209 Complex Number System

Standard 1: Understand that shapes in different categories (e.g., rhombi, rectangles, and others) may share attributes

While irrationals are uncountable infinite because beyond every countable number of decimal places, however, large it may be, there is a greater and infinite number. Any irrational number could be multiplied by some irrational number to get another irrational number. So pi could be multiplied by some irrational number to get e or root 2. There is so much more variation among irrationals and their ability to have infinite decimals explains why there are more of them in the universe.

Laura Spini, the writer and director of the short film 'You are Whole', answers for us the question, ‘What is the worst that can happen to a spirituality missionary who decides to go door to door spreading his knowledge of the stars?’

Now that we have the history of Pi out of the way, I bet you're intrigued on how people use Pi on an everyday basis . Pi is mostly used In circumference also known as perimeter . There is diameter and radius, diameter is solved by doing ‘3.14 * 12’ while radius is solved by ‘2*3.14*6’ squared . In case you didn’t know, diameter is the whole thing while radius is just half, which is why i used the example of twelve and six. Pi is also used in people’s everyday life , such as scientist when they search for new planets outside of our solar system .

My example of irrational numbers is the Golden ratio from the Fibonacci sequence. The Fibonacci sequence starts as 0, 1, 1, 2, 3, 5,... The numbers are added together to get the next sequence number. was made famous during the 13th century by Leonardo Pisano. The Golden ratio is when you take the sequence numbers and divide each by the previous number. The Golden ratio is the final set of numbers in, "the following set of numbers: 1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538..." (Knott, 2013). The Greek letter Phi is the represented symbol for the Golden ratio (1.61538...) This number is the reason for, " the spirals in the seedheads and the arrangements of leaves in many plants."(Knott, 2013). The

Have you ever thought about how there are infinite halfway points? Between any two points there is a halfway point. For example, say you are trying to walk to the door in the room that you are in. There is a halfway point between you and that door. There is another halfway point between the new point and the door and another halfway point between the newest point and the door, etc. This would go on for an infinite amount of times. However, you will eventually get to the door, which is a finite distance away. Therefore, when you add up infinitely many numbers, sometimes you can get a finite answer. This process is called converging. Diverging, the opposite, will add up infinitely many numbers and get an infinite answer. For example, what if you doubled the distance you walk instead of halving it? You would go twice the distance to the door, and then quadruple the distance to the door until you have reached an infinite distance.

Humans have many methods to describe large numbers mathematically. Ever since the invention of the concept of zero -- essential to the ancients for keeping good records of various transactions -- humans have been using mathematics to describe ever larger quantities of items. The story of the king and the chessboard of wheat demonstrates that while humans have been using large numbers since the dawn of history; using and comprehending are two entirely different things. In the story, a king tells a clever mathematician that he may ask for any reward he wishes for his services to the kingdom. And so the mathematician asks for one grain of rice to be placed on the first square of a chessboard, and then double that on the second square, and then double the second on the third, and so on. By the time all the rice would be added up, it would equal well over the production of rice in the entire modern world! Needless to say, the mathematician did not receive his reward . The same concept applies to time, at some point the human mind ceases to differentiate between larger and larger timescales.

Infinity is perhaps one of the most frequently encountered idea in today’s world. This concept of endlessness is what people normally associate with when discussing infinity, and it has become something that we have integrated into everyday language. Although it may seem to be simple concept, infinity is actually a widely debated and argued topic, and it has shown to be more complex than it initially appears to be. There are many types of infinities and over the years, people have studied, proved, analyzed, and applied the concepts of infinities. It is important to note that, although infinity is essential to mathematics, there exist types of infinity beyond the

Pi, a number older than the Great Wall of China, the Leaning Tower of Pisa and the Roman Colosseum, has been a number that has triggered my fascination since a young age. Pi, is an irrational number that is not only an infinite decimal, but it is also non-repeating (Baroody, 2002). I often find myself pondering the applications of Pi and its significance in mathematics. Sure, Pi is definitely a popular number in modern culture, it has been the central theme of numerous film adaptations, but what makes this number such an important number that all students around the world have to learn about? What makes Pi so integral to mathematics that other numbers cannot replace it? Therefore, the rationale behind this mathematics investigation is to satisfy my own queries about this unique number and to broaden my knowledge of Pi. This investigation will thus aim to identify the application and significance of Pi in mathematics but due to the extensive influence Pi has in mathematics, this investigation will focus specifically on Pi’s application in the fields of calculus, trigonometry, topics that fall within the standard level mathematics syllabus, but also differential geometry, a topic that is not in the standard level mathematics syllabus to broaden my knowledge of pi.

One of the oldest and most commonly known and used concepts in mathematics is that of Pi (π). In the earliest of know human civilizations, people realized the importance of finding the exact value of π for practical reasons. Even by todays standards, we still only need to know the exact value of π to a few decimal place values, although that hasn’t stopped mathematicians from pursuing a more accurate representation for its value throughout time.

\subsection{A primer on Interval Arithmetic} \label{prec_section} In this section, we present briefly Interval Arithmetic and focus on its interaction with floating point approximation of reals. For more details on Interval Arithmetic, the interested reader can consult a more extensive reference, such as~\cite{InterBook}.\medskip \subsubsection{Bird's eye view on Interval Arithmetic.} Interval arithmetic is a representation of reals by intervals that contain them. For instance, one can specify that a value $x$ is given with an error $\epsilon$ by considering the interval $[x-\epsilon, x+\epsilon]$, or even manipulating the transcendent constant $\pi$ as the interval $[3.14, 3.15]$. Interval arithmetic is crucial the context of numerical

which produces 1.618033989…, commonly used as the approximation 1.618. There is interesting fact that digits just keep on going to infinity, with no pattern. Mathematically, the Golden Ratio is tends to be an irrational number.

Likewise, in the set of ail Real numbers RU, 1.1, 1.11, 1.111,...} there are an infinite amount of members as weli. This sets cardinality is represented by t’ or, as Cantor proved in 1874, N1. it there are an infinite amount of Real values between each of the infinite amount of

Axis- a real or imaginary line through the center of an object, around which the object turns.

Numbers and equations simply to not concretely exist in our world. Maths is, after all, like any written language, glorified scribbling that somehow resonates with meaning in our minds. Take as an example '4 = 2 + 2 '. This string of symbols translates itself into what we understand is an equation: the two sides of the '= ' sign are the same. However, neither '4 ' not '2 + 2 ' exist. They are not visually identical. Yet somehow we accept that connecting them with the '= ' symbol indicates that their numerical value is equivalent. Like fiction, maths merely requires us to suspend our disbelief long enough for us to understand whatever story or message the author is trying to impart. This philosophical position is called Mathematical Fictionalism. Its proponents believe that mathematics are purely fictional but are still functional as long as its purpose is not meant to be a mirror of reality (Leng, 2016).