Imaginary and complex numbers

Imaginary and complex numbers When Are We Ever Going to Use This? – Imaginary and Complex Numbers The number √-9 may seem impossible, and it is when talking about real numbers. The reason is that when a number is squared, the product is never negative. However, in mathematics, and in daily life for that matter, numbers like these are used in abundance. Mathematicians need a way to incorporate numbers like √-9 into equations, so that these equations can be solvable. At first the going was tough, but as the topic gained more momentum, mathematicians found a way to solve what their predecessors deemed impossible with the use of a simple letter i, and today it is used in a plethora of ways. History of Imaginary Numbers During the early days of human mathematical history, when someone reached a point in a equation that contained the square root of a negative number, they froze. One of the first recorded instances of this was in 50 AD, when Heron of Alexandria was examining the volume of a truncated pyramid. Unfortunately for him, he came upon the expression which computes to . However, at his time, not even negative numbers were â€Å"discovered† or used, so he just ignored the negative symbol and continued on with his work. Thus, this first encounter with complex numbers was unsuccessful. It is not until the sixteenth century when the dilemma of complex numbers returns, when mathematicians attempt to solve cubic and other equations of higher-order. The Italian algebraist Scipione dal Ferro soon encountered these imaginary numbers when solving higher degree polynomials, and he said that finding the solution to these numbers was â€Å"impossible†. However, Girolamo Cardano, also Italian, gave this subject some hope. During his mathematical career, he opened up the realm of negative numbers, and soon began analyzing their square roots. Although he admitted that imaginary numbers were pretty much useless, he shed some light on the subject. Fortunately, this little bit of light would soon turn into a full beam. In 1560, the Bolognese mathematician Rafael Bombelli discovered a unique property of imaginary numbers. He found that, although the number √-1 is irrational and non-real, when multiplied by itself (squared), it produces both a rational and real number in -1. Using this idea, he also came up with the process of conjugation, which is where two similar complex numbers are multiplied together to get rid of the imaginary numbers and radicals. In the standard a+bi form, a+bi and a-bi are conjugates of each other. At this point, many other mathematicians were attempting to solve the elusive number of √-1, and although there were many more failed attempts, there was a little bit of success. However, although I have been using the term imaginary throughout this paper, this term did not come to be until the 17th century. In 1637, Rene Descartes first used the word â€Å"imaginary† as an adjective for these numbers, meaning that they were insolvable. Then, in the next century, Leonhard Euler finalized this term in his own Eulers identity where he uses the term ifor √-1. He then connects â€Å"imaginary† in a mathematical sense with the square root of a negative number when he wrote: â€Å"All such expressions as √-1, √-2 . . . are consequently impossible or imaginary numbers, for we may assert that they are neither nothing, not greater than nothing, nor less than nothing, which necessarily renders them imaginary or impossible.† Although Euler states that these numbers are impossible, he contributes with both the term â€Å"imaginary† and the symbol for √-1 as i. Although Euler does not solve an imaginary number, he creates a way to apply it to mathematics without much trouble. Throughout the years, there have been many skeptics of imaginary numbers; one is the Victorian mathematician Augustus De Morgan, who states that complex numbers are useless and absurd. There was a tug-of-war battle between those who believed in the existence of numbers such as i and those who did not. Soon after Rene Descartes contributions, the mathematician John Wallis produced a method for graphing complex numbers on a number plane. For real numbers, a horizontal number line is used, with numbers increasing in value as you move to the left. John Wallis added a vertical line to represent the imaginary numbers. This is called the complex number plane where the x-axis is named the real axis and the y-axis is named the imaginary axis. In this way, it became possible to plot complex numbers. However, John Wallis was ignored at this time, it took over a century and a few more mathematicians for this idea to accepted. The first one to agree with Wallis was Jean Robert Argand in 1806. He wrote the procedure that John Wallis invented for graphing complex numbers on a number plane. The person who made this idea widespread was Carl Friedrich Gauss when he introduced it to many people. He also made popular the use of the term complex number to represent the a+bi form. These methods made c omplex numbers more understandable. Throughout the 1800s, many mathematicians have contributed to the validity of complex numbers. Some names, to name a few, are Karl Weierstrass, Richard Dedekind, and Henri Poincare, and they all contributed by studying the overall theory of complex numbers. Today, complex numbers are accepted by most mathematicians, and are easily used in algebraic equations.

