Imaginary Numbers

[steve_bank]

The imaginary compared to real numbers can cause confusion. Imaginary numbers 'do not exist'. They exist more or less as do real numbers.

The complex plane is a horizontal axis labeled real, and a vertical axis called imaginary. Both are real number lines.

srrt(4) = 2 and is plotted on the real axis.
sqrt(-4) is i2 plotted on the imaginary axis.

A complex number z = r + i. which Segways into complex numbers. The relative magnitudes of r and i defining a phase angle a phase angle. It maps into actual physical systems both electrical and mechanical.


https://en.wikipedia.org/wiki/Imaginary_number

Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented 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 x-axis, a y-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. This vertical axis is often called the "imaginary axis" and is denoted iℝ,
I {\displaystyle \scriptstyle \mathbb {I} }
, or ℑ.
In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin. Multiplication by i corresponds to a 90-degree rotation in the "positive" direction (i.e., counterclockwise), and the equation i2 = −1 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 −i also solves the equation x2 = −1. 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.

https://en.wikipedia.org/wiki/Complex_number