TI-83 while driving to work -- and it is very hard to press [2nd] i on that thing with one hand), I came across this number:

4.8104773809655

Namely:

4.8104773809655^i => i

...at least on my TI-83. 4.81047738096535^i on the google calculator.

What is this number, and who has done work on it?

Regards,

Steve Rives
 
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  • #2
For:

[tex]x^i = i[/tex]

The solutions for x are:

[tex]e^{\frac{\pi}{2} + n\pi} \quad \forall n \in \mathbb{Z}[/tex]

So if you hadn't already guessed:

[tex]e^{\frac{\pi}{2}} \approx 4.8104773809653516554730356667 \ldots[/tex]

As for the equation:

[tex]i^x = 0[/tex]

Well there exists no complex solution for x. In fact the only time:

[tex]a^b = 0[/tex]

For a and b in complex numbers is when [itex]a=0[/itex] and [itex]b \neq 0[/itex].
 
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  • #3
Of coruse! Only, I found it the hard way just now:

ln(4.81047738...) is 1.5707963

And that's pi/2
 
  • #4
he did not ask when i^x = 0, he asked (what)^i -->0?

so we might look at a^i = e^(i ln(a)) = cos(ln(a)) + i sin(ln(a)).

but again it is clear that this number has absolute value 1, so cannot approach zero, at least not for real a.

now for complex a, we just need to solve for when e^z goes to 0.

but e^(-n)-->0 for example, so therefore e^i(in) -->0 too. so x = in satisfies

e^(ix) goes to zero as n goes to infinity, i.e. as x goes to i.infinity.
 

1. What does it mean to raise something to the i power?

Raising a number to the i power is a mathematical operation called complex exponentiation. It involves using the imaginary number i, which is equal to the square root of -1, as the exponent. The result is a complex number with a real part and an imaginary part.

2. How do you calculate a number raised to the i power?

To calculate a number raised to the i power, you can use the formula ai = ei*ln(a), where a is the base number and e is the mathematical constant 2.71828. First, find the natural logarithm (ln) of the base number, then multiply it by i. Finally, raise e to the power of the product.

3. What is the significance of 4.8104773809655 in raising something to the i power?

The number 4.8104773809655 is likely the result of a calculation involving complex exponentiation. It is a complex number with a real part and an imaginary part, and its exact value can vary depending on the base number and the exponent used.

4. What are some real-world applications of raising things to the i power?

Complex exponentiation has many practical applications, such as in electrical engineering, signal processing, and quantum mechanics. It is also used in solving differential equations and in understanding the behavior of systems with oscillating components.

5. How is raising something to the i power related to the concept of complex numbers?

Raising a number to the i power is one way to create a complex number, which has both a real and an imaginary component. Complex numbers are important in mathematics and physics, and they are used to represent quantities that cannot be expressed using only real numbers, such as the square root of a negative number.

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