Aligning Elementary School to Middle School Curriculum Essay

There has been an upsurge in curriculum alignment actions among educational institutions in an attempt to reinforce the efficiency and effectiveness of instructions and assessment programs (Marsh & Willis, 2003). The need to eliminate the differences in both learning and teaching has compelled leaders in educational settings to examine the concepts and skills taught in educational institutions (Marsh & Willis, 2003). Also, the No Child Left Behind (NCLB) program of 2002 justifies the significance of curriculum alignment activities (Clarke, Kayona, Ruebling & Stow, 2006). In curriculum development, there are three essential areas that need great considerations. One, the curriculum document needs to â€Å"give direction, focus, and accountability in the learning environment. (Clarke, Kayona, Ruebling & Stow, 2006). Two, the curriculum document should be revised on a regular basis by at least two educational leaders and three; student outcomes that recognize what the learners acquire prior to graduation should align with the standards set by the state. According to Clarke, Kayona, Ruebling & Stow, (2006) many educational institutions have not yet integrated successfully the essentials standards within their curriculum nor maintained a written document to guide instruction. Again, the important role of leaders in educational setting to develop and implement curriculum is evident with the alignment of taught, written and tested curriculum. Clarke, Kayona, Ruebling & Stow, 2006) also affirms that the lack of curriculum alignment can result to lower achievement where educators teach contents that is not part of the curriculum document and nor is it part of assessment material. This dissertation will analyze the curriculum articulation and alignment between elementary school and middle school social studies, specifically, fourth and seventh grade. Inclusive of this dissertation will be a plan to improve curriculum articulation and alignment between elementary school and middle school social studies as well as improve the academic progression of student and the consistency of the curriculum. Summary of Original Analysis The curriculum for the North Dakota Department of Public Instruction is a standards-based design that lays emphasis on the student’s ability. The curriculum is divided by subject, grade level and standard. It explains the core contents for each subject with proficiency descriptors. The curriculum document is relevant to the mission of the North Dakota Department of Public Instruction as well as the objectives for the subjects. The Social Studies curriculum document for North Dakota Department of Public Instruction has five major themes for both elementary school and middle school as well as benchmark expectations in each category. The main ideas and academic expectations are the same for both schools. The design of subject matter for adjoining educational stages includes concepts, skills, understandings, and the core content useful in assessment. The curriculum documents of North Dakota Department of Public Instruction for the elementary and middle school comprise vertical alignment between the adjoining educational levels. The concepts, skills and understandings become more complicated amid the two educational levels. Therefore, learners should understand the concepts and skills of the elementary school before learning the concepts and skills of middle school. The concentration of the elementary school is narrower than that of middle school. The curriculum documents of both elementary and middle school do not exhibit horizontal alignment. Improvement of curriculum alignment, student progression and curriculum consistency Curriculum alignment is necessary to guarantee harmony between the curriculum’s intentions and the instruction of the curriculum by testing what is taught. In effect, there is little or no space left for differences in curriculum implementation. The educational leaders thereafter, assess educators by the degree in which the learners meet the set standards through standardized examination (Marsh & Willis, 2003). The vertical alignment links subjects that express a relationship and a particular order and then match the curriculum to the particular order. In this, students will gain vital knowledge prior to the next subject. The alignment aids in material reinforcement and offers teachers knowledge in relation to students’ ability. In addition, Melvin (2007 notes that vertical alignment enhances learning for students as well as communication between teachers at various educational levels. Horizontal alignment transversely brings together curriculum objectives between subjects. Horizontal alignment benefits learners in that learning transmits to new situation where students realize the significance of the knowledge gained (Wraga, 1999). The social studies curriculum within the North Dakota Department of Public Instruction between elementary and middle school, puts emphasis on standards and outlines objectives needed by students to learn. So as to improve curriculum alignment, it is necessary for curriculum at adjoining educational levels to lay emphasis on these objectives at every appropriate occurrence but not only within a particular grade. Glatthorns (1999) suggests it is important that the curriculum contain objectives for constant development, which can center on learning as well as testing when probable. Again, the curriculum should include enhancement units for all learners. Educators can then assess the standards and align the taught, written and tested curriculum. By gathering the acknowledged state standards and applying resourcefulness, tutors can develop unit which, will be interesting to the students. Teachers’ flexibility can also be put in play where the curriculum can be adjusted to meet the requirements and capability of the learners. In this strategy, educators align the written curriculum with the tested curriculum and the taught curriculum with the written one. As they implement the