Photo by YouTube/UNSW
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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