There are many simple equations that have no real solution. For example:

x²= -1

x + 1 = x

1^x = 2

0x = 1

Why has a solution using the symbol i been created for the x² = -1 equation but no solutions for the others?

Some reasons are:

1. Imaginary numbers are useful when solving many real-world problems.
2. They create organized patterns.
3. They follow rules of mathematics in theoretical circumstances.

1. Imaginary numbers are useful when solving many real-world problems.

Example: when an artificial scale is used and the numbers do not represent the actual amounts.

10°C is in reality about 283°K

-10°C is in reality about 263°K

-4°C is in reality about 269°K. -9°C is in reality about 264°K

If we need to use a formula, such as the one below, that finds a value between the 2 original values, using the temporary value an imaginary number allows the equation to continue and produces a value between the original Centigrade values, similar to what was produced with the Kelvin values.

Electricity

Negative numbers can be useful to describe a situation to what occurs in electricity. Electricity can flow in every direction in 3 dimensional space. But in wiring, it is limited to 2 opposing directions. Hence, positive and negative values describe electricity flows in a wire quite well, with positive describing voltage moving in one direction and negative describing voltage moving in the alternating direction.

In the study of electricity and electronics, j  is used to represent imaginary numbers so that there is no confusion with i, which in electronics represents current.   It is also customary for scientists to write the complex number in the form a +jb.

The voltage in a circuit is 20 + j10 volts and the impedance is 4 + j5 ohms. What is the current?

So, the "sideways" forces of j10 and j5 had an effect when they were both used, producing a difference in the real force when divided. In effect it turned them in a direction where they could flow again in the direction the wire allowed and produced a change in the strength in the real value of the current.

Imaginary numbers are useful in real life situations where the direction changes with every other multiplication (or division), where its imaginary value eventually leads to a real result. Its intermediate (imaginary) state could possibly be thought of as "potential".

2. It creates organized patterns

The properties of i when raised to consecutive integer powers forms a repeating pattern, somewhat related to modulo 4 arithmetic; also similar to values of a sine wave at set points. The value of i when raised to powers are not limited to discrete values (such as integers 1 to 4) for the powers, but form continuous values that follow consistent patterns.

             

For example:    i^1.5 ≈ -0.707 + 0.707i     and      i^2.5 ≈ -0.707 - 0.707i

Finding the third root of a value will produce 3 results. The fourth root will produce 4 results, etc. Below are the roots of 1.

(1)(1)(1) = 1,

but it is also true that

([ -1 + i√ 3  ] / 2) ([ -1 + i√ 3  ] / 2) ([ -1 + i√ 3  ] / 2) = 1

and

([ -1 - i√ 3  ] / 2) ([ -1 - i√ 3  ] / 2) ([ -1 - i√ 3  ] / 2) = 1

Notice how the use of imaginary numbers allows the prediction of the exact number of roots.

The roots form a circle on the complex number plane.

The points are dense.

Most calculators will either give you an error message for the equations above, or they will provide only one solution, but not tell you that there are more.

In situations where the recurring pattern (no change, negative, no change, positive) occurs, imaginary numbers can provide the correct signs for values. The uses of imaginary numbers as an exponent illustrates where this pattern is useful:

3. It follows rules of mathematics in theoretical circumstances.

In a similar manner to the methods used in the e to the i pi equation above, but with other equations, the value of i can be defined for other, more abstract, mathematical manipulations - but retaining consistency and not leading to contradictions.

3ⁱ  ≈  0.45 + 0.89i

iⁱ  ≈  0.21

i!  ≈  0.5 - 0.15i

sin i  ≈  1.18i

if sin x = 2,
    x  ≈  1.57  ±  0.1.32i

if sin x = 3,
    x  ≈  1.57  ±  0.1.76i

if sin x = i,
    x  ≈  ±0.88i

There are some online calculators that will provide complex solutions to the abstract equations above. They each have strengths and weaknesses.

For example, the calculator below can calculate (-2)^1.9999, but cannot calculate many other imaginary results. The calculator on the right can calculate many other imaginary results, but cannot calculate (-2)^1.9999

Imaginary numbers, even when not multiplied to produce a real value, can allow the exploration of seemingly impossible situations. For example, consider a tachyon. No tachyons have been discovered, but the use of imaginary numbers allowed the theoretical exploration of their properties by modern physics.

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