# What does $\mathrm{Re}(x)$ mean?

I see this all the time in Mathematica output as well as in text, such as near the top of the Wikipedia Beta function page.

• For a complex number $x \in \mathbb{C}$, you can write it as $x = a+bi$ where $a$ and $b$ are real numbers and $i$ is the imaginary number. $Re(x) = a$, it is referring to the "real part" of $x$. Similarly, there is a function called $Im$ such that $Im(x)=b$. Oct 6 '11 at 2:48
• Sometimes, you'll see $\Re z$ and $\Im z$ used instead of $\mathrm{Re}(z)$ and $\mathrm{Im}(z)$. Oct 6 '11 at 3:23
• As a tiny Mathematica tip: whenever you see some function you don't quite understand in the output, highlight the name of the function (by double-clicking, for instance) and press the F1 key. Oct 6 '11 at 3:26

If a complex number $z$ is written as $z = a + bi$, then Re$(z) = a$ and Im$(z) = b$. (At risk of stating the obvious, "Re" stands for "Real" and "Im" stands for "Imaginary".)
If we visualize complex numbers as vectors in $\mathbb{R}^2$, Re is the projection onto the real axis, and Im is onto the imaginary axis. So $z = \mathrm{Re}(z) + \mathrm{Im}(z)i$.
• If I may add (that is someone correct me if im wrong), the $\text{IM}(\cdot)$ part also is the real number however it is the on the imaginary axis in the complex plane. Oct 6 '11 at 3:42
• @Tyler: $\mathrm{Im}(z)$ is certainly real; what you probably had in mind is that if one considers the Argand plane, $\mathrm{Im}(z)$ is equivalent to the vertical coordinate of the point $z$. Oct 6 '11 at 3:49