Complex Numbers

Sir William Rowan Hamilton

October 16, 1843: Sir William Rowan Hamilton comes up with the idea of “quaternion.” This is a mathematical term. Quaternions are non-commutative. Commutativity is the mathematical property referring to order. An example is 3+2 and 2+3 will both equal five and so they are commutative. However, 3-2 and 2-3 give different answers and they are non-commutative. Multiplication is commutative while division is not. Quaternions are a non-commutative number system that extends complex numbers.

Complex numbers in mathematics have no direct relationship to how the numbers are written out. They are simply an extension of real numbers. They are imaginary numbers and usually written as a lower case i. A simply written complex number is i2 = -1. All complex numbers can be written as a+bi (where a and b are real numbers and i is imaginary). In the sample above, 1*1 = 1 as well as -1*-1 = 1 since multiplying two negatives makes a positive. To square a number and get a negative, you need a little imagination. Real numbers can be rational such as 17 or -41/639, or irrational, such as pi. This is why math is hard, as Barbie once said.

Sir Hamilton was born in 1805 in Dublin. At age three he was sent to live with an uncle, a Trinity College alumnus who ran a school in Talbots Castle. The young boy showed a facility for learning languages and mastered about a dozen (some critics claim he had a bare understanding of many, rather than mastering them). Hamilton went to Trinity College and studied astronomy. He was appointed as Professor of Astronomy in 1827 even before he graduated. He contributed to both optics and classical mechanics, bringing fresh ideas and insights to the subjects.

Hamilton went on to higher mathematics and has several processes named for him. Quaternions are useful in applied mechanics of three-dimensional space. Theoretical and applied mathematics of the system are involved in three-dimensional rotations such as those used in 3D computer graphics. They are used in four-dimensional normed division algebra, which is the type of math used in Hurwitz’s theorem. Sir Hamilton’s advance of both astronomy and mathematics have led him to be celebrated as a hero – a leading Irish scientist.

“Pure mathematics is, in its way, the poetry of logical ideas.” – Albert Einstein

“The essence of mathematics is not to make simple things complicated, but to make complicated things simple.” – S. Gudder

“The highest form of pure thought is in mathematics.” – Plato

“There are things which seem incredible to most men who have not studied mathematics.” – Aristotle

This article first appeared at examiner.com in 2009. Editor’s update: Math may be hard, but keeping track of the number of items has been going on since prehistory. There are ancient artifacts found with tally marks cut into them. These have no concept of place value and are cumulative methods for tracking countable items. The first known number system with a place value was the Mesopotamian system from around 3400 BC. It was a base 60 system. The first base ten system (our own system is base ten) was from the Egyptians around 3100 BC. Zero as a number rather than a place holder is another concept that came later. Many early systems used 0 as a place holder and even the ancient Greeks had difficulty with the concept, pondering how nothing could be something. The Olmec people of Mexico began to use zero as a number as early as the 4^th century BC. In Europe the adoption was later. In fact, negative numbers were in use before the idea of nothing.

Also on this day: Cardiff Giant– In 1869, a petrified giant is found near Cardiff.
Planned Parenthood – In 1916, Margaret Sanger opened a clinic.
Disney – In 1923, Walt and Roy Disney signed a contract to produce the Alice Comedies.