write4u, here are some ideas on mathematics you might want to consider when you speak of mathematical patterns.
Mathematics is derived from the contributions of thinkers throughout the ages and across the globe. It gives us a way to understand patterns, to quantify relationships, and to predict the future. Math helps us understand the world — and we use the world to understand math.
The world is interconnected. Everyday math shows these connections and possibilities.
Algebra can explain how quickly water becomes contaminated and how many people in a third-world country drinking that water might become sickened on a yearly basis. A study of geometry can explain the science behind architecture throughout the world. Statistics and probability can estimate death tolls from earthquakes, conflicts and other calamities around the world. It can also predict profits, how ideas spread, and how previously endangered animals might repopulate. Math is a powerful tool for global understanding and communication. Using it, students can make sense of the world and solve complex and real problems. Rethinking math in a global context offers students a twist on the typical content that makes the math itself more applicable and meaningful for students.
For all of us to function in a global context, math content needs to help us get to global competence, which is understanding different perspectives and world conditions, recognizing that issues are interconnected across the globe, as well as communicating and acting in appropriate ways. In math, this means reconsidering the typical content in atypical ways, and explaining how the world consists of situations, events and phenomena that can be sorted out using the right math tools.
Math is often studied as a pure science, but is typically applied to other disciplines, extending well beyond physics and engineering. For instance, studying exponential growth and decay (the rate at which things grow and die) within the context of population growth, the spread of disease, or water contamination, is meaningful. It not only gives us a real-world context in which to use the math, but helps us understand global phenomena – a disease spreading in India, but we can’t make the connection without understanding how fast something like cholera can spread in a dense population. In fact, adding a study of growth and decay to lower level algebra – it’s most often found in algebra II – may give us a chance to study it in the global context than if it’s reserved for the upper level math that not all students take.
In a similar vein, a study of statistics and probability is key to understanding many of the events of the world, and is usually reserved for students at a higher level of math, if it gets any study in high school at all. But many world events and phenomena are unpredictable and can only be described using statistical models, so a globally focused math program needs to consider including statistics. Probability and statistics can be used to estimate death tolls from natural disasters, such as earthquakes and tsunamis; the amount of aid that might be necessary to help in the aftermath; and the number people who would be displaced.
Understanding the world also means appreciating the contributions of other cultures. In algebra, students could benefit from studying numbers systems that are rooted in other cultures, such the Mayan and Babylonian systems, a base 20 and base 60 system, respectively. They gave us elements that still work in current math systems, such as the 360 degrees in a circle, and the division of the hour into 60 minute intervals, and including this type of content can help develop an appreciation for the contributions other cultures have made to our understanding of math.
Mathematics is all about numbers. It involves the study of different patterns. And there are different types of patterns, such as
number patterns, image patterns, logic patterns, word patterns etc. Number patterns are very common in Mathematics. These are quite familiar to the students who study Maths frequently. Especially, number patterns are everywhere in Mathematics. Number patterns are all predictions. In this article, we will discuss what is a pattern, and different types of patterns like, arithmetic pattern, geometric pattern and many solved examples.
In Mathematics, a pattern is a repeated arrangement of numbers, shapes, colours and so on. The Pattern can be related to any type of event or object. If the set of numbers are related to each other in a specific rule, then the rule or manner is called a pattern. Sometimes, patterns are also known as a sequence. Patterns are finite or infinite in numbers.
For example, in a sequence 2,4,6,8,?. each number is increasing by sequence 2. So, the last number will be 8 + 2 = 10.
Few examples of numerical patterns are:
Even numbers pattern -: 2, 4, 6, 8, 10, 1, 14, 16, 18, …
Odd numbers pattern -: 3, 5, 7, 9, 11, 13, 15, 17, 19, …
Fibonacci numbers pattern -: 1, 1, 2, 3, 5, 8 ,13, 21, … and so on.
The arithmetic pattern is also known as the algebraic pattern. In an arithmetic pattern, the sequences are based on the addition or subtraction of the terms. If two or more terms in the sequence are given, we can use addition or subtraction to find the arithmetic pattern.
For example, 2, 4, 6, 8, 10, __, 14, __. Now, we need to find the missing term in the sequence.
Here, we can use the addition process to figure out the missing terms in the patterns.
