Why are complex numbers in the form a+bi?

  • Thread starter kramer733
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Does it have something to do with the quadratic form? What would i type on google to search for more information to get better search results?
 
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With that form one may describe a complex number as the sum of a real part (a) and an imaginary part (bi).

The form is known as "rectangular form."
 
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They are like that because complex numbers are constructed by taking the real numbers and then adding the new number i to them to form a new field of numbers. The results of the equations means that every complex number has a real and imaginary part and add as you would expect, so expressing them like that explains the idea that each complex number you have has a real part (the a) and an imaginary part (the b).

However, there are other ways you can express them. You can express them with polar coordinates like you can for points in the plane usually (each number is of the form r(cos([itex]\theta[/itex])+i.sin([itex]\theta[/itex])) where the r is the distance from the origin and the [itex]\theta[/itex] is the angle which you leave the origin from to go distance r to reach your point).

The neat thing about complex numbers is that r(cos[itex]\theta[/itex]+i.sin[itex]\theta[/itex])=[itex]re^{i\theta}[/itex] by something called Euler's formula, which allows you to express the number in the neat little form: [itex]z=re^{i\theta}[/itex]
 

HallsofIvy

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Does it have something to do with the quadratic form? What would i type on google to search for more information to get better search results?
To summarize what MisterX and Jamma said, we often write complex numbers like that because it is convenient (it fits our commonly used xy- coordinate system format) but there are many different ways in which we could write complex numbers: [itex]re^{i\theta}[/itex] is one. Engineers often use the format "[itex]r cis(\theta)[/itex]" which is short for "[itex]r(cos(\theta)+ i sin(\theta))= re^{i\theta}[/itex]" as Jamma said.

There is no "mathematical" reason- it is just a convention.
 
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But what about the "+" sign in between the real part and the imaginary part? Since graphing complex numbers on the complex plane is alot like graphing real numbers on the real plane, why didn't we use a comma inbetween the real parts and complex parts?

How do we know there are nice properties such as addition and multiplication for something in the form "A+Bi"? If we had instead used the convention "(A,Bi)" to graph complex numbers, then would it still have had addition and multiplication for complex numbers? It seems a bit odd to me. You can of course multiply a scalar with (a,bi) or add another complex number but it would've done differently.

with the current system, (a+bi)^2 = a^2 + 2(a)(bi) -b^2. would it still have resulted in the same if we would have multiplied (a,bi) with (a,bi)? How do we even do that?
 
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How do I go from cos(x+y) + sin(x+y)i

to

e^(i(x+y)

How do people on physicsforums put in proper notation instead of having to write down what i'm doing?
 

lurflurf

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Sometimes a comma is used. This is the type of thing done in and introduction to algebra. When we add a new element to a ring the addition and multiplication and elements are decided by the original ring. When we add i to R the only thing we can possibly get is C. The most general thing we can get by addition and multiplication of i by real numbers is the polynomial
z=p(i)=a0+a1 i+a2 i^2+...+an i^n
all we know about i is that i^2=-1
so we might as well gather up all the even and odd powers
z=a+bi
the same with
(a,b)(c,d)=(ac-bd,ac+bd)
there are no other choices
 

lurflurf

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exp(i x)=cos(x)+i sin(x)
might be a definition depending on how you have set up your system
It is the only reasonable result though.
suppose exp(i x)=A(x)+i B(x)
we want
exp(i (x+y))=exp(i x)exp(i x)
and
(A(x+y),B(x+y))=(A(x)A(y)-B(x)B(y),B(x)A(y)+A(x)B(y))
Thus the only reasonable choices for A and B are
A(x)=cos(x) B(x)=sin(x)
 

LCKurtz

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But what about the "+" sign in between the real part and the imaginary part? Since graphing complex numbers on the complex plane is alot like graphing real numbers on the real plane, why didn't we use a comma inbetween the real parts and complex parts?

How do we know there are nice properties such as addition and multiplication for something in the form "A+Bi"? If we had instead used the convention "(A,Bi)" to graph complex numbers, then would it still have had addition and multiplication for complex numbers? It seems a bit odd to me. You can of course multiply a scalar with (a,bi) or add another complex number but it would've done differently.

with the current system, (a+bi)^2 = a^2 + 2(a)(bi) -b^2. would it still have resulted in the same if we would have multiplied (a,bi) with (a,bi)? How do we even do that?
You might find this article helpful:

http://math.la.asu.edu/~kurtz/complex.html
 

symbolipoint

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Originally Posted by kramer733
Does it have something to do with the quadratic form? What would i type on google to search for more information to get better search results?
To summarize what MisterX and Jamma said, we often write complex numbers like that because it is convenient (it fits our commonly used xy- coordinate system format) but there are many different ways in which we could write complex numbers: [itex]re^{i\theta}[/itex] is one. Engineers often use the format "[itex]r cis(\theta)[/itex]" which is short for "[itex]r(cos(\theta)+ i sin(\theta))= re^{i\theta}[/itex]" as Jamma said.

There is no "mathematical" reason- it is just a convention.
Here is a thought, maybe helpful.

The same idea is done with Real Numbers, such as on a cartesian system (for example, in two dimensions). Consider a line equation. Some real number, C, can be expressed as Ax+By. We can say, Ax+By=C. Ax and By are ADDED and their result is a Real Number, C.

For Complex numbers, by comparable or corresponding positions, Ax is like a, By is like b, and C is like, z. Naturally, we might accept how simple addition is a good way to represent Complex Numbers. ax+bi=z for Complex, Ax+By=C for Reals.
 

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