Stellar Numbers Math Ib Sl Portfolio Investigation Mathematics Essay
Mathematics is a subject with simple purpose, which is to find and understand patterns and try to apply the patterns to real life situation. In this investigation however, we are dealing with sequences which in this specific case are not applicable to real life as we do not encounter stellar numbers and triangular numbers in everyday situations. The purpose of this investigation is to find and recognize progressive patterns.
For the general statement of the nth triangular number I have constructed a table of trials and improvements. I started by using the trial and improvement method then looked at it algebraically as well. This process was a lengthy one however I eventually was able to deduce the final correct general statement based on my trials. This is done by replacing Un(The general term which is substituted by a number i.e. 1,2,3,4,5,6,7,8) by the triangular number we are trying to find (in this case all positive integers work), and n is the same number as well, which is plugged into each equation.
To construct these stellar numbers manually or on a computer, one must first imagine a regular polygon that will represent the number of vertices. So for a 3 vertices stellar, a triangle, a 4 vertices stellar a square…. and a 20 vertices stellar a icosagon. Once you have imagined this shape, you can plot your stage 1 points, beginning with the shape and then a little further away from your shape and inbetween the points you add dots to make it star like. As demonstrated below(Thick black lines):
There are limitations to the formula as there cannot be a negative number for p or n as there are no negative stellars or 0 stellar numbers. P can also not take the values of 1and 2. Further limitations include irrational numbers, imaginary numbers, and fractions. Thus the general statement works in natural numbers above 3. The notation would be such: Sn= >3 where is the set of natural numbers. Irrational numbers such as π would not work as you cannot have a stellar numbers with π vertices or at stages of π. In this case π has no exact value and stellars have an exact value thus it is impossible to have a π stellar or an e stellar. Also it is impossible to have stellars which are fractions e.g.: 1/5 stellar: Sn=1/5 (2)2-1/5(2)+1=7/5. A 7/5 stellar cannot exist as stellars have to have a value which is a natural number greater than or equal to 3. Imaginary numbers in the form (X+I), where X is a complex real number and I is an imaginary number such as √-1.
I arrived at the first general statement for a 6 stellar through trial and error, aided with the use of a calculator to help deduce the final general statement. Furthermore as I understood the pattern for the 6 stellar, I tried the same type of formula for the 7 stellar, 5 stellar, 4 stellar, and 3 stellar. My reasoning was correct and through the analysis of the trials I was able to deduce the general formula of: Sn=p(n)2-p(n) +1.
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