pure imaginary number
Học thuậtThân thiện
Definition
Noun A pure imaginary number is a complex number of the form bi, where b is a real number and i is the imaginary unit (√-1). Its defining characteristic is that its real part is exactly zero.
Usage
The term is used in mathematics, specifically in complex number theory, to describe numbers that lie solely on the imaginary axis of the complex plane. * In the complex number a + bi, if a = 0, the result is a pure imaginary number. * The square of a pure imaginary number is a negative real number.
Examples
- The numbers , , and are all pure imaginary numbers.
- Zero () can be considered a pure imaginary number (as ), but it is also a real number.
- In the equation , the solutions are the pure imaginary numbers and .
Advanced Usage
- In electrical engineering, pure imaginary numbers are used to represent reactance in AC circuits.
- The set of all pure imaginary numbers is closed under addition and subtraction but not under multiplication (since , which is real).
Variants and Related Words
- Imaginary Number: A broader category; any number of the form where . A pure imaginary number is a specific type of imaginary number where .
- Complex Number: The general form , where and are real numbers.
- Real Number: A number that can be found on the number line; the counterpart to imaginary numbers.
Synonyms
- Purely imaginary number (variant phrasing)
Related Concepts
- Imaginary Unit (i): Defined as √-1, the fundamental building block of imaginary numbers.
- Complex Plane: A geometric representation where the horizontal axis is the real axis and the vertical axis is the imaginary axis. Pure imaginary numbers lie on the vertical (imaginary) axis.
Noun
- an imaginary number of the form a+bi where a is 0