argand diagram
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hi now in this tutorial what I want to do is show you how we can represent complex numbers on something called an Argand diagram normally when we get axis these axes are the x and y axis respectively but when it comes to complex numbers we replace the x axis with the real part of the complex number and the imaginary part replaces the y axis the real part ive just called re and the imaginary part I am you dont have to write re and I am some authors will just keep it as x and y but Im going to keep – re and I am so if we had a complex number lets just say we had the complex number Z 1 equaling 3 plus 2i we represent this on an Argand diagram by going 3 units in the positive real sense and 2 units in the positive imaginary sense and its represented by a line a line with an arrow so
it looks like a vector it has length and direction so this would be how we represent z1 then on an Argand diagram suppose I did the complex conjugate of said 1 often written z1 with a little star remember that when you have a complex conjugate of a complex number all you do is you switch the sign of the imaginary part so we end up with 3 minus 2i and if we were to draw this on the Argand diagram it would look something like this 3 units across 2 units down dont forget the arrow and this would be the complex conjugate of Z 1 and do you notice how its related to z1 its a reflection in the real axis now if we had another complex number lets say Z 2 Z 2 was equal to minus 4 plus I and we were to draw this on the Argand diagram then wed go minus 4 and plus 1 unit up there so
go from there to there would be our complex number Z 2 now if we were to add Z 1 and Z 2 together lets just do that Z 1 plus said to what do we get well 3 minus 4 is minus 1 and 2 I plus I is 3 I and if we were to represent this on the Argand diagram itd be something like this going from the origin up to there okay so that would be z1 plus z2 now its very interesting to relate this to vectors if I was to say call z1o a and z2 OB then if I was to do o to a + o2 B lets just put it down here Oh – a + o2 B remember in vectors the plus means followed by lets see what we get if I was to draw over z2 like so just take a copy of it and now translate it across here can you see that were
doing Z 1 followed by Z 2 what we get is Z 1 plus Z 2 so in other words if I called this point on the end here see Zed 1 + ed – Oh C then o – a perso to be equals o to see we didnt have to do this we could have even done Z 1 and we could have translated that across and done z2 followed by or plus z1 still the same result this diagonal of the resulting parallelogram so you can see I hope how complex numbers relate to vectors now having said that suppose we look at this Argand diagram and draw z1 and z2 back on again now if we do z1 – said – what do we get this time well 3 minus minus 4 is going to be 7 + 2 I – plus I is going to be plus I and if we represent 7 + I on the Argand diagram 7 units
unit up weve got this so this is the complex number Z 1 – said – but it in the same way if Z 1 is o2 a and Z 2 is o2 be if we were to take this vector we can translate this over to here and that fits in well with our vectors because if I think what B – a is B – a would be B – OH followed by o2 a and going from B 2 O is exactly the same as doing the reverse direction – o to be followed by out way and we can swap this round as oh – eh – OH – bo2 a being z1 then Oh to be being said – said one – said – well that brings us to the end of the tutorial but I hope its giving you an opportunity to see how we can represent then complex numbers on an Argand diagram and how they relate to vectors you
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Argand, diagram, complex, numbers, imaginary, roots, exam solutions, math, maths made easy
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