imaginary part

Thân thiện

A student plots the imaginary part of a complex number on a graph.

Definition

1. Noun:
   • The imaginary part of a complex number: In mathematics, specifically in complex number theory, the "imaginary part" refers to the component of a complex number that is multiplied by the imaginary unit i, where i is defined as the square root of -1. A complex number is expressed in the form a + bi, where a is the real part and bi is the imaginary part. The "imaginary part" itself is the real coefficient b.

Usage Examples

• Noun:
  • In the complex number 3 + 4i, the imaginary part is 4.
  • To plot the number on the complex plane, you use the real part for the x-axis and the imaginary part for the y-axis.
  • The function extracts the imaginary part of a given complex input.

Advanced Usage

• "Pure imaginary number": A complex number where the real part is zero, so the number consists only of an imaginary part (e.g., 5i).
  • The value 7i is a pure imaginary number; its imaginary part is 7.
• In engineering and physics, the imaginary part often represents a phase shift or an oscillatory component in waveforms and signal analysis.

Variants and Related Words

• Imaginary unit (i): The fundamental imaginary number equal to √-1.
• Complex number: A number of the form , containing both a real and an imaginary part.
• Real part: The component in the complex number .
• Imaginary axis: The vertical axis on the complex plane, representing the imaginary part.

Synonyms

• None direct: In strict mathematical terminology, "imaginary part" is the standard, precise term. Informally, one might refer to the "i-component" or the "coefficient of i".

Related Phrases

• "Extract the imaginary part": A common instruction in mathematics and programming to isolate the value from a complex number .
  • The software library includes a function to extract the imaginary part.
• "Imaginary part operator": Often denoted as Im(z). For a complex number z = a + bi, Im(z) = b.
  • Applying the imaginary part operator to 2 - 5i gives -5.

Related Idioms

• None: This is a technical mathematical term and is not used in idiomatic expressions.

Học thuật

Noun

1. the part of a complex number that has the square root of -1 as a factor

Từ đồng nghĩa

• imaginary part of a complex number