Which of the two numbers is smaller: -1 or +1? This is such an easy and straightforward question. Of course -1 is smaller than +1.
But, is it? Let me challenge a basic fundamental learning of your school education in this post. But before that, let’s first recall how we were taught mathematics in our schools.
So our school education starts with numbers. We are introduced to two types of numbers: the Roman numerals i.e. I, II, III, IV, V and so on. The other system which we are introduced to is the decimal system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and so on. Interestingly, the second system was introduced to me in my childhood as Arabic numerals which is a big lie. The decimal system is the Indian system of numerals. In fact, the very foundations of mathematics rests on ancient Indian mathematics. The mathematics travelled from India to Arab and then to the West. This translation and appropriation from Indian to Arabic to Western mathematics obviously resulted in distortions, mistakes, errors. On such error that I discovered recently is how children are taught about the negative numbers.
We are introduced to positive numbers first. It’s only when we reach class 6th that we are introduced to the negative numbers. And the way we are taught negative numbers is that they are numbers which are smaller than zero. This is very confusing to a child and even to many adults. How can be something smaller than zero? Think about it. 3 apples are less than 4 apples. Having 1 apple is still more than having zero apple. Now a mathematics teacher may explain to you that having -1 apple means you have to give 1 apple which you don’t have. So, if you get a dozen apples from the market, you will have to give out 1 because you had -1 apple beforehand. Hence, -1 is smaller than 0. But, still this is quite confusing and is actually a false-misleading explanation.
The original Indian system of mathematics taught negative numbers in a quite simple way. Negative numbers are not smaller than positive numbers. Negative numbers are just the opposite of positive numbers. This solves all confusion and problems in understanding negative numbers and various operations on them. For example, +5 apples means you will get 5 apples from the stock. -5 apples means you will return 5 apples to the stock. -5 is the opposite of +5. In the coordinates also, a reference point is chosen to be zero point and numbers to the right are denoted positive and numbers to the left are denoted negative. It doesn’t mean negative coordinates are smaller than the positive coordinates. Just means that negative coordinate is mirror opposite of the equivalent positive coordinate.
Positive charges are the opposite of negative charges. Having negative electric charge doesn’t mean the charge is lesser than the positive charge. Similarly, the two poles of a magnet are denoted positive and negative, again denoting the two opposites. This is true for all physical phenomena. Now lets try to understand certain mathematical results with this understanding.
- 1 + (-1) = 0
Now this makes sense. You have to take out 1 apple from the stock which is denoted by +1. You have to return 1 apple to the stock which is denoted by (-1). If you perform both the operations, you will have zero apples left.
- 1 – (-1) = 2
Negative numbers actually are the same as doing subtraction. Subtraction however doesn’t exist independently. It is performed upon numbers. Negative numbers do exist like positives. They are just opposite in nature. In this case, you have to take 1 apple from the stock. And then you have to subtract (-1) apple from your stock. Meaning you have to give back (-1) apple. So you give back (-1) apple to the stock. The stock has an extra (-1) apple. But then, the stock will have to give to you 1 apple. Remember our previous explanation of (-1) apple in this example where (-1) means giving 1 apple and (+1) means receiving 1 apple. So, you end up having 2 apples.
- 1 * (-1) =(-1)
Now this is where I got confused a lot in my teen days when I thought about the physical implications of this result. But with our current understanding, even this is not so difficult. All we have to do is be clear about our basic principles and understandings.
1 * 1 =1, 1 * 5 = 5. There are two ways to look at these results. 1 * 1 =1 means you have to take 1 apple from the stock 1 time which will leave you with 1 apple. 1 * 5 = 5 means you have to take 1 apple from the stock 5 times or 5 apples from the stock 1 time. In both the cases, you will end up with 5 apples.
1 * (-1) = (-1). The first explanation is straight forward. (-1) apples with you 1 time means you have to give 1 apple to the stock which is having (-1) apple with you. The second explanation is quite tricky to grasp which is you have to take 1 apple from the stock (-1) times. What does doing an operation (-1) times mean? (-1) is the opposite of +1. So the results yielded by doing an operation (-1) time will be equal and opposite to the results obtained by doing an operation (+1) time. (-1) is opposite of (+1). So doing something (-1) time will be equal and opposite to doing a task (+1) time. So 1 * (-1) also means doing the opposite of 1 * 1 i.e. giving back 1 apple to the stock 1 time which is having (-1) apple. Similarly, 1 * (-5) means taking (-5) apples one time which basically giving back 5 apples to the stock which is equivalent of having (-5) apples. Or, taking 1 apple from the stock (-5) times. Meaning giving back 1 apple to the stock 5 times which is basically having (-5) apples in our example.
- (-1) * (-1) = +1
(-1) * (-1) means (-1) apples from the stock (-1) times. Doing something (-1) time in our case is equal and opposite of doing the task 1 time. So (-1) * (-1) means giving (-1) apple back to the stock 1 time. But stock acquiring (-1) apple means stock has to return 1 apple to you. So you end up having 1 apple from the stock.
(-1) * (-5) = 5. This means taking (-1) apple from the stock (-5) times which in turn means giving (-1) apple back to the stock 5 times. Stock now has (-5) apples which means it has to give back to you 5 apples and so you end up having +5 apples. Alternatively, (-1) * (-5) can also mean taking (-5) apples from the stock (-1) time which in turn means giving (-5) apples to the stock 1 time. Stock now has (-5) apples, so it will return to you 5 apples and you end up having +5 apples which is the result.
You can think of your own examples, apply this basic shift in your understanding of positive and negative numbers.
Quoting from the film The Man from Earth, “Everything is so simple”.
Image credits of Brahmagupta: https://tinyurl.com/y4b23agq