Understanding Algebraic Variables with a Dash: Help with Math Term Explanation

[Ghost803]

What does it mean when an algebraic variable has a dash above it? _
---------------------------------------------------------------->z, like that?

 

[Hurkyl]

What does it mean when an algebraic variable has a dash above it? _
---------------------------------------------------------------->z, like that?

Sometimes it's just used as a decoration to create a new alphabetic symbol to use. (there aren't enough alphabets for the purposes of mathematics!)

Sometimes it's used to denote the complex conjugate function: \overline{a + bi} = a - bi.

Sometimes it's used to denote equivalence classes; e.g. when it's evident you're working modulo 3, the relation 2 \equiv 5 \pmod 3 can be expressed as \bar{2} = \bar{5}.

 

[matt grime]

If you want fixed width fonts, then you need to put in some tags. I think it might be 'code'. Otherwise attempting justification doesn't work since it is font specific, or machine dependent.

  _
  z

 

[Ghost803]

\overline{z}

 

[Ghost803]

it says the absolute value of z equals the absolute value of \overline{z}. This was in a list of properties of absolute value.

 

[HallsofIvy]

Okay, then you are talking about complex numbers and the "overline" denotes the complex conjugate

If z is a complex number, say z= a+ bi, so its absolute value is |z|= \sqrt{a^2+ b^2}. As Hurkyl said, then, \overline{z}= a- bi so that its absolute value is |\overline{z}|= \sqrt{a^2+ (-b)^2}= \sqrt{a^2+ b^2}= |z|.

One can also define absolute value of a complex number by |z|= \sqrt{z\overline{z}}.

 

[Ghost803]

Thx a lot. Makes sense now.