Mathematicians invented imaginary numbers to work out algebra problems that were otherwise unsolvable. When you square an imaginary number, you get a negative number. Though they may seem a little strange at first, imaginary numbers have many important practical uses in math, the sciences and engineering.
TL;DR (Too Long; Didn't Read)
When you square an imaginary number, the result is a negative number.
You typically deal with real numbers in everyday life – the temperature outside, the distance to a friend’s house, or the number of pennies in your change jar. These numbers represent real objects and phenomena. In addition to the whole numbers we use for counting, real numbers include zero and negative numbers. Some numbers are rational; you get them by dividing one whole number by another. Other numbers, like pi, e, and the square root of 2 are irrational. No whole number ratio exists for them. It can help to picture real numbers as marks on an infinitely long line, with zero being in the middle.
In the late 1500s, mathematicians discovered the existence of imaginary numbers. Imaginary numbers are needed to solve equations such as x^2 + 1 = 0. To distinguish imaginary numbers from real ones, mathematicians use the letter i, usually in italics, such as i, 3i, 8.4i, where i is the square root of -1 and the number before it serves as a multiplier. For example, 8.4i is the square root of -8.4. Some technical disciplines such as electrical engineering prefer to use the letter j instead of i. Not only are they different from real numbers, but also imaginary numbers have their own number "line.”
The Imaginary Number Line
In math, a line of imaginary numbers exists that is much like the real number line. The two lines sit at right angles to each other, like the x and y-axes of a graph. They intersect at the zero points of each line. These number lines help you picture how real and imaginary numbers work.
Complex Numbers: The Plane Truth
By themselves, the real and imaginary number lines, like any line in geometry, occupy one dimension and have infinite length. Together, the two number lines form what mathematicians call the complex number plane – two dimensions that describe any number, whether real, imaginary or complex. For example, 72.15 is a real number, and -15i is an imaginary number. For these two numbers, you can find a point on the complex number plane: 72.15, -15i. Note that this number is located on the plane, not directly on the imaginary or real number lines. It is like San Francisco, which has a latitude and longitude but is neither on the equator nor the prime meridian.
Rules for Imaginary Numbers
Imaginary and complex numbers work much like real ones. You can add, subtract, multiply and divide them in any combination. They follow the normal rules of math, with the wrinkle that imaginary numbers, when squared, give a negative answer.
Imaginary Numbers, Real Uses
Imaginary numbers are useful tools that help solve difficult math problems. In electronics, equations that describe AC circuits make use of imaginary and complex number math. Physicists use complex numbers when dealing with electromagnetic waves, which combine properties of electricity and magnetism. Quantum mechanics, the study of subatomic particles, also uses complex numbers. In geometry, the study of fractal shapes that meander and branch off in different directions involves complex number math.