complex line

In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers.{{citation

| last1 = Brass | first1 = Peter

| last2 = Moser | first2 = William

| last3 = Pach | first3 = János

| isbn = 9780387299297

| mr = 2163782

| page = 305

| publisher = Springer, New York

| title = Research Problems in Discrete Geometry

| url = https://books.google.com/books?id=cT7TB20y3A8C&pg=PA305

| year = 2005}}.{{citation|title=Introduction to Complex Analysis: Functions of Several Variables|volume=110|series=Translations of mathematical monographs|first=Boris Vladimirovich|last=Shabat|publisher=American Mathematical Society|year=1992|isbn=9780821819753|page=3|url=https://books.google.com/books?id=h5H4AwAAQBAJ&pg=PA3}} A common point of confusion is that while a complex line has complex dimension one over C (hence the term "line"), it has ordinary dimension two over the real numbers R, and is topologically equivalent to a real plane, not a real line.{{citation

| last1 = Miller | first1 = Ezra

| last2 = Reiner | first2 = Victor

| last3 = Sturmfels | first3 = Bernd

| isbn = 978-0-8218-3736-8

| location = Providence, RI

| mr = 2383123

| page = 9

| publisher = American Mathematical Society

| series = IAS/Park City Mathematics Series

| title = Geometric Combinatorics: Lectures from the Graduate Summer School held in Park City, UT, 2004

| url = https://books.google.com/books?id=W_SPdwfPTw8C&pg=PA9

| volume = 13

| year = 2007}}.

The "complex plane" commonly refers to the graphical representation of the complex line on the real plane, and is thus generally synonymous with the complex line, not the complex coordinate plane.