curriculum, In addition, teachers can include objectives for ongoing development and enhancement units. This process will allow learners to achieve high grades on tests while the ongoing students will have units of study that are comprehensive and thus will get their interest (Glatthorn, 1999). The North Dakota Department of Public Instruction maintains a standards-based curriculum that contains list of fundamental details and abilities at the lower end of Bloom’s Taxonomy, with no intention for connecting the fundamental details into more important outcomes of learning. Marsh and Willis (2003) contends that the progression that has been mandated by state would then be trussed by practical and fundamental knowledge with minimal consideration of lasting learning competency that is necessary to life as well as other areas of learning. A curriculum that lays out the necessity to address all students and concentrates on cognition development at a higher-level whilst keeping the standards-based curriculum is the most suitable to put into practice for the two adjoining educational stages (Wraga, 1999). A combined curriculum that is integrated, inclusive and student centered with holistic and multicultural aspects will enhance curriculum alignment both horizontally and vertically. In addition, this kind of curriculum will enhance learners’ academic progression and assist in curriculum consistency. Wraga (1999) admits that the standards-based curriculum must contain curriculum alignment of a larger curriculum that ensures lifelong learning. A curriculum that contains intrapersonal improvement encourages the learners to discuss learning and associate in relation to the learners’ life. Teaching and nurturing social skills through the curriculum allows cooperation of learners with other individuals, builds conflict resolution skills, and assists in team building. The curriculum within the North Dakota Department of Public Instruction shall integrate multicultural aspect so as to align the curriculum with the today’s society. Learners will get to value diverse culture and as a result develop communication among cultures. An all grade inclusive curriculum which is also adaptive is valuable to all learners; it will offer equal learning opportunities to students with disabilities as well as allow all students to embrace diversity among individuals. A student-centered curriculum focuses on the learners but not the curriculum materials, content to be covered or written goals. When there is implementation of a standards-based curriculum in association with a combined curriculum and high-stakes testing, students are assured of quality education. Layton & Lock (2007) asserts that student assessments are vital and in an effort to assist in academic progression of student educators should build tests that assess student learning correctly and adjust tests for the at-risk learners. Educators can unearth better ways of evaluating test results so as to identify learning needs of students and revise instruction. In addition this plan integrates authentic assessment. Authentic assessment demands the relevance and integration of skills vital to solve complicated problems and allows enhanced substantiation of the students’ ability to prosper in all situations. Learners have a chance to explain themselves allowing teachers to better understand the learning process and get instant feedback for learners. Furthermore, the application of authentic assessment sets up teacher collaboration and collegiality as well as expands the reflection and inquiry by the teacher. Conclusion Curriculum alignment involves matching a curriculum’s content to the use of tests so as to assess the learning of student (Marsh & Willis, 2003). Curriculum alignment calls for educational institutions to ensure consistence in the objectives, guides, textbooks, and tests that make up the curriculum (Marsh & Willis, 2003). Leaders in educational settings determine academic achievement through standard testing as a means to find out what happens in schools. Institutions of education emphasize effectiveness through increasing academic achievement where they term decline in academic achievement as a problem in the quality of the curriculum. Thus, aligning the intended curriculum with the taught curriculum seems to resolve this problem (Marsh & Willis, 2003). By integrating the previous aspects into the North Dakota Department of Public Instruction curriculum, a more significant learning experience can be achieved by both educators and students. Lastly, curriculum alignment will be seen, student academic progression as well as curriculum consistency will improve (Marsh & Willis, 2003). References Alexson, R. G. & Kemnitz, C. P. (2004): Curriculum articulation and transitioning student success: Where are we going wrong and what lessons have we learned? Educational Research Quarterly, Retrieved from ProQuest Education Journals database. Accessed on September 26, 2007 Biessman, J. B. (2006): An examination of curriculum articulation in Wisconsin high school and post-secondary introductory biology courses. Retrieved from ProQuest Digital Dissertations database. Accessed on September 25, 2007. Carnoy, M. (2005): Have state accountability and high-stakes test influenced student progression rates in high school? Educational Measurement, Issues and Practice, 24. Retrieved from ProQuest database. Accessed on September 26, 2007. Clarke, N. A. , Kayona, F. , Stow, S. , Ruebling (2006): Developing standards-based curricula and assessments: Lessons learned from the field. Retrieved from EBSCO host database. Accessed on September 26, 2007, Cumming, J. J. & Maxwell, G. S. (1999): Contextualizing authentic assessment. Assessment in Education, 6. Retrieved from ProQuest database. Accessed on September 26, 2007 Hodgkinson, H. (2000). Educational demographics: What teachers should know. Educational Leadership. John, B. (2003): Curriculum audits: achieving alignment for student success. Education Today. Retrieved from CBCA Complete database. Accessed on September 26, 