Thus far, it is. Finally we decide to increase the number of experiments. By a straightforward combination of established facts axiom I and without making any further assumptions we proofed that the theorem is true for any given number too. To prove that the theorem above is valid in general, we perform another, last real-word or thought experiment.
In other words, it is. Thus far, if axiom I is generally valid and thus far the foundation of a mathematics without any exception, the same is valid even if 0 is divided by 0.
In this case, a division of 0 by 0 cannot have any influence on the validity of axiom I. Given axiom I principium identitatis, lex identitatis, the identity law as generally valid, valid without any exemption, it is. In general it is. In general, we obtain. In general, until contrariwise proofed, it is.
Let p 0 A t denote the probability that an event 0 A t will occur or has occurred at the Bernoulli trial t. Let p R B t denote the probability that an event R B t will occur or has occurred at the Bernoulli trial t. Multiplying equation by p 0 A t , the probability that an event 0 A t will occur or has occurred, we obtain. The probability that an event 0 A t will occur or has occurred is equal to p 0 A t.
Let us assume that the probability that an event 0 A t at the Bernoulli trial t will occur or has occurred is independent of any other event, no matter what is the probability of the event 0 A t or of another event R B t. Mathematically, there is at least on mathematical operation which assures such an assumption. Under these conditions the probability of an event 0 A t will and must stay that what it is, i. This must not mean that the probability p 0 A t as associated with an event 0 A t , is and must be constant.
A probability p 0 A t as associated with an event 0 A t stays only that what it is, a third has no influence on the probability p 0 A t. Thus far, an event R B t , with its own probability of occurrence of p R B t can but must not have any influence of the probability of p R B t.
Only under these conditions an event R B t , with its own probability of occurrence of p R B t has no influence on the occurrence of the event 0 A t. The equation before changes to. In other words and as generally known, especially under conditions of independence and due to probability theory, it is.
According probability theory, every single event can possess a probability between 0. Probably the best way of understanding the law of independence of the probability theory is to accept as generally valid that. Today, the division of zero by zero is commonly not used and completely misleading.
Does a possible solution of the division of zero by zero exist? Of course, yes [35]. The aforementioned view is associated with the demand of a realistic approach to the solution of problems as associated with indeterminate forms.
In this context, it is worth to mention some points in detail. In this context, a more detailed view is necessary. Working with zero can lead to another problem too. Thus far, we started with something obviously incorrect, i. A straightforward conclusion could be to claim that a division of 0 by 0 is responsible for this contradiction and as such not allowed. Such an conclusion is inappropriate. The multiplication by zero must be differentiated in more detail.
Consequently, the division by zero is logically consistent and does not lead to any contradictions. It may be true that the demonstration that these false reasons concerning the division of 0 by 0 does not customarily lead to the abandonment or withdrawal of the prejudicial attitude.
Nonetheless, the phenomenon of the division of 0 by 0 suggests that over the long run, the sustaining of even prejudicial attitudes requires a kind of a logical justification. This thesis can be understood with richer nuance when we approach it as it is. Following our rules above, we obtain that. More recently, work on indeterminate forms has been an integral part of the development of modern mathematics and it has become a subject of extensive research in its own right.
Whether this line of thought and elaboration on indeterminate forms is strong and powerful enough to withstand the theoretical challenges and to make an end to the endless and ongoing battle against indeterminate forms may remain an open question. The need for a generally valid and logically self-consistent concept of independence in number theory and algebra is great.
In particular, it is easy to recognize that the above line of thought could be extended to a general and more complex version of indeterminate forms and can make a contradiction free connection to classical logic. While relying on axiom I as the starting point of further deduction it is assured, that the results are logically consistent from the beginning.
What are we to make of this? Against this, there is a long tradition of defining the result of the division of 0 by 0 and similar operations. It is uncontroversial though remarkable that this approach has not lead to the solution to the problem of indeterminate forms through centuries. In general, it will be helpful to begin any theorem with regards to indeterminate forms with axiom I.
In its simplest formulation, this should help us to achieve the desired goals. In this publication, it was demonstrated that the concept of independence under conditions of number theory can be derived from axiom I. Furthermore, evidence was provided that axiom I has the potential to serve as the foundation of the solution of the problems as associated with indeterminate forms. Finally, using axiom I, the problem of the division of zero by zero was solved in a logically consistent form.
