Title : Imaginary Numbers - Impossible is Possible (Lakshan)

Author: Lakshan Bandara

Created Date: 2020-07-06

Last Modified Date: 2020-07-06

Category: Education

Article Code: 88

Keywords: Imaginary Numbers, Mathematics

We Sri Lankans are capable of thinking differently.

Why don’t we use this skill to redesign our systems and for ideological inventions?

The world market is full of opportunities with problems.

If we can solve them, we can be rich as a country.

Note: patents and intellectual property are vital to USA economy

I solved the historical problem of "how to physically represent imaginary numbers?"

Please see attachment (pdf article).

I will provide here the historical significance of the problem:

1. The very first mention of trying to use imaginary numbers dates back to the 1st century.

2. In 50 A.D., Heron of Alexandria studied the volume of an impossible section of a pyramid.

Impossible was when he had to take √81-114. Then he gave up.

3. After negative numbers were invented, mathematicians tried to find, “could a squared number equal to a negative one?” Not finding an answer, they gave up.

4. In 1500 A.D., to solve 3rd and 4th degree polynomial equations, sometimes mathematicians required square roots of negative numbers. But, they too gave up.

5. In 1545 A.D., the first major work with imaginary numbers. Girolamo Cardano wrote a book titled Ars Magna. He solved the equation x(10-x)=40, finding the answer to be 5 plus or minus

√-15. Although he found the answer, he disliked imaginary numbers. He said that work with them would be, “as subtle as it would be useless”, and referred to working with them as “mental torture.”

6. In 1637 A.D., Rene Descartes came up with the standard form for complex numbers, which is a+bi. He termed “imaginary”. But, he too didn’t like complex numbers. He assumed that if Imaginary numbers were involved, you couldn’t solve the problem.

7. Issac Newton agreed with Rene Descartes, and Albert Girad. He even went as far as to call these, “solutions impossible”.

8. Rafael Bombelli was a firm believer in complex numbers. He had a wild idea that you could use imaginary numbers to get the real answers. Today, this is known as conjugation.

9. In 1685 A.D., John Wallis said that a complex number was just a point on a plane, where X-axis would be real numbers, and the Y-axis would be imaginary numbers. But he was ignored.

10. More than a century later, Caspar Wessel published a paper showing how to represent complex numbers in a plane, but was also ignored.

11. In 1777 A.D., Euler made the symbol i stand for √-1, which made imaginary numbers a little easier to understand.

12. In 1804 A.D., Abbe Buee thought about John Wallis’s idea to graph complex numbers, and agreed with him.

13. In 1806, Jean Robert Argand wrote how to plot complex numbers in a plane, and today that plane is called Argand diagram.

14. In 1831 A.D., Carl Friedrich Gauss made Argand’s idea popular to mathematical community, took Descartes’ a+bi notation, and termed it complex number.

15. In 1833 A.D., William Rowan Hamilton expressed complex numbers as pairs of real numbers (such as 4+3i being expresses as (4,3)).

16. Karl Weierstrass, Hermann Schwarz, Richard Dedekind, Otto Holder, Henri Poincare, Eduard Study, and Sir Frank Macfarlane Burnet all studied the general theory of complex numbers.

17. Augustin Louis Cauchy and Niels Henrik Able made a general theory about complex numbers accepted.

18. August Mobius made many notes about how to apply complex numbers in geometry.

19. In 1843 A.D., a mathematical physicist William Rowan Hamilton extended the idea of an axis of imaginary numbers in the plane to a three-dimensional space of quaternion imaginaries.

20. So far mathematicians and philosophers all over the world could not understand the philosophy of Imaginary numbers. They understood Imaginary numbers are not real numbers, because they could not plot them on a real number line. Though they couldn’t explain the physical existence of Imaginary number representations, they were confident that Imaginary numbers exist in mathematics, because the applications of Imaginary numbers exist in real world.

I can Imagine your friend.

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