Imaginary & complex Numbers

Math Encourages Impossible Questions!

We have seen many examples where math developed when “seemingly” impossible questions were asked.

This process was accelerated when math became symbolic, losing its obvious connection with reality.

When 5 – 2 was 3, someone asked the question what about 2 – 5? This gave rise to negative numbers.

In the 16th century, Italian mathematician Girolamo Cardano, while trying to find all solutions for a cubic equations in Algebra (which contain terms like x^3), came across solutions which contained square roots of negative numbers. The initial reaction of mathematicians was to reject such solutions.

Subsequently, they were accepted as a new type of numbers.

Rene Descartes called them “imaginary” numbers, in almost a derisive way. Leonard Euler called √(-1) as i, acknowledging the name imaginary.

We need to understand that this term is just a label. There is nothing imaginary about these numbers. They were numbers which could not be plotted on the "real number" line.

In one sense, all numbers, including Natural numbers are imaginary. They are a product of human imagination.

Interpretation of Imaginary Numbers

Later i was considered as a number in an axis which was perpendicular to the real number line and passing through the origin. We can imagine it as the y-axis of the cartesian plane.

This representation considered i as a vector and could represent multiplying by “i” as a 90 degree turn to the left.

Imagine a vector on the x-axis pointing to the right and hence signifying +1.

A 90 degree turn to the left, makes the vector point up. This can be seen as +i which is the product of +1 & i.

Another 90 degree turn to the left makes the vector point in the -x direction and hence equal to -1.

This can be seen as the product of +i & +i which is -1.

So this representation gives a way to represent imaginary numbers and also satisfies rules of operation with imaginary numbers.

Imaginary Numbers Describe Reality

In 1925, Edwin Schrodinger came out with his famous “wave equation” which describes correctly everything we know about the behaviour of atoms. It is the basis of all of chemistry and most of physics.

And this equation contains the imaginary number i. This surprised Schrodinger and all other physicists.

Complex Numbers

Descartes's idea of the cartesian plane can be reimagined to represent the “complex number” plane.

We have seen that “i” could be thought of as existing on a line which is perpendicular to the real number line!

A complex plane could be thought of the plane bound by the real number line and the vertical imaginary number axes!

Any point on the complex plane can be defined by 2 numbers; a real number (along the x-axis) and an imaginary number (along the imaginary number or y axis)

Numbers which were a combination of real numbers and imaginary numbers were called “complex” numbers. 3 + 5i is an example of a complex number. It is located 5 units along the imaginary number axis and 3 units along the x-axis.

So it was Rene Descartes who proposed the name imaginary and also was the inventor of the Cartesian plane.

Significance of Complex Numbers

The invention of complex numbers ended the search for a number system closed with respect for all arithmetic and algebraic operations.

Complex numbers appear in equations describing the behaviour of the real world phenomena; control theory, signal analysis, relativity, and fluid dynamics all use complex numbers.

Quantum Theory cannot be understood without the use of Complex Numbers.

This means that nature works with complex numbers and not with real numbers.

Giving up Math’s connection to reality seems to guide us to deeper truths about the way the universe works.

Here is a nice video on the history behind imaginary numbers. https://www.youtube.com/watch?v=cUzklzVXJwo