November 17, 2016 Thursday

Bedtime Story 


Complex Numbers Enable Solutions When There Are None

The point that may have got lost in my last night bed-time story is that the solutions to at least some of the cubic equations with rational coefficients end up in giving roots that are non-rational and non-real complex numbers.

A complex number is a number that is expressed in the form a + bi.

Here a and b are real numbers and i is the imaginary unit.

The imaginary unit (i) is that number whose square is -1.

The imaginary unit i satisfies the equation i² = -1.   

The need for such complex numbers arose when Gerolamo Cardano tried to solve the cubic equations.

The beauty of the complex numbers is that they allow solutions to certain equations for which simply no solutions exist in real numbers.

Take for example an equation:

(x + 1)² = -9  

This equation can only be solved if we bring in the idea of the imaginary unit i² = -1

The solution for the above equation then would be -1 + 3i and -1 and -3i

You can try it out by substituting for x these values and it will work out.

I want you to think something for yourself?

Where do you think we can place the complex numbers in the real number line?

Is it even possible?

Please think.

An algebraic number is a subset of complex number.

An algebraic number is a complex number that is a root of non-zero polynomial in one variable with rational coefficients.

On the other hand, a transcendental number is a real or complex number that is NOT a root of non-zero polynomial in one variable with rational coefficients.

The discussion of algebraic numbers and transcendental numbers is a long one and currently it is best to leave it at that as it will hinder our story on Cantor and basic notations of the modern set theory.

So now we have three sets namely the N, the Q and the R representing the natural, the rational and the real numbers.

The Real numbers include rational numbers such as integers and fractions and irrational numbers.

Complex numbers have a real part that is a real number and an imaginary part that is not a real number.

The letter P represents the set of prime numbers.

The prime numbers have their own story to tell.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night and my fellow cousin ape.

Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, may I suggest this large collection of Kids Songs:

https://www.youtube.com/channel/UCMX11Z5SJQ3kgwSsFJLRIcg