If we subtract two fractions with different denominators, ad - bc will be the form for the numerator, which is to say we "cross multiply" the numerator from each fraction with the denominator from the other fraction, then subtract the second one from the first.
If we have the points (0, 0), (a, b) and (c, d) on the plane as in the picture, we can create a parallelogram by including a fourth point at (a+c, b+d), and by finding the area of the big rectangle and subtracting away the areas of the rectangles and triangles that are not marked in yellow, we get the area of the parallelogram is just like the numerator of our fraction, ad - bc.
The formula is also known as the determinant of a 2x2 matrix where the entries look like
The determinant is an ancient idea, while the matrix is relatively modern, the first important work done in the mid 19th Century. The word matrix was chosen for its Latin root to "mother", with the implication that matrices were the mothers of determinants. The Wachoski brothers knew about this word derivation when they made The Matrix, which is both an array of numbers in a computer, but also that people inside The Matrix are actually in physical locations that look like the womb.
Any square matrix, 2x2 or 3x3 or 4x4, etc., has a determinant, but the calculation of the number gets messier as the matrix gets bigger. For instance, a 3x3 matrix of the form
a b c
d e f
g h i
has the determinant
aei + bfg + cdh - afh - bdi - ceg.
There are six terms here, because 6 = 3x2x1, or 3!, pronounced 3 factorial. A 4x4 matrix determinant has 4! = 24 terms, the 5x5 has 120 terms, and the numbers just get bigger and bigger. 10! for example is over three million. But because determinants are so useful, mathematicians found labor saving ways to calculate them, and some of the best were found by Charles Dodgson, known to the world as Lewis Carroll. The story goes that Queen Victoria was so taken with Alice's Adventures in Wonderland that she wrote to the author asking him to send her a copy of the next book he wrote. Carroll, loyal subject and nerdiest of nerds, dutifully sent the queen his next book, his treatise on determinants. The urban legend debunking website snopes.com considers this story to be false, given how carefully Dodgson tried to keep his pseudonym a secret. But it is true that the determinants book was the book he published only a year after Alice went into print.
24 November 2014
4 hours ago