Proposition: An idea isn’t thought of, it is received. First off, there’s no such thing as the universe, it doesn’t exist as such, nothing does. What exists is a fog of commonality and probability. The universe most people imagine is based on ancient mathematical principles (Greek) which themselves are ultimately based solely upon observation and memory - that’s the illusion of a universe not the reality of it. Our perception of the universe is based on memory, one moment to the next, limited by the speed at which the brain can interpret the key things that allow a memory to be preserved and the logic a context requires to recall it - that context is based on light, which is governed by our orbit around the sun. So what we actually perceive as opposed to what’s actually happening are two entirely different things - it’s the commonality and probability of one moment to the next and our perception of such that creates the universe as we currently understand it - that’s also possibly why we almost always attempt to create a ‘3d’ universe ‘in a box’ when we start out thinking about these concepts.
Proposition: Anything manifest can already be considered an event. It doesn’t matter exactly ‘how’ you get there, but by taking a look at the origins of maths there are some strange things happening. Firstly, we don’t have the number zero. It’s not used as a placeholder or as part of any type of mathematical expression until Babylonian times, and even then, only as an ‘indicator of context’. Read some more, and you discover that ancient mathematics is based around the subdivision of 1, and shows up over and over again as a guiding principle towards maths as it was understood at the time. Zero really took a hold from Greek Geology, at the time not even the ‘real’ Greek mathematicians were using ‘zero’, even as a placeholder.
Current mathematics make ‘zero’ synonymous with an event when perhaps that’s not really the case. My thoughts are that if ‘zero’ really means nothing then it doesn’t exist. For an event to take place however, there must always be the potential for it to manifest and in so puts emphasis on the the number one rather than ‘zero’.
Question:How would this affect modern maths?
Concepts of zero in math operations
One of the most difficult concepts in mathematics is doing operations involving zero. Contrary to popular opinion zero is far from being nothing, but what exactly it is can be difficult to explain and understand. When we add or subtract using zero we generally think that what we are adding nothing, and in this situation we would be correct in using that thinking. However there are some cases when zero means something or what it means can’t be accurately described.
Multiplication by zero does nothing to change the generally held concept of zero being “nothing”. When we multiply any number by zero the result is simply zero. While this result is understood as universal, multiplication in more advanced mathematics proves that this is not always so. Using exponents or raising a number to a power is one example where zero doesn’t mean nothing.
An exponent represents how many times a number is multiplied by itself. For example 2 x 2 would be 2 raised to the 2nd power or 2^2; 2 x 2 x 2 would be 2 raised to the 3rd power or 2^3, and 2 x 2 x 2 x 2 would be to the 4th power and so on. The use of zero as an exponent however, is defined as being; any number raised to the zero power is equal to 1. So this means that any number with “0” as an exponent is equal to “1”. This is the basis of the system of logarithms. Without this definition of “Zero” there could be no way to express numbers that are between “1” and “0” (fractions) in exponential form. The “zero power” allows the use of negative exponents to represent fractions. The use of zero makes negative exponents not only possible but sensible in that fractions can be express in an unlimited or “infinite” range of numbers because of negative exponents. This would not be possible if zero were just nothing.
Another use for zero that means something is zero as a reference point or marking an event. Zero is used to mark the center of the number line and the division between positive and negative numbers. Zero is neither positive or negative, but in the point that divides these two sets of numbers and provides a reference point from which to count. Besides accounting, where zero is the demarcation point for serveral conditions, like profit and loss, you can see it used to mark events. When you think of a space launch, the term t-minus is used during the countdown, this represent the time before launch. The time after launch is t-plus. When the countdown reaches “zero” we have the event (the launch). Once again zero is more than nothing, in fact it is the definition of an event.
One of the more interesting concepts involves the fact that division by zero is undefined. Proof of this statement can yeild some amazing and interesting results. Take for example the fact that the larger the number you divide into another number, the smaller the resulting answer. Conversely the smaller the number you divide into the number the larger the result. This is easy to see if we divide the same number by 5 and then by 3, the answer you get as a result of the division by three would be bigger. Continuing this same thought dividing 2 into the same number would give a larger answer, and dividing by 1/2 would result in a larger answer still. If you continue on this pattern it becomes clear that the smaller the fraction used to divide into a number the larger the answer becomes. As fractions become smaller and smaller they come closer and closer to zero, and at the same time the result of dividing them get larger and larger. 1/1,000,000 goes into 1 , 1 million times! So it would seem that if we divided a number by a fraction so small as to be almost zero the resulting anwser would be infinitely large. So if this is true why can’t division by zero be defined? We know that the answer should be infinity, right? Well not quite. Take a look at this simple equation to see the answer.
