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I am confused about complex numbers. Does $1-i$ lie outside the unit circle? How do I show that this is larger than $1$ in absolute value?

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9  
The question in the title made no sense, but the body of the post contained a different question which made sense. I've edited the title. –  Asaf Karagila Feb 26 '13 at 0:03
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|a| = sqrt(C² + R²) => |1 - i| = sqrt(1² + 1²) = sqrt(2) and |1| = 1. sqrt(2) > 1 => |1 - i| > |1| –  njzk2 Feb 26 '13 at 11:03

9 Answers 9

up vote 84 down vote accepted

Let me do something very different for me. Let me give a geometric proof:

Note that $1-i$ is the bottom-right corner of the square whose center is $0$, and each of each edges has length $2$. The other corners are $1+i; -1+i; -1-i$.

The diagonal running from $-1+i$ to $1-i$ is a straight line passing through $0$. Its length, by the Pythagoras theorem, is $2\sqrt2=\sqrt8$. Therefore the distance between $0$ and each of the corners is exactly half, i.e. $\sqrt2$.

And it is trivial to see that $\sqrt2>1$.

Here is a drawing:

$\hspace{5cm}$complex plane

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11  
I must be very tired if I am thinking about this in terms of geometry. –  Asaf Karagila Feb 25 '13 at 23:04
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...and now I added a drawing. The horror... (If anyone can improve this drawing somehow, please do. :-)) –  Asaf Karagila Feb 25 '13 at 23:18
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This is really the best demonstration of this property. +1 –  Arkamis Feb 25 '13 at 23:38
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+1 for being the only reasonable (read: geometric) answer! –  Tyler Feb 26 '13 at 3:53
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I concur, given the lack in intuitive understanding of complex numbers the OP showed, this really is the only reasonable answer. –  Christian Feb 26 '13 at 10:06

By definition, the absolute value of a given complex number $z=x+iy$ (for some $x,y\in \Bbb R$) is $\sqrt{x^2+y^2}$ and it is denoted by $|z|$.

If $z=1-i$, according to the definition we get $|1-i|=\sqrt{1^2+(-1)^2}=\sqrt {2}>1=|1|$.

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Hint: By definition, if $a+bi$ is a complex number ($a$ and $b$ being real numbers, as usual), $$|a+bi|=\sqrt{a^2+b^2}.$$


Also, it is really incorrect to say that "$1-i$ is larger than $1$"; the complex numbers have no ordering. You should just instead say that $|1-i|>1$.

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In complex number, you have to compute the module of a number to say if it is "larger" than something. In general the module is $$|a+bi|=\sqrt{a^2+b^2}$$

In your case $$|1-i|=\sqrt{1^2+(-1)^2}=\sqrt{2}$$

Then $$\sqrt{2}\geq|1|$$

And this imply that $1-i$ lies outside of the unit circle

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One of the properties of complex numbers is that we cannot compare them, but we can compare their modulus.

you can compare $|1-i|$ with $|1|$, equivalent with $\sqrt{1^{2}+1^{2}}\geq 1.$

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$|1|= 1$

$|1-i|= \sqrt{1^2+(-1)^2}=\sqrt{2}$

Since the modulus is radius in the Argand plane,

$|1-i|>|1|$

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Well as far as the title is concerned,

$|1-i|=\sqrt2$>$|1|$=1

$|1-i|$>$|1|$

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Here is another different approach:

The unique angle (in $[0,2\pi)$) between the vector $(1,i)$ in the complex plane and the x-axis is $\phi=\frac{7\pi}{4}$

Euler's identity yields now:

$1-i=L(e^{\frac{7\pi}{4}i})=L(\cos(\frac{7\pi}{4})+i\sin(\frac{7\pi}{4}))=L(cos(-\frac{\pi}{4})+isin(-\frac{\pi}{4}))=L(cos(\frac{\pi}{4})-isin(\frac{\pi}{4}))=L(\frac{1}{\sqrt{2}}-i\frac{1}{\sqrt{2}}) $

Hence $L=\sqrt{2}$

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This is a small variant of Asaf Karagila's geometric proof: If $a$ and $b$ are two points in the complex plane, the distance between them is $|a-b|$, so if you draw the (right isosceles) triangle with $a=1$, $b=i$ and $c=0$, you see that $|1-i|\gt1$.

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