Sum of Natural Numbers Not a Natural Number

Numberphile  Uploaded on Jan 9, 2014 MAIN VIDEO IS AT: http://youtu.be/w-I6XTVZXww Ed Copeland and Tony Padilla are physicists at the University of Nottingham. Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile Google Plus: http://bit.ly/numberGplus Tumblr: http://numberphile.tumblr.com Videos by Brady Haran Brown papers: http://bit.ly/brownpapers A run-down of Brady’s channels: http://bit.ly/bradychannels

Uploaded on Jan 9, 2014

MAIN VIDEO IS AT: http://youtu.be/w-I6XTVZXww

Ed Copeland and Tony Padilla are physicists at the University of Nottingham.

Website: http://www.numberphile.com/
Numberphile on Facebook: http://www.facebook.com/numberphile
Numberphile tweets: https://twitter.com/numberphile
Google Plus: http://bit.ly/numberGplus
Tumblr: http://numberphile.tumblr.com
Videos by Brady Haran

Brown papers: http://bit.ly/brownpapers

A run-down of Brady’s channels: http://bit.ly/bradychannels


paper size

Many paper size standards conventions have existed at different times and in different countries. Today there is one widespread international ISO standard (including A4, B3, C4, etc.) and a local standard used in North America (including letter, legal, ledger, etc.). The paper sizes affect writing paper, stationery, cards, and some printed documents. The standards also have related sizes for envelopes. A series The […]

Many paper size standards conventions have existed at different times and in different countries. Today there is one widespread international ISO standard (including A4, B3, C4, etc.) and a local standard used in North America (including letter, legal, ledger, etc.). The paper sizes affect writing paper, stationery, cards, and some printed documents. The standards also have related sizes for envelopes.

A series

The international paper size standard, ISO 216, is based on the German DIN 476 standard for paper sizes. ISO paper sizes are all based on a single aspect ratio of square root of 2, or approximately 1:1.4142. The base A0 size of paper is defined to have an area of 1 m2. Rounded to millimetres, the A0 paper size is 841 by 1,189 millimetres (33.1 in × 46.8 in).

Successive paper sizes in the series A1, A2, A3, and so forth, are defined by halving the preceding paper size across the larger dimension. The most frequently used paper size is A4 measuring 210 by 297 millimetres (8.3 in × 11.7 in).

The significant advantage of this system is its scaling: if a sheet with an aspect ratio of \sqrt{2} is divided into two equal halves parallel to its shortest sides, then the halves will again have an aspect ratio of \sqrt{2}. Folded brochures of any size can be made by using sheets of the next larger size, e.g. A4 sheets are folded to make A5 brochures. The system allows scaling without compromising the aspect ratio from one size to another—as provided by office photocopiers, e.g. enlarging A4 to A3 or reducing A3 to A4. Similarly, two sheets of A4 can be scaled down and fit exactly 1 sheet without any cutoff or margins.

The behavior of the aspect ratio is easily proven: Let a and b be the long side and the short side of the paper respectively. The imposed initial condition is that a/b=\sqrt{2}, let cbe the length of the short side after folding it in half. That is c=a/2, if we take the ratio of the newly folded paper we have that:

  \frac{b}{c}=\frac{b}{\frac{a}{2}}=\frac{2}{\frac{a}{b}}=\frac{2}{\sqrt{2}}=\sqrt{2}

Therefore the aspect ratio is preserved for the new dimensions of the folded paper.

Weights are easy to calculate as well: a standard A4 sheet made from 80 g/m2 paper weighs 5 g (as it is one 16th of an A0 page, measuring 1 m2), allowing one to easily compute the weight—and associated postage rate—by counting the number of sheets used.

The advantages of basing a paper size upon an aspect ratio of \sqrt{2} were first noted in 1786 by the German scientist and philosopher Georg Christoph Lichtenberg.[2] Early in the 20th century, Dr Walter Porstmann turned Lichtenberg’s idea into a proper system of different paper sizes. Porstmann’s system was introduced as a DIN standard (DIN 476) inGermany in 1922, replacing a vast variety of other paper formats. Even today the paper sizes are called “DIN A4″ (IPA: [di?n.?a?.fi???]) in everyday use in Germany and Austria. The term Lichtenberg ratio has recently been proposed for this paper aspect ratio.

According to some theorists, ISO 216 sizes are generally too tall and narrow for book production (see: Canons of page construction). European book publishers typically use metricated traditional page sizes for book production

1

Asking what a number (1) minus a non-exising value .9 repeating is meaningless. But if we assume the “infinitiness” of .9 repeating then we must also assume that taking 1 – “this” will equal some infinitely small number that is also greater than 0. This fictitious “number” would be the same as “the smallest real number that’s […]

Asking what a number (1) minus a non-exising value .9 repeating is meaningless. But if we assume the “infinitiness” of .9 repeating then we must also assume that taking 1 – “this” will equal some infinitely small number that is also greater than 0. This fictitious “number” would be the same as “the smallest real number that’s larger than zero”, which doesn’t exist.

http://en.wikipedia.org/wiki/0.999

An ‘infinitesimal’ is a 17th century idea that was never mathematically rigorous. 200 years later that idea was entirely replaced with the modern definition of a limit.

