# Why do Mathematicians use $u$ and $v$ as variables?

I'm sure this has happened to you as well: you are reading some hand-written work, the variables used are $u$ and $v$, and at some point the handwriting becomes unclear and you cannot distinguish the $u$s and $v$s at all. This happens to me quite often --- even distinguishing them when they are typeset can sometimes be irritating.

Is there any particular reason, perhaps historical or traditional, why the two most similar-looking letters in the Western alphabet are also one of the most frequently used pairs of variables?

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And $\upsilon$, while you're at it! :) (For those not in the know, that's upsilon. Who nu?) – Ted Shifrin Feb 23 '14 at 22:00
I have always thought that $\upsilon$ not $\epsilon$ should be used for error. When you make a mistake: Oopsilon! – SpamIAm Feb 23 '14 at 22:01
My first year prof had invented the unified letter: $i,\iota,j,e,l,1,r,v,u,\upsilon,\nu,n,m,w,\omega$ all looked virtually indistinguishable on the blackboard. – Hagen von Eitzen Feb 23 '14 at 22:02
The reason is of course historical: When $x$ was introduced as unknown (I once knew where that occured first) and then $y$ and $z$ as natural candidates for the next unknowns/variables, it turned out to be necessary to step back a bit for the next unkowns $u,v,w$. And traditionally from this there is the association $x\leftrightarrow u$, $y\leftrightarrow v$, $w\leftrightarrow z$, e.g. from 3D vector algebra or complex ananlysis ($z=x+iy\mapsto w=u+iv$). – Hagen von Eitzen Feb 23 '14 at 22:07
@HagenvonEitzen Descartes used $x$ and $y$ (and also $z$), whereas Viète had used $A$, $E$ and the other vowels. – egreg Feb 23 '14 at 22:11