By virtue of everyday usage, the fact that (-1) x (-1) = 1 has been engraved onto our heads. But, only recently did I actually sit down to explore why, in general negative times negative yields a positive number !
Let’s play a game called “continue the pattern”. You would be surprised, how intuitive the results are:
2 x 3 = 6
2 x 2 = 4
2 x 1 = 2
2 x 0 = 0
2 x (-1) = ?? (Answer : -2 )
2 x (-2 ) = ?? (Answer : -4 )
2 x ( -3) = ?? (Answer : -6 )
The number on the right-hand side keeps decreasing by 2 !
Therefore positive x negative = negative.
2 x -3 = -6
1 x -3 = -3
0 x -3 = 0
-1 x -3 = ?? (Answer : 3)
-2 x -3 = ?? (Answer : 6)
The number on the right-hand side keeps increasing by 3.
Therefore negative x negative = positive.
Pretty Awesome, right? But, let’s up the ante and compliment our intuition.
The Number Line Approach.
Imagine a number line on which you walk. Multiplying x*y is taking x steps, each of size y.
Negative steps require you to face the negative end of the line before you start walking and negative step sizes are backward (i.e., heel first) steps.
So, -x*-y means to stand on zero, face in the negative direction, and then take x backward steps, each of size y.
Ergo, -1 x -1 means to stand on 0, face in the negative direction, and then take 1 backward step. This lands us smack right on +1 !
The Complex Numbers Approach.
The “i” in a complex number is an Instruction! An instruction to turn 90 degrees in the counterclockwise direction. Then i * i would be an instruction to turn 180 degrees. ( i x i = -1 ). where i = √-1
Similarly ( -1 ) x i x i = (- 1 ) x ( -1 )= 1. A complete revolution renders you back to +1.
We can snug in conveniently with the knowledge of complex numbers. But, complex numbers were established only in the 16th century and the fact that negative time negative yields a positive number was well established before that.
Hope you enjoyed the post and Pardon me if you found this to be rudimentary for your taste. This post was inspired by Joseph H. Silverman’s Book – A friendly Introduction to Number Theory. If you are passionate about numbers or math, in general it is a must read.
There are several other arithmetic methods that prove the same, if you are interested feel free to explore.
Have a Good Day!