In the pattern, the rule used is “Add 2 to the previous term to get the next term”.
In the example given above, take the second term (4). If we add “2” to the second term (4), we get the third term 6.
Similarly, we can find the unknown terms in the sequence.
First missing term: The previous term is 10. Therefore, 10+2 = 12.
Second missing term: The previous term is 14. So, 14+2 = 16
Hence, the complete arithmetic pattern is 2, 4, 6, 8, 10,
12, 14,
16.
The geometric pattern is defined as the sequence of numbers that are based on the multiplication and division operation. Similar to the arithmetic pattern, if two or more numbers in the sequence are provided, we can easily find the unknown terms in the pattern using multiplication and division operation.
For example, 8, 16, 32, __, 128, __.
It is a geometric pattern, as each term in the sequence can be obtained by multiplying 2 with the previous term.
For example, 32 is the third term in the sequence, which is obtained by multiplying 2 with the previous term 16.
Likewise, we can find the unknown terms in the geometric pattern.
First missing term: The previous term is 32. Multiply 32 by 2, we get 64.
Second missing term: The previous term is 128. Multiply 128 by 2, we get 256.
Hence, the complete geometric pattern is 8, 16, 32,
64, 128,
256.
The Fibonacci Pattern is defined as the sequence of numbers, in which each term in the sequence is obtained by adding the two terms before it, starting with the numbers 0 and 1. The Fibonacci pattern is given as 0, 1, 1, 2, 3, 5, 8, 13, … and so on.
Third term = First term + Second term = 0+1 = 1
Fourth term = second term + Third term = 1+1 = 2
Fifth term = Third term + Fourth term = 1+2 = 3, and so on.
To construct a pattern, we have to know about some rules. To know about the rule for any pattern, we have to understand the nature of the sequence and the difference between the two successive terms.
Finding Missing Term: Consider a pattern 1, 4, 9, 16, 25, ?. In this pattern, it is clear that every number is the square of their position number. The missing term takes place at n = 6. So, if the missing is xn, then xn = n2. Here, n = 6, then xn = (6)2 = 36.
Difference Rule: Sometimes, it is easy to find the difference between two successive terms. For example, consider 1, 5, 9, 13,……. In this type of pattern, first, we have to find the difference between two pairs of the sequence. After that, find the remaining elements of the pattern. In the given problem, the difference between the terms is 4, i.e.if we add 4 and 1, we get 5, and if we add 4 and 5, we get 9 and so on.
In Discrete Mathematics, we have three types of patterns as follows:
- Repeating – A type of pattern, in which the rule keeps repeating over and over is called a repeating pattern.
- Growing – If the numbers are present in the increasing form, then the pattern is known as a growing pattern. Example 34, 40, 46, 52, …..
- Shirking – In the shirking pattern, the numbers are in decreasing form. Example: 42, 40, 38, 36 …..
Understanding mathematical patterns allows someone to identify such patterns when they first appear. After all, you can not gain the benefit of patterns if you can't see them and you can only see them if you understand them.
Patterns provide a sense of order in what might otherwise appear chaotic. When you notice that things happen in a certain pattern - even something as mundane as a bus always stopping at a certain corner at 5pm - order is provided.
Patterns allow someone to make educated guesses. Much science is based on making a hypothesis and hypothoses are often based on understanding patterns. Similarly, we make many common assumptions based on recurring patterns.
Understanding patterns aid in developing mental skills. In order to recognize patterns one need to have an understanding of critical thinking and logic and these are clearly important skills to develop.
Patterns can provide a clear understanding of mathematical relationships. This can be seen in a very evident manner in the form of multiplication tables. 2 x2, 2 x 4, 2 x 6 are clearly examples of the relationship pattern found in multiplication.
Understanding patterns can provide the basis for understanding algebra. This is because a major component of solving algebra problems involves data analysis which is deeply related to the understanding of patterns. Without being able to recognize the appearance of patterns the ability to be proficient in algebra will be limited.
Understanding patterns provide a clear basis for problem solving skills. In a way, this is related to critical thinking but more directed towards mathematics specifically. Patterns essentially provide a means of recognizing the broader aspects that can be shored down in order to arrive at the specific answer to a particular problem.
