Benton Jones Math 2 Imaginary and Complex Numbers Imaginary and complex numbers are not actually complicated, it’s just in the name. It’s pretty easy to understand as long as you know the basics. Complex numbers are made up of real and imaginary numbers, in the form of a+bi. Imaginary numbers are in the form of i, where i is equal to the square root of negative one. See, you already have the basics of imaginary and complex numbers, but there is more that can help you as well. Imaginary and complex numbers have shortcuts and other helpful things imbedded in them. Imaginary and complex numbers have patterns, can be simplified, have classification, and can also be solutions. First of all, imaginary numbers have a specific pattern that they …show more content…
If you are asked to give an exact solution for a quadratic equation that does not have x-intercepts, then you will answer that question using the variable i. Say you need to find the square root of negative sixty-four. We know that the perfect square of positive sixty-four is eight. What we are going to do is take out i, resulting in the square root of positive sixty-four, which we know is eight, therefore the answer is plus or minus eight i. It is plus or minus because square roots can be positive or negative because a negative times a negative is also equal to a positive. You can also simplify an equation if there is a constant before the negative under the radical. You do the same thing in terms of simplifying as you would do without the constant. After you get your imaginary number you put the constant in the correct position, and then you are left with a complex number, such as a+bi. You can also simplify i when it is involved in a polynomial. If you multiply out two polynomials that have imaginary numbers in them you may end up with i with an exponent attached. You can use your previous knowledge of patterns to simplify the equation. Say you end up with a term along the lines of thirty-six i squared. We know that i to the second power is the same as negative one. From there we can multiply negative one and thirty-six to result in the product of negative thirty-six. As you have …show more content…
There is a chain that leads up to a top classification. Everything under one classification is recognized as part of that set as well as being independently it’s own set. For a set of numbers an operation is either closed or open. Closed means that performing this operation using terms out of that set and getting a result that is a part of that same set. For an operation to be open means that when performing this operation using numbers from that set would not result in a number included in that set.The first classification is natural numbers. These are commonly referred to as counting number because they are the most common numbers that you count with. These numbers are all positive, whole numbers that are greater than zero. The symbol for this is N. The next classification is whole numbers. This is all whole numbers excluding negative numbers, but including zero. This is recognized as W. The next is integers. These are whole numbers that can be negative, zero, or positive. The symbol is Z. The fourth classification is rational numbers. These are any positive or negative number that can be written as a fraction, including zero, and is commonly known as Q. Not above, but beside rational numbers are irrational numbers. These are numbers that can not be written as a fraction, such as decimals that continue forever, such as pi. The symbol is R/Q, which represents real numbers excluding
o What we call things and where we draw the line between one class of things and another depends upon the interests we have and the purpose of the classification.
In their most basic and natural settings, these two concepts can simply be defined as such:
Part 1. (a) Define each, (b) Explain its significance, (c) where indicated with this symbol * provide an example.
What is complexity? How does it differ from complicated issues? Complicated systems are exactly what Microsoft Word said. They are multifaceted and intricate. They have several parts, but there is only one determined way for these systems to work. This means that a change in one area will always create a change in another area, because the parts are connected in a predictable manner.
The more common notion of numeracy, or mathematics in daily living, I believe, is based on what we can relate to, e.g. the number of toasts for five children; or calculating discounts, sum of purchase or change in grocery shopping. With this perspective, many develop a fragmented notion that numeracy only involves basic mathematics; hence, mathematics is not wholly inclusive. However, I would like to argue here that such notion is incomplete, and should be amended, and that numeracy is inclusive of mathematics, which sits well with the mathematical knowledge requirement of Goos’
As well as converting degrees to radians, converting radians to degrees and determining an exact value. It is very essential to memorize and know how to read the unit circle. Being able to identify cosine and sine is a math skill I learned that is very essential to find the exact value of an expression. However, the unit circle has been useful and suitable for other chapters of the course. Furthermore, I learned that the Rule of Logarithms is very important to rewrite expressions and solve equations. The five rules are the inverse, product, quotient, power, and the base of change formula. As well as the tangent, cotangent, secant and cosecant reciprocal identities are concepts I learned from
A categorical analysis is an analysis of qualitative variables or non-numerical values (Mirabella, 2011). There are three different types of variables used in categorical analysis according to Malcolm Campbell. The categories he lists are dichotomous, nominal, and ordinal (Campbell, 2016). Dichotomous variables only have two categories such as yes and no or true and false. Nominal and ordinal variables can have more than two categories. The difference between the two is one has no natural order present and the other does (Campbell, 2016). The textbook states that a nominal ordinal can have numerical values as well as non-numerical values as categories, but the values have no quantitative value (Mirabella, 2011).
