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Fisher-Price Discovering Numbers | Book by Igloo Books | Official Publisher Page | Simon & Schuster
Sign in. All Football. To multiply two numbers with 1 billion digits requires 1 billion squared, or 10 18 , multiplications, which would take a modern computer roughly 30 years. For millennia it was widely assumed that there was no faster way to multiply.
Then in , the year-old Russian mathematician Anatoly Karatsuba took a seminar led by Andrey Kolmogorov, one of the great mathematicians of the 20th century. Kolmogorov asserted that there was no general procedure for doing multiplication that required fewer than n 2 steps. Karatsuba thought there was — and after a week of searching, he found it.
Mathematicians Discover the Perfect Way to Multiply
The method saves time because addition takes only 2 n steps, as opposed to n 2 steps. When dealing with large numbers, you can repeat the Karatsuba procedure, splitting the original number into almost as many parts as it has digits.
And with each splitting, you replace multiplications that require many steps to compute with additions and subtractions that require far fewer. First, it introduced the use of a technique from the field of signal processing called a fast Fourier transform.
The technique has been the basis for every fast multiplication algorithm since. Then last month, Harvey and van der Hoeven got there.
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Their method is a refinement of the major work that came before them. It splits up digits, uses an improved version of the fast Fourier transform, and takes advantage of other advances made over the past forty years.
Establishing that this is the best possible approach is much more difficult. At the end of February, a team of computer scientists at Aarhus University posted a paper arguing that if another unproven conjecture is also true, this is indeed the fastest way multiplication can be done. In addition, the design of computer hardware has changed. Two decades ago, computers performed addition much faster than multiplication.
The speed gap between multiplication and addition has narrowed considerably over the past 20 years to the point where multiplication can be even faster than addition in some chip architectures.