Last year, a student in my first year mathematics class asked me an apparently innocent question:

Why does -5 x -5 = 25 ?

After some uncomfortable silence and a couple of minutes of moving by head around from hand to the other and scratching my chin (in a vain attempt to appear intelligent and considered)I resorted to the standard rescue line used by the intellectually challenged:

"Good question ! I'll have to think about that and get back to you."

My intellectual limitations aside, this is a good question. The other major rules of arithmetic seem obvious and intuitive, for example, if I have five dollars and you give me another five dollars, I indeed now have ten dollars (5 + 5 = 10). If you ask me to divide that amount evenly among five people, that will result in five piles of two dollars ( 10/5 = 2, 5 x 2 = 10). If from this ten dollars you take eight, I clearly have two left (10 - 8 = 2). If I owe you a further ten dollars, than it also follows that I am now eight dollars in debt (2 - 10 = -8). If I keep acumulating a debt of five dollars for five days, than I would owe twenty five dollars after five days (-5 x 5 = -25). Furthermore, if I now split this debt among twenty five people, they would each owe one dollar (-25/25 = -1).

Up to this point, these basic procedures of adding, subtracting, dividing and multilplying seem entirely consistent with our experience of the world.

However, the idea of -5 x -5 is hard to express in terms of a simple business transaction or an everyday example of counting. In fact, the idea of multiplying two negtives is quite abstract.

Do we accept that -5 x -5 = 25 by convention or is there a deeper reason ?

In fact, there are very good logical grounds why we accept that multiplying two negatives results in a positive. The argument goes as follows.

Multiplying numbers that are added (or subtracted) is the same as multiplying the numbers separately and than adding (or subtracting) the terms together (e.g. (2+5) x 6 = (2 x 6) + (5 x 6)).

Therefore, we can construct the following argument about multiplying any unknown (signified by y) with a negative:

(-1) x y = (-1) x y + y - y (yes, a bit of algebraic fiddling but true !)

= (-1 + 1)y - y (using the principle above)

= - y

Therefore, multiplying any number by a negative results in that nunmber changing signs and it follows:

-5 x -5 must produce a positive.

Logic has provided an answer where intuition and common experience have failed !

This matching of mathematical ideas with "intuition" and common experience becomes difficult as more advanced ideas of mathematics are introduced to students. The inherent abstraction associated with mathematics is one of its great attritubutes, as these abstractions lead to wonderful insights and ideas, but also can provide a barrier in learning. I think to a certain degree struggling with these ideas is a sign that you are really thinking about the issues, or at least that is my excuse !

Why does -5 x -5 = 25 ?

After some uncomfortable silence and a couple of minutes of moving by head around from hand to the other and scratching my chin (in a vain attempt to appear intelligent and considered)I resorted to the standard rescue line used by the intellectually challenged:

"Good question ! I'll have to think about that and get back to you."

My intellectual limitations aside, this is a good question. The other major rules of arithmetic seem obvious and intuitive, for example, if I have five dollars and you give me another five dollars, I indeed now have ten dollars (5 + 5 = 10). If you ask me to divide that amount evenly among five people, that will result in five piles of two dollars ( 10/5 = 2, 5 x 2 = 10). If from this ten dollars you take eight, I clearly have two left (10 - 8 = 2). If I owe you a further ten dollars, than it also follows that I am now eight dollars in debt (2 - 10 = -8). If I keep acumulating a debt of five dollars for five days, than I would owe twenty five dollars after five days (-5 x 5 = -25). Furthermore, if I now split this debt among twenty five people, they would each owe one dollar (-25/25 = -1).

Up to this point, these basic procedures of adding, subtracting, dividing and multilplying seem entirely consistent with our experience of the world.

However, the idea of -5 x -5 is hard to express in terms of a simple business transaction or an everyday example of counting. In fact, the idea of multiplying two negtives is quite abstract.

Do we accept that -5 x -5 = 25 by convention or is there a deeper reason ?

In fact, there are very good logical grounds why we accept that multiplying two negatives results in a positive. The argument goes as follows.

Multiplying numbers that are added (or subtracted) is the same as multiplying the numbers separately and than adding (or subtracting) the terms together (e.g. (2+5) x 6 = (2 x 6) + (5 x 6)).

Therefore, we can construct the following argument about multiplying any unknown (signified by y) with a negative:

(-1) x y = (-1) x y + y - y (yes, a bit of algebraic fiddling but true !)

= (-1 + 1)y - y (using the principle above)

= - y

Therefore, multiplying any number by a negative results in that nunmber changing signs and it follows:

-5 x -5 must produce a positive.

Logic has provided an answer where intuition and common experience have failed !

This matching of mathematical ideas with "intuition" and common experience becomes difficult as more advanced ideas of mathematics are introduced to students. The inherent abstraction associated with mathematics is one of its great attritubutes, as these abstractions lead to wonderful insights and ideas, but also can provide a barrier in learning. I think to a certain degree struggling with these ideas is a sign that you are really thinking about the issues, or at least that is my excuse !

Note: It can be argued that the distributive law of muliplication (the centre of the argument I presented) can be viewed as a convention of a particular system or rather than as a "law" and that it is possible to construct different systems built around quite different conventions. Barry Mazur explains this interesting view in his excellent book "Imagining Numbers" (Penguin, London, 2003).

There's a flaw in your explanation:

ReplyDeleteSuppose y was (-z):

Then if we follow your explanation:

(-1) x y = (-1) x y + y - y (yes, a bit of algebraic fiddling but true !)

= (-1 + 1)y - y (using the principle above)

= - y

We'll end up with -(-z).

Now the task of multiplying negative with a negative still remains - multiplying (-1) with (-z)...

At this moment you have to do the real multiplication which is not well defined that how it will convert to +z. It still relies just on the assumption that multiplication of two negatives results in positive assumption. Since it has worked so far..we are good and noone questions :)