2X + 6 = X + 3
Solve this equation for X and you get
X = -3
But use the rules of algebra and factor out 2 from 2X + 6 the result is:
2(X + 3) = X + 3
Divide both sides by X + 3 and the result is:
2 = 1
The use of algebra is absolutely correct, so how could this answer be? Well the fact is that the factor (x + 3) that was used to divide into both sides of the equation is actually equal to ZERO, since X = -3 and (-3 + 3)! So what we did in finding a solution is divide both side by Zero. If you tried to define division by Zero you would find that problems like this prove why you can’t do it. Division by Zero would make anything equal to anything else!
Zero is a lot of things, but the uses indicated in this paper show that there is a lot more to zero than nothing!
Pick a Digit, Any Digit
One of the most amazing mathematical results of the last few years was the discovery of a surprisingly simple formula for computing digits of the number pi. Unlike previously known methods, this one allows you to calculate isolated digits—without computing and keeping track of all the preceding numbers.
“No one had previously even conjectured that such a digit-extraction algorithm for pi was possible,” says Steven Finch of MathSoft, Inc. in Cambridge, Mass.
The only catch is that the formula works for hexadecimal (base 16) or binary digits but not for decimal digits. Thus, it’s possible to determine that the 40 billionth binary digit of pi is 1, followed by 00100100001110. . . . However, there’s no way to convert these numbers into decimal form without knowing all the binary digits that come before the given string.
In hexadecimal form, the number pi is written as 3.243F6A8885A308D313198A2E0. . . , where the letters stand in for the hexadecimal equivalent of the base-10 numbers 10 (A), 11 (B), 12 ©, 13 (D), 14 (E), and 15 (F). It’s straightforward to convert a hexadecimal expression into binary form but not into decimal form.
The novel scheme for computing individual hexadecimal digits of pi was found by David H. Bailey of the NASA Ames Research Center in Mountain View, Calif., Peter B. Borwein of Simon Fraser University in Burnaby, British Columbia, and Simon M. Plouffe, now at the University of Quebec in Montreal.
The method is based on the following new formula for pi:
Computing individual hexadecimal digits using that formula relies on a venerable technique known as the binary algorithm for exponentiation. Bailey, Borwein, and Plouffe provide details of how that procedure works in a soon-to-be-published report titled “On the rapid computation of various polylogarithmic constants.”
A year after the discovery of the formula, Fabrice Bellard, a student at the Ecole Nationale Supérieure des Télécommunications in Paris, used it to calculate the100 billionth hexadecimal digit of pi: 9, followed by C381872D2. . . . Last September, he computed the trillionth digit: 8, followed by 7F72B1DC. . . . In binary form, the corresponding result is 1, followed by 000011111110111. . . . The main calculation required a month of computation on more than 20 workstations and personal computers.
The basic Bailey-Borwein-Plouffe algorithm can also be used to compute the nth digit of certain other transcendental numbers, such as log(2) and (pi)2. Log(2), for example, can be determined from the series: 1/2 + 1/8 + 1/18 + . . ., where the kth term is 1/k2k.
Recently, physicist David John Broadhurst of the Open University in Milton Keynes, England, reported the discovery of formulas for determining isolated hexadecimal or binary digits of several additional numerical constants of considerable interest to mathematicians, including Catalan’s constant, zeta(3), and zeta(5).
“It’s a spectacular piece of work,” says Simon Fraser’s Jonathan Borwein. “His tools were a marvellous combination of experiment, numeric and symbolic computation, followed by a mix of computer and human proofs.”
The decimal digits of Catalan’s constant, named for the Belgian mathematician Eugène Charles Catalan (1814–1894), can be calculated from the series: 1 – 1/9 + 1/25 – 1/49 + . . ., where the kth term is (–1)k/(2k + 1)2. Broadhurst found a formula that could be used to obtain isolated hexadecimal and binary digits. Interestingly, mathematicians have not yet proved that Catalan’s constant (0.915965594177. . .) is an irrational number—that is, not expressible as a fraction, or rational number. The number itself is now known to 12.5 million decimal digits.
“It illustrates that there’s a world of difference between being able to come up with methods of computing a constant quickly and necessarily being able to prove something about its transcendance or irrationality,” Borwein says. “This highlights the difference between what we can prove, what we have good evidence for, and what we just know is true even though a proof appears hopelessly out of our reach.”
What apparently makes it feasible to find formulas for certain constants and not others is that each successfully tackled number, including pi, is the value of a logarithm, Borwein says. Broadhurst’s interest in the formulas lies in the context of applying the theory of mathematical knots to physics, specifically quantum field theory.
Mathematically, the big question is whether a similar approach could be used to compute individual decimal digits of such numbers as pi. “We’re decimal creatures,” Borwein remarks. “It would be an order of magnitude more stirring to compute decimal digits in this way.”
Plouffe, for one, has found a way to compute individual decimal digits of pi without calculating preceding digits. That makes it possible to compute a particular decimal digit of pi using a pocket calculator, Plouffe says. However, his algorithm is fairly slow and clearly impractical for determining the millionth or billionth decimal digit of pi. Details are available at http://www.lacim.uqam.ca/plouffe/Simon/articlepi.html.