Further and more detailed research is possible and necessary to solve the problems of indeterminate forms and to enable a generally valid mathematics without any exception. While relying on axiom I, this goal appears to be achievable. Chelsea Pub. Millar, London, 6. The Carus Mathematical Monographs, No The Mathematical Association of America Inc.
Dialectica, 2, In: Schilpp, P. VII, Evanston, Illinois, International Journal of Applied Physics and Mathematics, 7, Dover Publication, Mineola, NY. MacMillan and Co. A Theory of Energy, Time and Space. Lulu, Morrisville, Leon Lichfield Academix Typographi, Oxonii, In Tres Tomos Distributa. Tomus primus. Castillionues, Juris consultus, Lusannae et Genevae, 4.
Erster Theil. Bei der kayserlichen Akademie der Wissenschaften, St. Petersburg Russia , Books on Demand, Hamburg-Norderstedt, Harvard University Press, Cambridge, 5. Oxford University Press, Inc. Ratio, 6, Berliner Tageblatt, Morgen-Ausgabe, Supplement 4, 1. Chez Jean Schreuder, Amsterdam, Wissenschaftsverlag, Hamburg, Scientia, Wilhelmshaven, New Statistical Methods.
Books on Demand, Hamburg, Norderstedt, Physics Procedia, 22, International Journal of Applied Physics and Mathematics, 6, Journal of Applied Mathematics and Physics, 4, Journal of Biosciences and Medicines, 5, Journal of Biosciences and Medicines, 6, Home Journals Article. Zero Divided by Zero Equals One.
DOI: Abstract Objective: Accumulating evidence indicates that zero divided by zero is equal to one. Share and Cite:. Journal of Applied Mathematics and Physics , 6 , Introduction The question of the nature of independence and the plausibility of scientific methods and results with respect to some theoretical or experimental investigations of objective reality is many times so controversial that no brief account of it will satisfy all those with a stake in the debates concerning the nature of truth and its role in accounts of classical logic and mathematics.
Material and Methods If not otherwise stated, the standard notation for various sets of numbers, mathematical operations et cetera is used. Definitions Definition 0. Definition 1. Definition 2.
Bernoulli Trial. Methods In the spring , a graduate Student of history J. Thought Experiments Thought experiments [17] play a central role both in natural sciences and in philosophy and are valid devices of the scientific [18] investigation.
Counter Examples A system of axioms or basic laws and conclusions derived in a purely logically deductive manner from such axioms together form what is called a theory. Axioms There have been many attempts to define the foundations of logic and science as such in a generally accepted manner.
Axiom I Lex Identitatis. Principium Identitatis. Results 3. The n-th trial Finally we decide to increase the number of experiments. Theorem Probability Theory and Independence Let p 0 A t denote the probability that an event 0 A t will occur or has occurred at the Bernoulli trial t. Discussion Today, the division of zero by zero is commonly not used and completely misleading.
Conflict of Interest Declaration I have no conflict of interest to declare. Conflicts of Interest The authors declare no conflicts of interest. References [ 1 ] Kolmogorov, A. Journals Menu. Contact us. All Rights Reserved.
Kolmogorov, A. Kac, M. Einstein, A. Schilpp, P. However, in the case of -1 0 , the negative sign does not signify the number negative one, but instead signifies the opposite number of what follows. So we first calculate 1 0 , and then take the opposite of that, which would result in Another example: in the expression - -3 2 , the first negative sign means you take the opposite of the rest of the expression.
Why does zero with a zero exponent come up with an error?? Please explain why it doesn't exist. In other words, what is 0 0? Answer: Zero to zeroth power is often said to be "an indeterminate form", because it could have several different values. But we could also think of 0 0 having the value 0, because zero to any power other than the zero power is zero. So laws of logarithms wouldn't work with it.
So because of these problems, zero to zeroth power is usually said to be indeterminate. However, if zero to zeroth power needs to be defined to have some value, 1 is the most logical definition for its value. This can be "handy" if you need some result to work in all cases such as the binomial theorem.
See also What is 0 to the 0 power? How is that proved? What is the difference between power and the exponent? Varthan The exponent is the little elevated number. In this case, 3 is the exponent, and 2 3 the entire expression is a power. Math Lessons menu. Hint: it has to do with a "recipe" that many math lessons follow. The do's and don'ts of teaching problem solving in math Advice on how you can teach problem solving in elementary, middle, and high school math.
How to set up algebraic equations to match word problems Students often have problems setting up an equation for a word problem in algebra.
This article explains some of those relationships.
0コメント