Minus Times Minus
Discovery: It can be negative
For hundreds of years, teachers have taught that minus times minus is plus. Mathematicians have described this rule as being not an assumption but a necessary logical consequence of the fundamental properties of multiplication and the laws of algebra, such as the commutative law.
In the Ars Magna, in 1545, Girolamo Cardano affirmed the rule that minus times minus is plus. But years later, in a little-known work, Cardano changed his mind. He argued that it should be minus. Similarly, a few other mathematicians rejected the rule, including Thomas Harriot and Francis Maseres, but they failed to formulate a coherent alternative.
In my book Negative Math (Princeton University Press, 2005), I analyzed the history of negative numbers and I formulated a coherent algebra in which minus times minus is minus. In this new algebra, multiplication of numbers with different signs is non-commutative, which introduces multiple symmetries that are missing in traditional numerical algebra. I further explained this system in a chapter of my book, The Cult of Pythagoras (University of Pittsburgh Press, 2012).
“Can Minus Times Minus Be Minus?” Negative Math: How Mathematical Rules Can Be Positively Bent (Princeton: Princeton University Press, 2006), 131-165.
“Inventing Mathematics?” The Cult of Pythagoras: Math and Myths (Pittsburgh: University of Pittsburgh Press, 2012), 181-200.
“Unity in Mathematics,” Negative Math: How Mathematical Rules Can Be Positively Bent (Princeton: Princeton University Press, 2006), 166-173.
“Making a Meaningful Math,” Negative Math: How Mathematical Rules Can Be Positively Bent (Princeton: Princeton University Press, 2006), 174-185.
“Designing Numbers and Operations,” Negative Math: How Mathematical Rules Can Be Positively Bent (Princeton: Princeton University Press, 2006), 186-219.
“Physical Mathematics?” Negative Math: How Mathematical Rules Can Be Positively Bent (Princeton: Princeton University Press, 2006), 220-234.
REVIEWS
"Martínez shows that it’s possible to construct a fully consistent system of arithmetic in which minus times minus makes minus. It’s a wonderful vindication...”
JAKE RANDELL
AMERICAN SCIENTIST
"Martínez can certainly take credit for having produced by far the best and most detailed account of this important strand in Einstein's early work.”
JÜRGEN RENN
MAX PLANCK INSTITUTE
“An essential text for any future reserarch on Bruno, Galileo and The Inquisition. ...possibly the most important book of the year for the history of astronomy.”
CLIFFORD CUNNINGHAM
ASTRONOMICAL HISTORY
BOOKS
BIO
Alberto Martinez is originally from San Juan, Puerto Rico. As a professor at UT Austin, he investigates the history of science, especially Einstein and relativity theory, history of math, historical myths, and Giordano Bruno and Galileo. He is a Fellow of the American Physical Society.
He also researches myths in political news media, The Eyes of Texas, and episodes in the history of money and corruption.
Now, he's finishing writing a novel about Albert Einstein.