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.