imaginary number

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imaginary number

A student writes an imaginary number on a chalkboard.

Definition
  1. Noun:
    • A complex number with a real part of zero: An "imaginary number" is specifically a complex number where the real component (a) is equal to zero, resulting in the form bi, where b is a real number and i is the square root of -1.
    • A number whose square is a non-positive real number: More broadly, it is a number that, when squared, yields a negative real number. The fundamental imaginary unit is i, defined as √-1.
Usage Examples
  • Noun:
    • In the equation z = 3i, the term '3i' is an imaginary number.
    • The solution to x² + 1 = 0 is the imaginary number i.
    • Electrical engineers often use imaginary numbers to represent phase differences in alternating current circuits.
Advanced Usage
  • Pure imaginary number: This is a more precise term synonymous with the core definition of an imaginary number (of the form where b is real and b ≠ 0).
    • The value 5i is a pure imaginary number.
  • Role in Complex Numbers: An imaginary number is a fundamental building block of complex numbers, which have the form a + bi.
    • Every complex number combines a real number and an imaginary number.
Variants and Related Words
  • Imaginary unit (i): The fundamental constant defined as √-1, from which all imaginary numbers are derived.
  • Complex number: A number of the form a + bi, where a and b are real numbers, and i is the imaginary unit. An imaginary number is a specific type of complex number where a=0.
  • Real number: A number that can be found on the number line, in contrast to an imaginary number.
Synonyms
  • Pure imaginary number: (Technical synonym)
  • Non-real number: (Broader category, includes all numbers not on the real number line)
Related Phrases and Concepts
  • Imaginary axis: In the complex plane, the vertical axis used to represent the imaginary part of a complex number.
    • The point 3i lies on the positive imaginary axis.
  • Complex conjugate: For an imaginary number , its complex conjugate is .
imaginary number

A student writes an imaginary number on a chalkboard.

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

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