Geometric interpetation of a complex number in R^2

[Valhalla]

For this problem i am given two complex numbers [tex]Z_1 , Z_2[/tex] and then a third which is the sum of the first two complex numbers [tex] Z_3 [/tex]. I am then asked to find the geometric interpetation of these numbers in [tex] \mathbb{R}^2 [/tex]. I am fine when graphing them in the complex plane but unsure of what they look like in [tex] \mathbb{R}^2 [/tex]. Do I just take the real part and graph a point in the [tex] \mathbb{R}^2 [/tex]? If so how do I determine which axis it would be on? Do you think this might be a typo?

 

[HallsofIvy]

The standard "complex plane" is to plot the complex number a+ bi as the point (a, b). That is, the x-axis is the "real axis" and the y-axis is the "imaginary axis". It might occur to you that the sum of complex numbers, (a+ bi)+ (c+ di)= (a+c)+ (b+d)i looks a lot like (a+c, b+d), the sum of vectors. And that might lead you to think about a parallelogram.

 

[Valhalla]

The standard "complex plane" is to plot the complex number a+ bi as the point (a, b). That is, the x-axis is the "real axis" and the y-axis is the "imaginary axis". It might occur to you that the sum of complex numbers, (a+ bi)+ (c+ di)= (a+c)+ (b+d)i looks a lot like (a+c, b+d), the sum of vectors. And that might lead you to think about a parallelogram.

Yes, I understand that the complex numbers add like vectors. However, what I am confused about is that I thought that to graph them they needed to be in the complex plane. Like you said the x-axis is the real and the y-axis is the imaginary. When I look at the complex number (a+bi) and consider where that would be on R^2, I get confused.

So if I get what your saying then the complex number (a+bi) would just be the vector (a,b) in R^2? Am I overthinking this?

 

[HallsofIvy]

Yes, that's exactly what I am saying!

 

[Valhalla]

Ok I think I got it then. Thanks HallsofIvy!