Mathematics and Physics for Programmers [History of Numbers]
A Brief History of Numbers.
I have been reading the 2nd Edition of Mathematics and Physics for programmers by John Flynt, who also wrote Software Engineering for Game Developers and decided to share a few interesting points that I learned after reading it. Just to note that the book was not written as a textbook but more as a reference to Game Programmers when tackling mathematical and physics related problems in the development process.
The first chapter fittingly starts by describing the history of numbers. How numbers took millennia to evolve to the point we know now. The first category of numbers is the Natural Number which all of you would have learned as children, 1,2,3,4. These numbers were made to define a certain number of things, and then zero was made to define the absence of these things.
Evidence of natural number arithmetics have been seen on an Ishango bone dating to 18,000 BC .
But it was not until 200BC , or BCE if you want to use the common era format , that we have found evidence of negative numbers, some can be seen in the Chinese number rod system representing positive numbers as red and negative numbers as black
In 620AD evidence in india from Brahmagupta (598–670) came who used the idea of fortunes and debts for positive and negative. By this time a system based on place-value was established in India, with zero being used in the Indian number system, Brahmagupta used a special sign to represent positive and negatives and stated the rules for dealing with positive and negative quantities as follows.
The rules of positive and negative quantities.
A debt minus zero is a debt; A fortune minus zero is a fortune; Zero minus zero is zero A debt subtracted from zero A fortune subtracted from zero is a debt The product of zero multiplied by a debt or a fortune is zero. The product of zero multiplied by zero is zero. The product or quotient of two fortunes is one fortune. The product or quotient of two debts is one fortune. The product or quotient of a debt and a fortune is a debt. The product or quotient of a fortune and a debt is a debt
Brahmagupta, India, 620AD
Why did it take a while after positive numbers were invented to come up with negative numbers?I am going to explain why in another example.
Say for instance you have a two friends Pascal and Kelvin, Pascal has one marble and Kelvin has 4 , Now Kelvin gives Pascal 2 marbles. Kelvin now has 2 marbles and Pascal has 3. So the deal is that Kelvin giving Pascal 2 marbles means that Pascals marbles goes up by 2 but more importantly Kelvins goes down by 2, how can we represent this number of things, the minus sign followed by the change. In summary Kelvin giving you two marbles was
“ the other way around” to Kelvin gaining two marbles. So if we put this in context again extending to multiplication, if I was to -2 marbles -2 times firstly its the other way around to -2 marbles +2 times so instead of removing 4 marbles you are gaining 4.
As said in the book, if that seemed confusing to you that explains why it took centuries or even millennia to accept this “other way around” counting. As British mathematician Francis Maseres said in 1758.
“[negative numbers] darken the very doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple”
Francis Masares , London Uk , 1758
Masares and his contemporaries did not even believe that they existed it was not until Dr Morgan Peacock and others began to investigate the ‘laws of arithmetic’ in terms of logical definitions thus the problem of negative numbers was finally sorted out.
Henceforth the Integers represented as Z. I like to think its because it is Positive-Negative numbers which are like A-Z ,(but really its because it named after the German word Zahlen which means number),more specifically it comprises of the Natural Numbers Negative Numbers and Zero. In fact it took so long for people to accept this negative counting that fractions, which is where you take the quotient of two integers, was accepted sooner. First came the fractions of 1 eg 1/2 , 1/3 , 1/4, in many cases these were just the reciprocals of the natural numbers which can be represented as quotients of 1 , ie the reciprocal of 2/1 is 1/2 , 3/1 is 1/3 and so on. After the reciprocals the simple fractions occurred which is just fractions between 0 and 1 like 2/3 , 5 / 8 , 9/11 and so on. Then came the vulgar fractions which had top heavy fractions greater than 1 , so 5/4 , 11/3 . Finally after accumulating all of these together you get Rational Numbers, represented as Q. I like to think its represented as Q because it is Integers + Quotients and turns out thats exactly why its called Q.
After Rational numbers came the ones that didn’t fit into this category, Numbers which could not be found by taking the quotient of two integers . the Irrational numbers. Many for a while thought that these could not exist however the man him self Pythagorus in 550 BC found out that at least the sqrt(2) is an irrational number a^2 + b^2 = c^2 now take a and b as 1 , and you get that c is an irrational number. Another famous irrational number is pi.
This cannot be found by taking the quotients of two integers , but rather it is the ratio of the circumference of a circle to its diameter. Ie if you are to take a circle of diameter 1 , its circumference would be Pi.
Rational and Irrational numbers together make Real Numbers. Which is represented by the R symbol for self explanatory reasons. Lastly we come across the last set of numbers.
Unreal numbers?
No i joke, these are known as complex numbers which are essentially multiples of i , the sqrt(-1). We wont discuss these numbers just yet, but once I get onto Quaternions Spacial rotation I will.
