Why is a ‘minus times a minus equal to a plus’?
Among my friends I am the resident mathematics nerd and so I get asked all sorts of weird and wonderful little questions ranging from the trivial to the impossible to answer. One such question that came up while on a night out was ‘why is a minus times a minus equal to a plus?’
I have to admit that I couldn’t come up with an answer that satisfied my non-mathematical friend which bugged me no end so I asked around….’How would you explain it?’. I asked on Twitter, I asked in my office, I asked some math teachers I know. I even came up with a ‘proof’ (I am an ex-physicist so I apologise if this is considered to be far from rigorous) as follows.
From the distributive law of multiplication we have
a*(b+c) = a*b + a*c
which holds for all real numbers. So it will hold if we put a=-1, b=1 and c=-1 right? let’s do it!
(-1)*(1+(-1)) = (-1)*1 + (-1)*(-1)
1+(-1) = 0 and -1*1 = -1 so we get
(-1)*0 = -1 + (-1)*(-1)
0 = -1 + (-1)*(-1)
rearrange to get
1 = (-1)*(-1)
Simple enough but not exactly intuitive is it? By the way, would any proper mathematicians care to comment as to whether this constitutes a proof or not?
Ideally I wanted an explanation that was completely intuitive and so far the best I have come across is the explanation from my friend Paul over at Crossed Streams who’s fiancee asked us this very question while we were all on a night out. Other good explanations involved ideas such as vectors and direction but when we tried those out on our target we were met with suspicion – ‘I thought we were talking about numbers and you explain using vectors!’
What fascinated me was that my tweets asking for explanations were possibly the most popular I have ever made. I had people phone me at my office to ask ‘So did anyone come up with a good explanation then?’
A minus times a minus is a plus….everyone knows it….but how would you explain it such that even the least mathematical person would grok it?