 

 

 

billions

Published on May 24, 2012 We discuss millions, billions, trillions and centillions is this film about the long and short scales. We also touch on quadrillions, sextillions, milliards, billiards and the Greek myriad. Featuring Dr James Grime and Dr Tony Padilla. More about them athttp://www.numberphile.com/team/index… Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile In English, the original […]

Published on May 24, 2012

We discuss millions, billions, trillions and centillions is this film about the long and short scales. We also touch on quadrillions, sextillions, milliards, billiards and the Greek myriad.

Featuring Dr James Grime and Dr Tony Padilla. More about them athttp://www.numberphile.com/team/index…

Website: http://www.numberphile.com/
Numberphile on Facebook: http://www.facebook.com/numberphile
Numberphile tweets: https://twitter.com/numberphile

In English, the original naming of numbers ends at 1000. So each place has a name up to 1000, (units, tens, hundreds, thousands). Then it ends. So the full ‘life’ of numbers ends at 1000. Since 1000 is the end of numbering, it is in the greek an eon. The furthest ‘life’ of the numbers.

From there we count how many thousands. 1, 2, up to 999 thousand.
So we say one hundred thousand meaning that there are (100) thousands
After that it just sounds funny to say a thousand thousand, So a name was invented for (1000) thousands,
“milli” means 1000. For example there are 1000 millimeters in a meter.
Milli (means 1000) eon (means 1000)
So, milli eon means (1000) thousands = million
Now, continuing to count thousands, we get to (1000 x 1000) thousands, so how many thousands are there? Since our count of how many thousands we have says thousand twice (1000 x 1000) we say bi meaning twice. Therefore lets name it bi-milli (twice thousand) ion (thousand) which is condensed to billion.
So, billion means (thousand thousand) thousands. Because we are naming it based on how many thousands we have (and NOT the TOTAL power of 1000 the number is), and in this case we have (thousand thousand) thousands, thus ‘thousand said twice’ worth of thousands.

English numbers ended at 1000, and after that we are counting how many thousands we have. From there the logic continues:
(thousand thousand thousand) thousands = (tri milli) eon = trillion
(thousand thousand thousand thousand) thousands = (quadra milli) eon = quadrillion.

The numbers are named based on how many thousands we have, not based on what power 1000 is brought to to get the final number. So, the total power of 1000′s will be 1000 to the prefix power (the count of how many thousands) + 1 (for the thousand, which is that which is being counted by the name itself).
So, a centillion is (thousand thousand …..repeated 100 times) thousands. = 1000 ^ (100+1).

Taxi cab numbers

Fermat’s Last Theorem near misses? In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubesin n distinct ways. The concept was first mentioned in 1657 by Bernard Frénicle de Bessy, and was made famous in the early 20th century by a story involving Srinivasa Ramanujan. […]

Fermat’s Last Theorem near misses?

In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubesin n distinct ways. The concept was first mentioned in 1657 by Bernard Frénicle de Bessy, and was made famous in the early 20th century by a story involving Srinivasa Ramanujan. In 1938, G. H. Hardy and E. M. Wright proved that such numbers exist for all positive integers n, and their proof is easily converted into a program to generate such numbers. However, the proof makes no claims at all about whether the thus-generated numbers are the smallest possible and thus it cannot be used to find the actual value of Ta(n).

The restriction of the summands to positive numbers is necessary, because allowing negative numbers allows for more (and smaller) instances of numbers that can be expressed as sums of cubes in n distinct ways. The concept of a cabtaxi number has been introduced to allow for alternative, less restrictive definitions of this nature. In a sense, the specification of two summands and powers of three is also restrictive; a generalized taxicab number allows for these values to be other than two and three, respectively.

Ta(2), also known as the Hardy–Ramanujan number, was first published by Bernard Frénicle de Bessy in 1657 and later immortalized by an incident involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy [1]:

I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. “No”, he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways.”

The subsequent taxicab numbers were found with the help of supercomputersJohn Leech obtained Ta(3) in 1957. E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel found Ta(4) in 1991. J. A. Dardis found Ta(5) in 1994 and it was confirmed by David W. Wilson in 1999.[1][2] Ta(6) was announced by Uwe Hollerbach on the NMBRTHRY mailing list on March 9, 2008,[3] following a 2003 paper by Calude et al. that gave a 99% probability that the number was actually Ta(6).[4] Upper bounds for Ta(7) to Ta(12) were found by Christian Boyer in 2006.[5]