complex number

Học thuật
Thân thiện
complex number

A student plots a complex number on the coordinate plane.

Definition
  1. Noun:
    • A complex number is a number that can be expressed in the form ( a + bi ), where ( a ) and ( b ) are real numbers, and ( i ) is the imaginary unit, defined by the property ( i^2 = -1 ). The component ( a ) is called the real part, and ( b ) is called the imaginary part.
Usage
  • Complex numbers are fundamental in advanced mathematics, physics, and engineering because they provide a complete system for solving equations that have no solutions within the set of real numbers alone.
  • They are used to represent two-dimensional quantities, such as electrical impedance, signal phases, and quantum states.
Examples
  • Noun:
    • The solutions to the equation ( x^2 + 1 = 0 ) are the complex numbers ( i ) and ( -i ).
    • In electrical engineering, alternating current (AC) circuits are often analyzed using complex numbers to simplify calculations involving phase and magnitude.
    • The number ( 3 + 4i ) is a complex number with a real part of 3 and an imaginary part of 4.
Advanced Usage
  • Complex Plane (Argand Plane): A geometric representation where a complex number ( a + bi ) corresponds to the point with coordinates (a, b). The horizontal axis represents the real part, and the vertical axis represents the imaginary part.
  • Complex Conjugate: For a complex number ( z = a + bi ), its complex conjugate is ( \overline{z} = a - bi ). Multiplying a complex number by its conjugate yields a real number: ( (a+bi)(a-bi) = a^2 + b^2 ).
  • Modulus (Absolute Value): The modulus of a complex number ( z = a + bi ) is its distance from the origin in the complex plane, given by ( |z| = \sqrt{a^2 + b^2} ).
Variants and Related Words
  • Real Number: A number that can be found on the number line, such as 5, -3.2, or π. It is the ( a ) component in ( a + bi ) when ( b = 0 ).
  • Imaginary Number: A number that is a real multiple of the imaginary unit ( i ), such as ( 2i ) or ( -5i ). It is the ( bi ) component in ( a + bi ) when ( a = 0 ).
  • Complex Analysis: A major branch of mathematical analysis that investigates functions of complex numbers.
Synonyms
  • None (Technical Term): As a precise mathematical concept, "complex number" does not have direct synonyms. In informal contexts, one might refer to "numbers involving ( i )" or "numbers with real and imaginary parts."
Related Phrases
  • Complex Arithmetic: The rules for adding, subtracting, multiplying, and dividing complex numbers.
    • To add two complex numbers, you add their real parts and their imaginary parts separately.
  • Set of Complex Numbers: Denoted by ( \mathbb{C} ), this is the set containing all numbers of the form ( a + bi ).
complex number

A student plots a complex number on the coordinate plane.

Noun
  1. (mathematics) a number of the form a+bi where a and b are real numbers and i is the square root of -1

Từ đồng nghĩa

Từ chứa "complex number"