Question Video: Identifying the Imaginary Part of Complex Numbers on Argand Diagrams | Nagwa Question Video: Identifying the Imaginary Part of Complex Numbers on Argand Diagrams | Nagwa

Question Video: Identifying the Imaginary Part of Complex Numbers on Argand Diagrams Mathematics • Third Year of Secondary School

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What is the imaginary part of the complex number shown?

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Video Transcript

What is the imaginary part of the complex number shown?

In this question, we’re given the complex number 𝑍 on an Argand diagram and we need to use this to determine the imaginary part of our complex number 𝑍. To do this, let’s start by recalling what we mean by an Argand diagram. In an Argand diagram, the horizontal coordinate of our point represents the real part of our complex number and the vertical coordinate of our number represents the imaginary part of our complex number.

And it might be worth reiterating what we mean by the real and imaginary parts of a complex number. For a complex number 𝑍 given in algebraic form, that’s the form 𝑎 plus 𝑏𝑖 where 𝑎 and 𝑏 are real numbers, we see the real part of our complex number 𝑍 is the value of 𝑎 and the imaginary part of our complex number 𝑍 is equal to our value of 𝑏. And we have notation for this. So in our Argand diagram, the horizontal coordinate of our point represents the value of 𝑎 and the vertical coordinate represents the value of 𝑏.

We want to find the imaginary part of our complex number 𝑍, so we need to find its vertical coordinate. We can see that this is equal to negative four. And this is enough to answer our question. The imaginary part of our complex number will always just be equal to its vertical coordinate on an Argand diagram, in this case, negative four.

However, we can actually find more than this. We can also find the horizontal coordinate of this point. This is given by negative two. And this means we know both the imaginary and real parts of our complex number 𝑍. So, we can represent 𝑍 in algebraic form. 𝑍 will be equal to the complex number negative two minus four 𝑖. Therefore, given a complex number 𝑍 on an Argand diagram, we were able to find the imaginary part of our complex number 𝑍. All we had to do was find its vertical coordinate on our Argand diagram. We found that this was equal to negative four.

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