section_path: Definition and basic operations
section_name: Definition and basic operations
paragraph_text: Definition and basic operations
  
  A complex number is an expression of the form a + bi, where a and b are real numbers, and i is an abstract symbol, the so-called imaginary unit, whose meaning will be explained further below. For example, 2 + 3 i is a complex number.
  
  For a complex number a + bi, the real number a is called its real part, and the real number b (not the complex number bi) is its imaginary part. The real part of a complex number z is denoted Re(z), R e (z) {\displaystyle {\mathcal {Re}}(z)}, or R (z) {\displaystyle {\mathfrak {R}}(z)}; the imaginary part is Im(z), I m (z) {\displaystyle {\mathcal {Im}}(z)}, or I (z) {\displaystyle {\mathfrak {I}}(z)}: for example, Re (2 + 3 i) = 2 {\textstyle \operatorname {Re} (2+3i)=2}, Im (2 + 3 i) = 3 {\displaystyle \operatorname {Im} (2+3i)=3}.
  
  A complex number z can be identified with the ordered pair of real numbers (ℜ (z), ℑ (z)) {\displaystyle (\Re (z),\Im (z))}, which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the complex plane or Argand diagram,. The horizontal axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical axis, with increasing values upwards.
  
  A real number a can be regarded as a complex number a + 0 i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. As with polynomials, it is common to write a + 0 i = a, 0 + bi = bi, and a + (− b) i = a − bi; for example, 3 + (−4) i = 3 − 4 i.
  
  The set of all complex numbers is denoted by C {\displaystyle \mathbb {C} } (blackboard bold) or C (upright bold).
  
  In some disciplines such as electromagnetism and electrical engineering, j is used instead of i, as i frequently represents electric current, and complex numbers are written as a + bj or a + jb.