2007. Odden, A. (2000): The costs of sustaining educational change through comprehensive school reform. Ornstein, A. C. & Levine, D. U. (1990): School effectiveness and reform: Guidelines for action. Porter, A. C. (2005): Getting the Content of Instruction Right. Retrieved from CBCA Complete database. Accessed on September 25, 2007 Layton, C. A. & Lock, R. H. (2007): Use authentic assessment techniques to fulfill the promise of no child left behind. Intervention in School and Clinic. Retrieved from ProQuest database. Accessed on September 25, 2007 Marsh, C. J. , & Willis, G. (2003): Curriculum: Alternative approaches, ongoing issues, 3rd edition. Upper Saddle River, NJ: Pearson. Wraga, W. G. (1999). The educational and political implications of curriculum alignment and standards-based reform. Journal of Curriculum and Supervision. Retrieved from ProQuest database. Accessed on September 25, 2007 Young, M. L. (2004): High-stakes assessment and school accountability: A multicase study describing and comparing the best practices of five K–6 elementary schools that have demonstrated significant increases in student achievement as measured by the norm-referenced SAT-9 assessment and the Academic Performance Index of California. Retrieved from ProQuest Digital Dissertations database. Accessed on September 25, 2007

Gender Separation in Public Restrooms - 1233 Words

For most of the general population, the decision of which restroom to use when in public is really not much of a decision at all. No thought needs to be put into it, people just choose one and go. For a transgender person deciding upon which restroom to use can be an agonizing decision to make, one that can lead to severe anxiety. Most businesses and schools have very clearly defined â€Å"men’s† and â€Å"women’s† restrooms. Within the transgender community, it is not that black and white. A transgender person is defined as somebody whose gender identity is different than the sex they were assigned with at birth (â€Å"New Law†). In simpler terms, it is a person who identifies as the opposite sex. This makes choosing which restroom to use when in†¦show more content†¦Some people think that it should remain as such. Although this may be a sound argument, it lacks any real substance. Men and women have lived together since the beginning of time and to say that each doesn’t know what the other does in the restroom is absurd. Another issue some people have is that bathrooms are often used for socializing, hence why women frequently use the restroom together. Many people are concerned that integrating restrooms could cause the loss of the same-sex socialization. However, there is no reason that there can’t still be socializing in a mixed-gender bathroom. Lack of privacy is yet another concern of people opposed to gender-neutral restrooms. Moreover, a lack of gendered privacy seems to be the concern. Many people already have a level of discomfort when using a public restroom. For women, it tends to lean towards the lack of femininity that is associated with the elimination of body waste. Expelling excrement has long been viewed as taboo and something that girls just don’t do. Many women find issue with being able to use the facilities when only women are in there. Therefore the thought of having to do it with men in the same room can be a very uncomfortable idea. However, the reality is people on a subconscious level know what goes on in the bathroom. Men know that women do the same things that they do. While it is something that is seldom discussed, everyone knows what theShow MoreRelatedGen 205 Fall 2015 Final Exam Essay Questions1422 Words   |  6 Pagesfor your final paper? I do not understand. What perspective? Question 2 Identify 3 cultural symbols of gender privilege and describe how interactional and structural levels of society reinforce this gender privilege. How is this privilege shaped by intersectionality? Gender Binary Hegemonic masculinity Intersectionality The interactional and structural levels of society reinforce this gender privilege is mentioned in Messner’s article. 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Gender Inequality Since The Beginning Of Time Mankind Has...

Marichelle Arnold Professor Lee March Political Science 1100 6 March 2015 Gender Inequality In America Since the beginning of time mankind has been divided and categorized based on gender. Starting from birth males and females are assigned roles based are expected to act and adhere to stereotypes given to that sex. Only in recent times have either gender stepped outside of the gender boundaries. However, despite the new sense of individuality society tries to restrict them to gender archetypes. Males are expected to be masculine and females are expected to me feminine and if either tries to be the other they are told its wrong. There is so much inequality in society today, even beyond the binary gender . Transgender persons also experience inequality in equal parts as cisgendered people. In most cases though when the topic of equality is brought up the conversation transitions to feminism, almost ignoring transgender. Inequality equally affects everyone not just females. Despite the inequalities genders face, improvement can be seen. Although there are a rguments that say there are areas in American society where there has been very little advancement, there are some areas where there has been major increase in gender equality especially for women rights, women in government, women in the medical field, and transgender equality. While gender equality as a whole is improving greatly in America there are arguments that there has been little to no improvement. The reasonsShow MoreRelatedGender-Based Disaster Relief And Ngo Efforts: A Case Study10156 Words   |  41 Pages Gender-Based Disaster Relief and NGO Efforts: A case study of the majhi community in the 2015 Earthquake. By RABINA KC (201529240001) SCHOOL OF SOCIAL DEVELOPMENT AND PUBLIC POLICY BEIJING NORMAL UNIVERSITY SUPERVISOR YANG LICHAO 1. Introduction 1.1. 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