How to draw an Argand diagram for this complex number? (FP1)

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amelienine
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#1
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#1
If the number were 5/(1-i) let's say, how would I plot that on an Argand diagram?
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ValerieKR
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(Original post by amelienine)
If the number were 5/(1-i) let's say, how would I plot that on an Argand diagram?
The easiest method for that one is to turn it into the co-ordinate complex number form by 'realising the denominator'

worked example below
Spoiler:
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5/(1-i) = ((1+i)*5)/(1+i)(1-i)=(5+5i)/2= 2.5 + 2.5i
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amelienine
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(Original post by ValerieKR)
Turn it into the co-ordinate complex number form by 'realising the denominator'
Spoiler:
Show
5/(1-i) = ((1+i)*5)/(1+i)(1-i)=(5+5i)/2= 2.5 + 2.5i
Oh right I forgot about the conjugations!! Thank you!
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RDKGames
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(Original post by ValerieKR)
The easiest method for that one is to turn it into the co-ordinate complex number form by 'realising the denominator'

worked example below
Spoiler:
Show
5/(1-i) = ((1+i)*5)/(1+i)(1-i)=(5+5i)/2= 2.5 + 2.5i
That's called rationalising the complex number, I think. When it's rational it's then under the real category, without the imaginary part.
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ValerieKR
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(Original post by RDKGames)
That's called rationalising the complex number. When it's rational it's then under the real category, without the imaginary part.
Thanks for the info - I thought about calling it that but noticed that the method still works for things like (2-2i)/(1-i)
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RichE
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(Original post by RDKGames)
That's called rationalising the complex number, I think. When it's rational it's then under the real category, without the imaginary part.
I've seen it called both. I think actually realising is a better term for complex numbers as it would not make a lot of sense to apply the term rationalising if the denominator is

e + pi i

Rationalising is certainly what you're doing when dealing with surds.
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mik1a
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I prefer realising the denominator.

Rationalising is too strict, you might still want something like i/sqrt(5) with a real but irrational denominator.
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RDKGames
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(Original post by RichE)
I've seen it called both. I think actually realising is a better term for complex numbers as it would not make a lot of sense to apply the term rationalising if the denominator is

e + pi i

Rationalising is certainly what you're doing when dealing with surds.
Mhm, just the word 'realising' throws me off a bit. I think we need a proper phrase for this.

I'll just call it "Nullifying the imaginary part" from now on.
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ValerieKR
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(Original post by RDKGames)
Mhm, just the word 'realising' throws me off a bit. I think we need a proper phrase for this.

I'll just call it "Nullifying the imaginary part" from now on.
'That multiply by 1 technique'
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Zacken
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(Original post by RDKGames)
Mhm, just the word 'realising' throws me off a bit. I think we need a proper phrase for this.

I'll just call it "Nullifying the imaginary part" from now on.
Agree with Rich_E - realising makes the most sense.

Nullifying makes no sense at all, you're not nullifying the imaginary part.

(Original post by ValerieKR)
'That multiply by 1 technique'
There's two techniques, which I've normally heard it called as: "add 0 creatively" and "multiply by one creatively".
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RDKGames
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(Original post by Zacken)
Nullifying makes no sense at all, you're not nullifying the imaginary part.
Are you practically not? (a+bi)(a-bi)=a^2-b^2i^2=a^2+b^2 so the imaginary part disappears, hence this product can be expressed as a complex number (a^2+b^2)+0i where the imaginary part is null.
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atsruser
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(Original post by RichE)
I've seen it called both. I think actually realising is a better term for complex numbers as it would not make a lot of sense to apply the term rationalising if the denominator is

e + pi i

Rationalising is certainly what you're doing when dealing with surds.
I agree. My previous rant on the use of this terminology:

http://www.thestudentroom.co.uk/show...45&postcount=7
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atsruser
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(Original post by RDKGames)
Mhm, just the word 'realising' throws me off a bit. I think we need a proper phrase for this.

I'll just call it "Nullifying the imaginary part" from now on.
The obvious choice might be "reifying" but that's already got too many uses, so how about "decomplexifying"?
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Zacken
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(Original post by RDKGames)
Are you practically not? (a+bi)(a-bi)=a^2-b^2i^2=a^2+b^2 so the imaginary part disappears, hence this product can be expressed as a complex number (a^2+b^2)+0i where the imaginary part is null.
Huh? Take \frac{1}{5-i} = \frac{1}{2}(1+i) which certainly has an imaginary part. Unless you want to call it "nullifying the imaginary part of the denominator", which is a mouthful.
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RDKGames
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(Original post by Zacken)
Huh? Take \frac{1}{5-i} = \frac{1}{2}(1+i) which certainly has an imaginary part. Unless you want to call it "nullifying the imaginary part of the denominator", which is a mouthful.
\frac{1}{5-i} \not= \frac{1}{2}(1+i) but yeah I meant it as far as the denominator is concerned in my comment. Meh, I wouldn't say it's too much of a mouthful.
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RichE
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(Original post by atsruser)
The obvious choice might be "reifying" but that's already got too many uses, so how about "decomplexifying"?
That would seem to imply real numbers aren't complex (which they are).
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atsruser
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(Original post by RichE)
That would seem to imply real numbers aren't complex (which they are).
Maybe, but you could make the same objection to the term "realising", so I don't think it's any worse than that. I suspect that it's a pointless discussion anyway, as too many people are happy to misuse "rationalise" in this context.
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RichE
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(Original post by atsruser)
Maybe, but you could make the same objection to the term "realising", so I don't think it's any worse than that. I suspect that it's a pointless discussion anyway, as too many people are happy to misuse "rationalise" in this context.
I don't see your point as most complex numbers aren't real. You are making real a typically complex denominator.
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atsruser
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(Original post by RichE)
I don't see your point as most complex numbers aren't real. You are making real a typically complex denominator.
I'm afraid I don't understand what you're saying here. Your second sentence doesn't seem to agree with what you said in your previous post:

"That would seem to imply real numbers aren't complex (which they are)."

Anyway, I don't think the current usage will ever change, so this will be my final contribution on this particular prickly question of nomenclature.
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RichE
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(Original post by atsruser)
I'm afraid I don't understand what you're saying here. Your second sentence doesn't seem to agree with what you said in your previous post:

"That would seem to imply real numbers aren't complex (which they are)."

Anyway, I don't think the current usage will ever change, so this will be my final contribution on this particular prickly question of nomenclature.
I said initially that real numbers are complex, which is true, and secondly said most complex numbers aren't real which is also true.
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