Countless time teachers encounter students that struggle with mathematical concepts trough elementary grades. Often, the struggle stems from the inability to comprehend the mathematical concept of place value. “Understanding our place value system is an essential foundation for all computations with whole numbers” (Burns, 2010, p. 20). Students that recognize the composition of the numbers have more flexibility in mathematical computation. “Not only does the base-ten system allow us to express arbitrarily large numbers and arbitrarily small numbers, but it also enables us to quickly compare numbers and assess the ballpark size of a number” (Beckmann, 2014a, p. 1). Addressing student misconceptions should be part of every lesson. If a student perpetuates place value misconceptions they will not be able to fully recognize and explain other mathematical ideas. In this paper, I will analyze some misconceptions relating place value and suggest some strategies to help students understand the concept of place value.
While numeracy and mathematics are often linked together in similar concepts, they are very different from one another. Mathematics is often the abstract use of numbers, letters in a functional way. While numeracy is basically the concept of applying mathematics in the real world and identifying when and where we are using mathematics. However, even though they do have differences there can be a similarity found, in the primary school mathematics curriculum (Siemon et al, 2015, p.172). Which are the skills we use to understand our number systems, and how numeracy includes the disposition think mathematically.
For a normal quadratic equation there is a well known formula to find the roots. There is a formula to find the roots of a 3rd and fourth degree equation but it can be troubling to find those roots, but if the function f is a polynomial of the 5th degree there is no formula that can enable us to find the root...
What is math? If you had asked me that question at the beginning of the semester, then my answer would have been something like: “math is about numbers, letters, and equations.” Now, however, thirteen weeks later, I have come to realize a new definition of what math is. Math includes numbers, letters, and equations, but it is also so much more than that—math is a way of thinking, a method of solving problems and explaining arguments, a foundation upon which modern society is built, a structure that nature is patterned by…and math is everywhere.
The early acquisition of mathematical concepts in children is essential for their overall cognitive development. It is imperative that educators focus on theoretical views to guide and plan the development of mathematical concepts in the early years. Early math concepts involve learning skills such as matching, ordering, sorting, classifying, sequencing and patterning. The early environment offers the foundation for children to develop an interest in numbers and their concepts. Children develop and construct their own meaning of numbers through active learning rather than teacher directed instruction.
One of the most debated questions throughout human history concerns whether or not math, one of the most useful areas of knowledge, was discovered or invented. Of course, many people have evidence and theories to what they think proves either side, yet the question still remains unanswered. When analyzing this question, it is important to fully understand the difference between the terms “discovered” and “invented”. Discovered can be defined as finding something or someone either unexpectedly or while searching. This is contrasted against the definition for invented which can be defined as creating, designing or producing something that has not existed before. With such different meanings, one must question how it is still unknown whether or not mathematics was found among nature or created in the mind. When developing something so basic as the number systems, did the human race invent math or simply discover the coding already written into the universe? Was math created to describe occurrences in nature or was it the reason patterns in nature occur? To put it more simply, does or does not math exist independent of humans? Did it exist before humans came along, and will it continue to exist after their extinction? There does not seem to be an agreed upon conclusion for these questions, however, it is important to analyze both sides of the argument. Overall, the best question to ask is, how do we know whether or not math is discovered or invented? Using various ways of knowing, examples, theories, and ideas both sides are exposed and reveal that they both are supported by plausible evidence and theories.
The Nature of Mathematics Mathematics relies on both logic and creativity, and it is pursued both for a variety of practical purposes and for its basic interest. The essence of mathematics lies in its beauty and its intellectual challenge. This essay is divided into three sections, which are patterns and relationships, mathematics, science and technology and mathematical inquiry. Firstly, Mathematics is the science of patterns and relationships. As a theoretical order, mathematics explores the possible relationships among abstractions without concern for whether those abstractions have counterparts in the real world.