Applied Mathematics is a way to use math to solve problems in the real world, like in economics, engineering, and medicine. 

I researched on the fundamentals of computing algebra, and found out the arithmetic operations in the computer. This inspired me to see what happens to the counting number sequence after it has undergone a binary transformation.

Ever wanted to explore the mysteries of binary numbers? In this project, you'll discover patterns in binary sequences and even invent your own!

Project Title: Exploring Binary Sequences

Objectives/Goals

My first intention on the goal of this project was to find something new, like
an invention to innovate this
society. I had no trouble starting with a topic, since my head is always full of
questions. I was more
interested in computers and binary numbers, and so I studied binary numbers in
depth.

Methods/Materials

I researched on the fundamentals of computing algebra, and found out the
arithmetic operations in the
computer. This inspired me to see what happens to the counting number sequence
after it has undergone a
binary transformation. My original hypothesis was that I wouldn't get any
noticeable patterns, or maybe
one that won't be helpful. The Inverse Sequence came directly from computer
subtraction algorithm, but I
invented the Reverse Sequence (a.k.a. #the Richard Sequence,# which I happily
named it after myself).
Converting a number to binary, and converting it back to decimal after a string
operation could be a
tedious duty. And so I prepared my procedures accordingly: write all programs on
my TI-89 graphing
calculator, generate sequences for at least the first 1000 terms, observe
sequences, and try to prove or
explain patterns if any.

Results

After generating the sequences, I noticed astounding patterns. The Inverse
sequence was generated by
inverting the binary (0=1 1=0) of the natural number sequence, and the results
came out as expected (the
subtraction sequence). But the Richard Sequence did surprise me. Created by
reversing the string of the
binary of the natural number sequence, it holds many patterns, some obvious and
some hidden. I derived
formulas of some patterns for both sequences. (Although the formulas look
complex, they are the only
mathematical explanation of the observed patterns.) Uncovering some patterns
required grouping of
terms, summing, and manipulation of entire sequences.

Conclusions/Discussion

After my whole research, I concluded that I discovered something entirely new
that no one has ever done,
thus I am the one who should be able to find a pertinent use for this invention.
Overall, I think the Richard
Sequence is a very interesting sequence, and proofs will be needed to follow-up
my patterns, formulas,
and theories. I will try to work on proofs in the future, and maybe someday, my
sequence would become
an innovation others can base their work on.

Summary Statement

I invented a new binary sequence and found many interesting properties.
Help Received
N/A

Exploring Binary Sequences

Exploring binary sequences

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Math is all about numbers, patterns, and solving problems. Computer science is about using math and technology to create computer programs and solve real-world problems. Together, they help us understand how things work, how to solve problems, and how to use technology to make our lives better!

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Exploring Binary Sequences

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