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tahanson43206

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Registered: 2018-04-27

Posts: 19,382

Mathematics Pure and Applied

For SpaceNut .... no topic existed with the word mathematics in the title ....

https://getpocket.com/explore/item/this … ket-newtab

This article is about what is reported to be a way of multiplying integers that is said to be faster than traditional methods.

Pocket worthyStories to fuel your mind
This Guy Found a Faster Way to Multiply

Because the method you learned in middle school is ridiculously slow.
Popular Mechanics

    David Grossman

Read when you’ve got time to spare.

Popular Mechanics
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person speaking in front of a whiteboard filled with math figures

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From grade school onward, complex multiplication has been a headache. But an assistant professor from the University of New South Wales Sydney in Australia has developed a new method for multiplying giant numbers together that's more efficient than the "long multiplication" so many are taught at an early age.

“More technically, we have proved a 1971 conjecture of Schönhage and Strassen about the complexity of integer multiplication,” associate professor David Harvey says in the video below.

The Schönhage–Strassen algorithm, developed by two German mathematicians, was actually the fastest method of multiplication from 1971 through 2007. Although a faster method was developed in 2007, it's rarely used today.

Schönhage and Strassen predicted that an algorithm multiplying n-digit numbers using n * log( n) basic operations should exist, Harvey says. His paper is the first known proof that it does.

Harvey picks the example of 314 multiplied by 159—a large equation, sure, but much larger ones are used every day in real-life scenarios. To solve the problem, most people are taught to multiply each individual number together, and then add up the sums:

9 is multiplied by 4, 1, and 3; then 5 is multiplied by 4, 1, and 3, and so on. The result ends up with 9 digit-by-digit products.

For some, seeing a word in the middle of a math problem might be enough to make their eyes glaze over like they did in algebra class. As a refresher: log is short for logarithm, which helps people decipher exponents that make numbers squared or cubed or even something higher. So 2⁵ is 32, but expressed logarithmically, it would look like log₂(32)=5. While it may seem like a mouthful, logarithms are crucial when numbers get much larger.

The Schönhage-Strassen method is very fast, Harvey says. If a computer were to use the squared method taught in school on a problem where two numbers had a billion digits each, it would take months. A computer using the Schönhage-Strassen method could do so in 30 seconds.

But if the numbers keep rising into the trillions and beyond, the algorithm developed by Harvey and collaborator Joris van der Hoeven at École Polytechnique in France could find solutions faster than the 1971 Schönhage-Strassen algorithm.

“It means you can do all sorts of arithmetic more efficiently, for example division and square roots," he says. "You could also calculate digits of pi more efficiently than before. It even has applications to problems involving huge prime numbers.

“People have been hunting for such an algorithm for almost 50 years," he continues. It was not a forgone conclusion that someone would eventually be successful. It might have turned out that Schönhage and Strassen were wrong, and that no such algorithm is possible.

“But now we know better,” he says.

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This post originally appeared on Popular Mechanics and was published October 18, 2019. This article is republished here with permission.

This topic is available for NewMars members to post tips about mathematics required for the Mars Settlement problem, or news about mathematics discoveries or advances in general.

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tahanson43206

Moderator

Registered: 2018-04-27

Posts: 19,382

Re: Mathematics Pure and Applied

For Mars_B4_Moon in particular .... Because you've been searching the Internet in so many topic areas, I am hoping you will keep this topic in mind as you review new items you find.  Success in settling Mars will require successful execution of vast sets of numerical calculations.

We already have several project ideas that have the potential to be developed further with applied mathematics:

1) Large Ship - a Unitary Rotation space vessel design
2) Practical Ship - A dual counter rotating habitat design using a single axle
3) Inline habitat ship - a dual counter rotating habitat design using parallel axles held in place by an exterior frame
4) Solar energy powered plant to manufacture hydrocarbon fuels from air and water inputs
5) Nuclear fission powered plant to manufacture hydrocarbon fuels from air and water inputs

For all ... New Mars forum now offers the option to store completed work in Dropbox folders.  This insures your work is readily available for access when it is needed.  Please contact one of the Admins for details.

(th)

tahanson43206

Moderator

Registered: 2018-04-27

Posts: 19,382

Re: Mathematics Pure and Applied

The article at the link below is about research into the factorability of equations...

https://getpocket.com/explore/item/in-t … ket-newtab

The work increases the number of "prime" equations (recognized by mathematicians) and while practical applications may be few, they are not zero ... It appears that choosing equations that cannot be factored for encryption algorithms is a good idea.

(th)

tahanson43206

Moderator

Registered: 2018-04-27

Posts: 19,382

Re: Mathematics Pure and Applied

The local Linux group met this evening, via Zoom, as usual.

The group includes a mathematician.

We were looking at some software running on a Canadian machine, and I noticed that libreOffice 7.3 (and later) includes a Math package.  I asked the mathematician to bring up that package on his machine, and we spent the next half hour exploring how LibreOffice allows the operator to format equations for display in a document (for publication or just for homework presentation).  Somewhere along the line, someone suggested putting the ASCII translation of the equations into wolfram.com, which has an equation evaluation feature.

At that point, I really got to see a collaboration at work in real time ... the mathematician lead the process, by making selections from the examples provided by LibreOffice, or by composing his own equations, and submitting them to Wolfram.

The result was a realtime display of the equations as understood by the Wolfram math engine, including graphical displays showing the results over the span of integration specified by the mathematician.

For the first time in decades, I felt a sense of recognition of the processes of calculus.
If there is someone in the NewMars readership whose knowledge of calculus is a bit rusty, this is a way to restore some of the original gleam.

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