Nonetheless, there’s no evidence yet that an efficient algorithm for computing isolated decimal digits of pi doesn’t exist.
[b]Are the digits of Pi Random? Lab Researcher May Hold The Key[/B}
http://www.lbl.gov/Science-Articles/Archive/pi-random.html
BERKELEY, CA — David H. Bailey, chief technologist of the Department of Energy’s National Energy Research Scientific Computing Center (NERSC) at Lawrence Berkeley National Laboratory, and his colleague Richard Crandall, director of the Center for Advanced Computation at Reed College, Portland, Oregon, have taken a major step toward answering the age-old question of whether the digits of pi and other math constants are “random.” Their results are reported in the Summer 2001 issue of Experimental Mathematics.
Pi, the ubiquitous number whose first few digits are 3.14159, is irrational, which means that its digits run on forever (by now they have been calculated to billions of places) and never repeat in a cyclical fashion. Numbers like pi are also thought to be “normal,” which means that their digits are random in a certain statistical sense.
Describing the normality property, Bailey explains that “in the familiar base 10 decimal number system, any single digit of a normal number occurs one tenth of the time, any two-digit combination occurs one one-hundredth of the time, and so on. It’s like throwing a fair, ten-sided die forever and counting how often each side or combination of sides appears.”
Pi certainly seems to behave this way. In the first six billion decimal places of pi, each of the digits from 0 through 9 shows up about six hundred million times. Yet such results, conceivably accidental, do not prove normality even in base 10, much less normality in other number bases.
In fact, not a single naturally occurring math constant has been proved normal in even one number base, to the chagrin of mathematicians. While many constants are believed to be normal – including pi, the square root of 2, and the natural logarithm of 2, often written “log(2)” – there are no proofs.
The determined attacks of Bailey and Crandall are beginning to illuminate this classic problem. Their results indicate that the normality of certain math constants is a consequence of a plausible conjecture in the field of chaotic dynamics, which states that sequences of a particular kind, as Bailey puts it, “uniformly dance in the limit between 0 and 1” – a conjecture that he and Crandall refer to as “Hypothesis A.”
“If even one particular instance of Hypothesis A could be established,” Bailey remarks, “the consequences would be remarkable” – for the normality (in base 2) of pi and log(2) and many other mathematical constants would follow.
This result derives directly from the discovery of an ingenious formula for pi that Bailey, together with Canadian mathematicians Peter Borwein and Simon Plouffe, found with a computer program in 1996. Named the BBP formula for its authors, it has the remarkable property that it permits one to calculate an arbitrary digit in the binary expansion of pi without needing to calculate any of the preceding digits. Prior to 1996, mathematicians did not believe this could be done.
The digit-calculation algorithm of the BBP formula yields just the kind of chaotic sequences described in Hypothesis A. Says Bailey, “These constant formulas give rise to sequences that we conjecture are uniformly distributed between 0 and 1 – and if so, the constants are normal.”
Bailey emphasizes that the new result he and Crandall have obtained does not constitute a proof that pi or log(2) is normal (since this is predicated on the unproven Hypothesis A). “What we have done is translate a heretofore unapproachable problem, namely the normality of pi and other constants, to a more tractable question in the field of chaotic processes.”
He adds that “at the very least, we have shown why the digits of pi and log(2) appear to be random: because they are closely approximated by a type of generator associated with the field of chaotic dynamics.”
For the two mathematicians, the path to their result has been a long one. Bailey memorized pi to more than 300 digits “as a diversion between classroom lectures” while still a graduate student at Stanford. In 1985 he tested NASA’s new Cray-2 supercomputer by computing the first 29 million digits of pi. The program found bugs in the Cray-2 hardware, “much to the consternation of Seymour Cray.”
Crandall, who researches scientific applications of computation, suggested the possible link between the digits of pi and the theory of chaotic dynamic sequences.
While other prominent mathematicians in the field fear that the crucial Hypothesis A may be too hard to prove, Bailey and Crandall remain sanguine. Crandall quotes the eminent mathematician Carl Ludwig Siegel: “One cannot guess the real difficulties of a problem before having solved it.”
Among the numerous connections of Bailey’s and Crandall’s work with other areas of research is in the field of pseudorandom number generators, which has applications in cryptography.
“The connection to pseudorandom number generators is likely the best route to making further progress,” Bailey adds. “Richard and I are pursuing this angle even as we speak.”
For more about the normality of pi and other constants, visit David Bailey’s website. The BBP algorithm for calculating binary digits of pi was found using the PSLQ algorithm developed by Bailey and mathematician-sculptor Helaman Ferguson; it is discussed at Bailey’s website and also in the Fall 2000 issue of Berkeley Lab Highlights.
The Berkeley Lab is a U.S. Department of Energy national laboratory located in Berkeley, California. It conducts unclassified scientific research and is managed by the University of California.
Proposition: If pi can be accurately predicted and yet subject to chaos dynamics, wouldn’t it make for a more natural ‘zero’?
