- #1
Lonewolf
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How do we express complex powers of numbers (e.g. 21+i) in the form a+bi, or some other standard form of representation for complex numbers?
Exploring Representations of Complex Powers
- Thread starter Lonewolf
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In summary, to express complex powers of numbers in the form a+bi or some other standard form of representation for complex numbers, we can use Euler's expression exp(ix) = cos(x) + i sin(x) and logarithm rules to manipulate the expression and solve for the desired form. For example, for 21+i, we can use the equation e^(1+i)ln2 = 21+i and apply Euler's identity to get the final form of 2 cos(ln2) + 2i sin(ln2). This method can be applied to any complex power of a number.
- #2
HallsofIvy
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First, of course, 21+i= 2*2i so the question is really about 2i (or, more generally, abi).
Specfically, look at eix.
It is possible to show (using Taylor's series) that
e^(ix)= cos(x)+ i sin(x).
a^(bi)= e^(ln(a^(bi))= e^(bi*ln(a))= cos(b ln(a))+ i sin(b ln(a))
= cos(ln(a^b))+ i sin(ln(a^b))
For your particular case, 2^i= cos(ln(2))+ i sin(ln(2))
= 0.769+ 0.639 i.
2^(1+i)= 2(0.769+ 0.639i)= 0.1538+ 1.278 i.
Specfically, look at eix.
It is possible to show (using Taylor's series) that
e^(ix)= cos(x)+ i sin(x).
a^(bi)= e^(ln(a^(bi))= e^(bi*ln(a))= cos(b ln(a))+ i sin(b ln(a))
= cos(ln(a^b))+ i sin(ln(a^b))
For your particular case, 2^i= cos(ln(2))+ i sin(ln(2))
= 0.769+ 0.639 i.
2^(1+i)= 2(0.769+ 0.639i)= 0.1538+ 1.278 i.
- #3
Lonewolf
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Now why didn't I see that? Oh well, thanks for pointing it out.
- #4
chroot
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You're no doubt familiar with Euler's expression
exp(i x) = cos(x) + i sin(x)
You're probably also familiar that logarithms can be expressed in any base you'd like, like this:
loga x = ( logb x ) / ( logb] a )
For example, if your calculator has only log base 10, and you want to compute log2 16, you could enter
log10 16 / log10 2
We can put these facts together to good use.
To start with, let's try a simple one: express 2i in the a + bi form. We can express 2i as a power of e by solving this equation:
2i = ex
i ln 2 = x
We've just used the logarithm rule I described above in "reverse." So we've just changed the problem to expressing exp(i ln 2) in a + bi form. Now we can just apply Euler's identity, and we get
exp(i ln 2) = cos(ln 2) + i sin(ln 2).
Thus 2i = cos(ln 2) + i sin(ln 2), as we wished to find.
Now let's try 21 + i. I'm going to skip all the fanfare and just show the steps.
21+i = ex
(1+i) ln 2 = x
e(1+i) ln 2 = 21+i
eln 2 + i ln 2
eln 2 ei ln 2
2 ei ln 2
2 [ cos(ln 2) + i sin(ln 2) ]
2 cos(ln 2) + 2 i sin(ln 2)
Hope this helps.
- Warren
exp(i x) = cos(x) + i sin(x)
You're probably also familiar that logarithms can be expressed in any base you'd like, like this:
loga x = ( logb x ) / ( logb] a )
For example, if your calculator has only log base 10, and you want to compute log2 16, you could enter
log10 16 / log10 2
We can put these facts together to good use.
To start with, let's try a simple one: express 2i in the a + bi form. We can express 2i as a power of e by solving this equation:
2i = ex
i ln 2 = x
We've just used the logarithm rule I described above in "reverse." So we've just changed the problem to expressing exp(i ln 2) in a + bi form. Now we can just apply Euler's identity, and we get
exp(i ln 2) = cos(ln 2) + i sin(ln 2).
Thus 2i = cos(ln 2) + i sin(ln 2), as we wished to find.
Now let's try 21 + i. I'm going to skip all the fanfare and just show the steps.
21+i = ex
(1+i) ln 2 = x
e(1+i) ln 2 = 21+i
eln 2 + i ln 2
eln 2 ei ln 2
2 ei ln 2
2 [ cos(ln 2) + i sin(ln 2) ]
2 cos(ln 2) + 2 i sin(ln 2)
Hope this helps.
- Warren
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FAQ: Exploring Representations of Complex Powers
What is the purpose of exploring representations of complex powers?
The purpose of exploring representations of complex powers is to gain a deeper understanding of how complex numbers behave when raised to different powers. This allows for a better understanding of complex functions and their graphs, as well as their applications in various fields such as physics, engineering, and computer science.
How are complex powers represented in mathematics?
In mathematics, complex powers are represented using the notation zn, where z is a complex number and n is a real number. This is also commonly written as (a + bi)n, where a and b are real numbers and i is the imaginary unit.
What is the difference between real and complex powers?
The main difference between real and complex powers is that real powers result in real numbers, while complex powers can result in either real or complex numbers. Additionally, real powers can only be raised to whole number exponents, while complex powers can be raised to any real number exponent.
How do you graph complex powers?
To graph complex powers, you can first convert them to polar form using the formula zn = rn(cos(nθ) + i sin(nθ)). Then, plot the values of rn on the radial axis and the values of nθ on the angular axis. The resulting graph will be a spiral shape known as a polar graph.
What are the applications of exploring representations of complex powers?
Exploring representations of complex powers has many applications in various fields such as physics, engineering, and computer science. For example, in physics, complex powers are used in the study of electrical circuits and quantum mechanics. In engineering, they are used in signal processing and control systems. In computer science, they are used in image processing and data encryption.
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