Can Complex Numbers in Polar Format Be Equated Like Real Numbers?

[KrayzBlu]

Hi,

We know that if we have two complex numbers in polar format (i.e., magnitude and exponential), that for two complex vectors

z1 = A*exp(iB)
z2 = C*exp(iD)

If z1 and z2 are equal, then A = C and B = D. However, this is assuming these values are all real. What if they are complex? I.e. can we say if we have two complex numbers

z3 = (a+ib)*exp(c+id)
z4 = (e+if)*exp(g+ih)

If z3 and z4 are equal, can we say that (a+ib) = (e+if) and (c+id) = (g+ih)?

Thanks

 

[SteveL27]

Hi,

We know that if we have two complex numbers in polar format (i.e., magnitude and exponential), that for two complex vectors

z1 = A*exp(iB)
z2 = C*exp(iD)

If z1 and z2 are equal, then A = C and B = D.

Not quite true. B and D must differ by an integer multiple of 2pi. You'll need to take that into account when working out the rest of this.

 

[KrayzBlu]

Not quite true. B and D must differ by an integer multiple of 2pi. You'll need to take that into account when working out the rest of this.

Thanks for pointing this out, SteveL27, I should have said B = D +/- n*2*π, where n is any integer.

 

[AlephZero]

If z1 and z2 are equal, then A = C and B = D. However, this is assuming these values are all real.

You can have A = -C, if B and D are different by an odd multiple of π

z3 = (a+ib)*exp(c+id)
z4 = (e+if)*exp(g+ih)
If z3 and z4 are equal, can we say that (a+ib) = (e+if) and (c+id) = (g+ih)?

It should be easy to see why that is false. For example take
a = 1, b = c = d = 0, e = 0, f = 1, and find g and h to make z3 = z4.

If you convert z3 = x3 + i y3 and z3 = x4 + i y4, you only have 2 equations (x3 = x4 and y3 = y4) but 8 unknowns (a through h). You need 6 more equations before you can hope there is a unique solution.

 

[CellCoree]

for your question! The answer is yes, we can say that (a+ib) = (e+if) and (c+id) = (g+ih) if z3 and z4 are equal. This is because complex numbers follow the same rules of equality as real numbers. Just like how we can equate two real numbers if they have the same value, we can also equate two complex numbers if they have the same value.

In the case of z3 and z4, we can break them down into their real and imaginary parts and equate them separately. This is because the real and imaginary parts of complex numbers behave independently from each other, much like how the x and y components of a vector behave independently.

Therefore, if z3 and z4 are equal, it follows that their real parts (a+ib) and (e+if) must also be equal, and their imaginary parts (c+id) and (g+ih) must also be equal. This is a fundamental property of complex numbers and is known as the "equality principle."

In conclusion, we can equate complex numbers just like we can equate real numbers, as long as they have the same value. This allows us to perform operations on complex numbers with ease and use them in various mathematical and scientific applications.