Imaginary Number Definition Imaginary Number Definition An imaginary number is a number whose square is less than or equal to zero. For example, is an imaginary number and its square is . An imaginary number can be written as a real number multiplied by the imaginary unit , which is defined by its property . An imaginary number can be added to a real number to form a complex number of the form , where and are called, respectively, the real part and the imaginary part of the complex number. Imaginary numbers can therefore be thought of as complex numbers whose real part is zero. The name "imaginary number" was coined in the 17th century as a derogatory term, as such numbers were regarded by some as fictitious or useless, but today they have a variety of essential, concrete applications in science and engineering. Geometric interpretation :- Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented
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perpendicular to the real axis. One way of viewing imaginary numbers is to consider a standard number line, positively increasing in magnitude to the right, and negatively increasing in magnitude to the left. At 0 on this -axis, a -axis can be drawn with "positive" direction going up; "positive" imaginary numbers then increase in magnitude upwards, and "negative" imaginary numbers increase in magnitude downwards. In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin. Multiplication by corresponds to a 90-degree rotation in the "positive" direction (i.e., counterclockwise), and the equation is interpreted as saying that if we apply two 90-degree rotations about the origin, the net result is a single 180-degree rotation. Note that a 90-degree rotation in the "negative" direction (i.e. clockwise) also satisfies this interpretation. This reflects the fact that also solves the equation — see imaginary unit. In general, multiplying by a complex number is the same as rotating around the origin by the complex number's argument, followed by a scaling by its magnitude. Let us show you an example of imaginary number. If we have a negative number in the Square root then we cannot solve it until we remove the negative sign and we can do that but when we remove the negative sign we have to put a imaginary unit before the square root. So in imaginary numbers definition, we have an example: √ -36 which is an imaginary number and it’s square root is -36 but imaginary number can be written as a real number by putting an imaginary unit before the square root. In imaginary number, we put an imaginary unit because of the property i2 = -1.
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Imaginary number definition has one more important Point to share and that is a complex number can be equal to an imaginary number when the real part of the complex number is zero. Imaginary numbers definition says that imaginary number has many useful applications in complex number field. Imaginary number has many applications and they are: Imaginary number is useful in construction of non-real complex numbers, which plays a very important role in applications of science and technology and it has many other applications in control theory, fluid dynamics, quantum mechanics, vibration analysis, electromagnetism, signal processing and cartography.
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