Knowledge of patterns is transferred into science fields where they prove very helpful. Understanding animal patterns has been used to help endangered species. Understanding weather patterns not only allows one to predict the weather but also predict the common impact of weather which can aid in devising the appropriate response in an emergency situation.
One of the lesser known aspects of patterns is the fact that they often form the basis of music. For example, there are various patterns of notes that provide the basis for proper harmony on a piano. If you don't believe patterns are important when playing a piano simply walk up to the nearest piano and start banging away randomly at the keys. You probably won't hear any songs that you recognize!
Patterns provide clear insight into the natural world. While animals and certainly plants are far from thinking beings they do have certain habits that exist in patterns and understanding these behavioral patterns provides a clearer understanding of all living things.
It is safe to say that the benefits of understanding patterns open many doors where this knowledge can be applied. Of course, that is a commonality with all forms of learning mathematical logic: there is a deep application that can be provided that we often do not realize when we first study the material. With understanding patterns - and other forms of math - sometimes you really need to stick with it for the long term, but with that practice comes skill. Researchers have found that pattern skills can be learned relatively quickly.
Regular expression patterns can capture simple but com- mon and important properties easily, even though they are not as powerful as languages in more sophisticated frame- works. The combined power and simplicity of regular ex- pressions contribute to their wide use in computing, from languages and compilers, to database and web information retrieval, to operating systems and security, etc.
Parametric regular path queries extend the patterns with variables, called parameters, which significantly increase the expressiveness by allowing additional information along sin- gle or multiple paths to be captured and related, and the amount of such information is not bounded by the size of the pattern. This extension enables analysis of significantly many more important properties about dependencies, con- currency, resource usage, etc. The regular expression pat- terns used to analyze these properties are simple, easy to write, and succinct.
General algorithms have been studied for solving simpler regular path queries, in particular, queries involving uncor- related paths and queries containing no variables. A method was also proposed to code parametric regular path queries using logic programs. What have been lacking are complete algorithms and data structures for solving parametric regular path queries directly, efficiently, and with precise complexity analysis.
See:
https://byjus.com/maths/patterns/
See:
https://www.mathworksheetscenter.com/mathtips/mathpatterns.html
See:
https://asiasociety.org/education/understanding-world-through-math
Getting back to Max Tegmark again...
In an article entitled “Parallel Universes” in the May 2003 issue of Scientific American (
https://www.scientificamerican.com/magazine/sa/2003/05-01/), Tegmark presents a clear and comprehensive picture of the parallel-universe idea. What Tegmark describes is actually a set of related concepts which have in common the notion that there are universes beyond the familiar observable one that astronomers can see parts of directly with telescopes and other instruments. Some of these parallel universes are completely unlike our own; others are nearly identical; still others are identical up to a point and then split off into might-have-been worlds of choices not made.
Tegmark’s main argument is that, far from being a shadowy, speculative corner of cosmology, the parallel-universe idea has been increasingly confirmed by recent experiments, and we should get used to it because it appears that it will be around for a while.
If true, this is not good news for proponents of intelligent design such as William A. Dembski. In his recent book No Free Lunch, Dembski is at considerable pains to show why the parallel-universe idea is basically a non-starter. He recognizes the threat that parallel universes pose to the concept of specified complexity. Simply expressed, if literally anything can happen, it will, including the most unlikely and designed-looking things such as earth, life, and humanity. If certain forms of the parallel-universe idea are true, then chance, not design, becomes omnipotent.
Cosmology has always bordered upon metaphysics. Questions of ultimate origin and destiny began as metaphysical questions, and only in the last century has science begun seriously to address some of these issues with theories based on empirical evidence. It is still not always easy, therefore, to distinguish cosmology based on empirical evidence from a philosophical position disguised as empirical cosmology. Tegmark’s article deals largely with theories whose main feature, namely, multiple universes, cannot be verified by observation or experiment even in principle. The experimental tests he proposes for these theories really consist in making the philosophical presuppositions required for believing in the theories, and then verifying that the theories agree with already-known data about the present visible universe. So far from being a legitimate way to inflate probabilistic resources to defeat arguments in favor of intelligent design, Tegmark’s parallel universes represent an array of philosophical arguments disguised as science. While the philosophical arguments may have merit on their own, it is illegitimate to claim that they are empirically verified in the conventional scientific sense, as Tegmark sometimes does.
Hartmann352