What is the conjugate of a real number, a?

Question

I understand that $a=a+0i$, and hence the conjugate is just

$a-0i=a$.

But why can’t we say that $a=0i+a$ (by the commutative property),

hence the conjugate is $0i-a=-a$?

Answer 1

Conjugation does not mean "turn the $+$ into a $-$ and vice versa". Conjugation means "send $i$ to $-i$ and vice versa" (roughly). Therefore $\overline{a + 0 i} = a - 0i$.

Answer 2

The complex conjugation map $\sigma :\mathbb{C}\to \mathbb{C}$ is defined by $x+iy\mapsto x-iy$ for $x,y\in \mathbb{R}$. If $a\in \mathbb{R}\subseteq \mathbb{C}$, then $a=a+0\cdot i$. By definition, then $\sigma(a)=a-0\cdot i=a.$

Answer 3

The conjugate of a complex number is the number with equal real part and opposite (i.e. negated) imaginary part. If you prefer to think of modulus and argument representation, it has the same modulus but opposite (negated) argument. A key property of the conjugate is that $z\overline z=|z|^2$.

Any of these ways of defining it force $\overline a=a$ if $a\in\mathbb R$.

Answer 4

Let $a$ be a real number.

Then,

$$\overline{a} = \overline{a + 0i} = \overline{a} + \overline{0i} = a - 0i = a.$$