We have all known this since middle school. But could you explain why ‘less for less is more’?
Is the mathematical language we have constructed consistent with logic or is it logic that drives the mathematical language? Let us see it together.
Gardner’s explanation
The best explanation of why less for less makes more is Martin Gardner‘s.
Consider a large hall where there are two types of people, good people and bad people. We define addition as ‘getting people into the room’, subtraction as ‘getting people out of the room’, ‘positive number’ as ‘good person’ and ‘negative number’ as ‘bad person’. Now, adding a positive number means getting a good person into the room, which obviously leads to an increase in the total amount of goodness in the room.
Adding a negative number means bringing in a bad person, so the net amount of goodness decreases. Subtracting a positive number means taking out a good person: the net goodness in the room decreases. Subtracting a negative number means taking out a bad person: the net goodness increases. Multiplication is nothing more than repeated addition (or subtraction).
Minus four times minus six? Bring out six bad people. Repeat this four times. The end result? A net increase in goodness of 24. Someone might object and ask what happens if the bad people who have been asked to leave the room decide not to do so. But that is a matter outside mathematics.
The mathematical explanation
Let us put ourselves in the position of someone who knows the rules of mathematics, has already defined negative numbers and is in doubt about how to define multiplication involving them. What we are looking for is consistency.
Suppose we have to do a calculation:
3 * -2
If we are sure that the number is 6, what will be the sign? We reason in this way:
3 * (2 - 2) = 3 * 0 = 0
This is because we want every number that is multiplied by 0 to result in 0. We also want the properties of multiplication, such as the distributive property, to apply.
But I can also write:
3 * (2 - 2) = 6... = 0
And what is missing if not a -6?
3 * (2 - 2) = 6 - 6 = 0
So:
3 * 2 = 6 e 3 * - 2 = -6
If we want the multiplication of negative numbers to be consistent with the properties of multiplication, in this case the distributive one, we must convince ourselves that + times – must equal -.
And we come to the multiplication of two negative numbers, reasoning in the same way, since we now know that + times – must equal -. We look for the sign of -4 * -5, since the numerical value is 20:
-4 * (5 - 5) = -4 * 0 = 0
But also:
-4 * (5 - 5) = -20... = 0
And clearly there is a +20 here:
-4 * (5 - 5) = -20 + 20 = 0
So:
-4 * 5 = -20 e -4 * -5 = +20
This explains why – for – must do +: it’s consistent with the other rules that already apply to multiplication.
The intuitive explanation
We also give an intuitive explanation. If we imagine ourselves on the number line and know that, starting from 0, we move to the right towards the positive numbers and to the left towards the negative numbers.
Now, in multiplication, the sign of the first factor tells me where to look (to the right or to the left) and the sign of the second factor tells me in which direction to go (in the same direction as I am looking or in the opposite direction). Let’s look at some examples:
+2 * +3
More and more, then I look to the right (towards the positive numbers) and walk in the same direction, taking 3 steps of length 2:
And this is where it is:
+2 * +3 = +6
We will introduce the negatives and do the math:
-3 * +2
Minus and plus, then I look to the left (towards the negative numbers) and walk in the same direction, taking 2 steps of length 3:
And this is where it is:
-3 * +2 = -6
And so we have arrived at the multiplication of two negative numbers:
-2 * -2
Less and less, then I look to the left BUT I walk in the other direction, so like a shrimp, to the right, taking 2 steps backwards of length 2:
So:
-2 * -2 = +4
This means that – multiplied by – is +.
A trick of the trade
There is a trick to remembering the sign rule. It is particularly useful for primary school children. Consider positive numbers as friends and negative numbers as enemies:
- Multiplying a negative number by a positive number sounds like saying “my friend’s enemy”, who is obviously also my enemy, and therefore a negative number.
- Multiplying a negative number by a negative number sounds like saying “the enemy of my enemy”, who is of course my friend, and therefore a positive number.
Sources and Data
- Why does less for less make more? – Roba Scientifica – Quora
- Perché meno per meno fa più? La spiegazione di Martin Gardner
