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 November 25th, 2009, 10:14 PM #1 Newbie   Joined: Nov 2009 Posts: 1 Thanks: 0 Complex numbers questions So I missed the last week of class because I've been sick and with my textbooks stuck in my school lockers, I can't really get caught up on work. My friend was able to send me two sample questions they've been learning in class, but I have no idea what it means? Can someone please explain and teach me how to solve these questions? Unsure of the notation too, the second z in the first question has a line over it, I don't think it's supposed to be z hat, but that's all I know. 1) Find the complex number z such that $(2+i)z + (4-i)\hat{z}= -1 + 5i$. 2) If $u=-3 + 2i$ and$v= 3 + 5i$ are complex numbers, calculate the modulus of $\frac{u}{v}$. (Also, is modulus supposed to be like the programming modulus operation, or is it related? (i.e. 15 mod 10 = 5)) Thanks a lot, guys!
 November 26th, 2009, 04:06 AM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 How much complex numbers stuff do you know already? $\bar z$ is the complex conjugate of $z,$ sometimes written $z^\ast$ (or rarely $z^\dag$). It represents the unique complex number satisfying $\text{Re}(\bar z)=\text{Re}(z),\ \text{Im}(\bar z)=-\text{Im}(z).$ In other words, if $z=a+bi$ where $a,b\in\mathbb{R},$ then $\bar z=a-bi.$ It is a useful exercise (i.e. you should do this ) to check directly that the complex conjugate satisfies the following properties for $w,z\in\mathbb{C}:$ $\bar{\bar z}=z,\\ \bar{(w\pm z)}=\bar w\pm\bar z,\\ \bar{wz}=\bar w\bar z,\\ \bar{\left(\frac wz\right)}=\frac{\bar w}{\bar z},\quad z\neq0,\\ z=\bar z\text{ if and only if }z\in\mathbb{R},\\ z\bar z\in\mathbb{R},\ z\bar z\,\geq\,0,\\ z+\bar z=2\text{Re}(z).$ The modulus $|z|$ (which is different to the modulo), otherwise known as the absolute value, is then defined to be the unique real, non-negative number satisfying $|z|^2=z\bar z.$ This means that if $z=a+bi,$ then $|z|=\sqrt{a^2+b^2}.$ You should check this, too. The modulus satisfies the following properties for $w,z\in\mathbb{C},$ among others: $|z|=|\bar z|,\\ |z|\,\geq\,0,\text{ and } |z|=0\text{ iff }z=0,\\ |wz|=|w||z|,\\ |z|\,\geq\,\text{Re}(z),\ |z|\,\geq\,\text{Im}(z),\\ |w+z|\,\leq\,|w|+|z|\text{ (Triangle inequality)},\\ 1/|z|=|1/z|,\quad z\neq0.$ You should check these, too - most are easy when you express $|z|$ in terms of $z$ and $\bar z$ and use the properties of the conjugate. I'll show you how to do the triangle inequality one, since this is probably the trickiest: \begin{align}|w+z|^2&=(w+z)\bar{(w+z)}\\ &=(w+z)(\bar w+\bar z)\\ &=w\bar w+z\bar z+w\bar z+\bar w z\\ &=|w|^2+|z|^2+w\bar z+\bar{w\bar z}\\ &=|w|^2+|z|^2+2\text{Re}(w\bar z)\\ &\leq\.|w|^2+|z|^2+2|w\bar z|\\ &=|w|^2+|z|^2+2|w||z|\\ &=(|w|+|z|)^2.\end{align} Therefore $|w+z|\,\leq\,|w|+|z|.$ Use the properties of the modulus and conjugate to have a go at your problems, and then come back if you have any further questions!

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