Dunham, The Calculus Gallery, Princeton University Press, 2008, p. Guy, The Book of Numbers, Copernicus, 1996 For others, they do, but the definitions and the results differ. For many of those, the arithmetic operations make no sense. It is prudent then to deal with various infinities one at a time. Various infinities are defined differently and are subject to different operations and different laws. Besides ∞, other symbols, for example, and ω, are in circulation that denote infinities very much different from the one (or ones?) that ∞ usually stands for. There are many infinities and, say, ∞ is not a common notation used to denote each of them. Thirdly, even assuming there is a useful definition of infinity that deserves a recognition and a symbol of its own, it is not yet obvious that for this infinity it is possible in a reasonable manner to define arithmetic operations with more common numbers.įinally, there are many infinities in mathematics and this is true in more than one sense. To be understood by others, it is imperative to fall in line and use the common language.) We are now about 150 year past this landmark, at the time when the standards of mathematical thought and education have evolved dramatically. After Leibniz's death, it took about 250 years to set Calculus on a reasonably solid foundation. (As an aside, one of the inventors of Calculus - Gottfried Wilhelm Leibniz (1646-1716) - was used to express divergence of the harmonic series thus:Ĭlearly, at the beginning, intuition played a more important role than rigor. Each slide contains a question with multiple choice answers allowing your Year 5 class to then correctly select the answer. ![]() So, in the absence of any preliminary understanding or common knowledge, the best (but seldom expected or acceptable) answer is, What do you mean by ∞ or, for that matter, by dividing by 0? This Multiplying and Dividing by 10 and 100 including Decimals Multiple Choice PowerPoint Quiz focusses on curriculum link 5MD-1, Multiplying and dividing by 10 and 100. But all the rest (meaning other numbers and operations, like subtraction, multiplication, division) require a definition. Addition for natural numbers is an outgrowth of counting and may be easy to define (or accept) for the numbers 1, 2, 3. Secondly, even for those numbers that are perceived natural, the concept of division is not fundamental and has to be defined in the course of a study. Indeed, while counting numbers have been realized by various cultures, zero and negative numbers were very long in coming, not to mention decimals and complex numbers. But, for one, it is hard to insist that even number is a natural concept. Why this is a difficult question? It is difficult because it is usually asked by those who feel that infinity is a natural concept, like a number is, and everyone, especially a math teacher, should not have difficulty answering the question. ![]() And then of course comes another thoughtful extension: Is 1/∞ = 0? 10^6 / 80).One of the most difficult questions a curious student may ask a math teacher is whether 1/0 = ∞ or not. If your input always follows the pattern of (only) being in the form of #.#e+006 then you can make this much simpler by taking the value of 'e+') and multiplying it by 12500 (i.e. Note that this is a simplified example that will not handle negative exponents. If your XSLT processor does not recognize scientific notation, you will have to do the work yourself - for example:
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