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.įor millennia it was widely assumed that there was no faster way to multiply. The carrying method works well for numbers with just a few digits, but it bogs down when we’re multiplying numbers with millions or billions of digits (which is what computers do to accurately calculate pi or as part of the worldwide search for large primes). So three-digit numbers require nine multiplications, while 100-digit numbers require 10,000 multiplications. The grade school or “carrying” method requires about n 2 steps, where n is the number of digits of each of the numbers you’re multiplying. If you’re multiplying two two-digit numbers, you end up performing four smaller multiplications to produce a final product. We stack two numbers, multiply every digit in the bottom number by every digit in the top number, and do addition at the end. Most everyone learns to multiply the same way. “If you want to know how fast computers can solve certain mathematical problems, then integer multiplication pops up as some kind of basic building brick with respect to which you can express those kinds of speeds.” “In physics you have important constants like the speed of light which allow you to describe all kinds of phenomena,” van der Hoeven said. Van der Hoeven describes their result as setting a kind of mathematical speed limit for how fast many other kinds of problems can be solved. The complexity of many computational problems, from calculating new digits of pi to finding large prime numbers, boils down to the speed of multiplication. “Everybody thinks basically that the method you learn in school is the best one, but in fact it’s an active area of research,” said Joris van der Hoeven, a mathematician at the French National Center for Scientific Research and one of the co-authors. The paper marks the culmination of a long-running search to find the most efficient procedure for performing one of the most basic operations in math. On March 18, two researchers described the fastest method ever discovered for multiplying two very large numbers. Try the following now.Four thousand years ago, the Babylonians invented multiplication. Once again, use an area model to calculate the same product and see if you can find where the numbers of intersections appear in the area model. There is one possible glitch in this lines & intersections algorithm, as illustrated by the computation 246 × 32 below:Īlthough the answer 6 thousands, 16 hundreds, 26 tens, and 12 ones is absolutely correct, one needs to carry digits and translate this as 7,872. Can you see where the values 2, 2, 6, and 6 are in the area model? Use an area model like Vera's to calculate the same product. Count the number of intersections in each and add the results diagonally as shown: There are four sets of intersection points. To compute 22 × 13, for example, draw two sets of vertical lines, the left set containing two lines and the right set two lines (for the digits in 22) and two sets of horizontal lines, the upper set containing one line and the lower set three (for the digits in 13). Here’s an unusual way to perform multiplication. Other Algorithms: Lines and Intersections That is, the area of the 4 x 3 rectangle is 12 square units. Since each unit was actually a square, we refer to the area using "square units". The product, 12, is the total number of squares in that rectangle. So we can think about 4 × 3 as a rectangle that has length 3 and width 4. Using the repeated addition perspective of multiplication, we can picture 4 × 3 as 4 groups of 3 squares each, as shown below.īut if we stack up the groups, we would have 4 rows, with 3 squares in each row. So, you could describe 3 units using three squares. Suppose our basic unit is one square, as shown below. It is also important that we understand why the area perspective of multiplication makes sense. But it is important to explicitly note what area actually is: a measurement describing two-dimensional space. We know that multiplication and area are related. In mathematics, we see this in the fact that often the first way we model relationships between numbers abstractly is with the number line. That is because one of the first forms of measurement we learn are based on a linear measurement model. When people refer to "measurement", often they think of using a ruler, yardstick, or tape measure.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |