Zero is my favorite number after i. This number, zero is strange. The concept is mind blowing. Same as the infinity, thanks to God, the infinity is not a number but a concept. In contrast, zero is a number. So we need to deal with it everyday. That is why, zero becomes one of my favorite numbers. And it is possible to talk about zero for hours.
Lets check basic four operators; addition, subtraction, multiplication and division against this strange number, zero.
Addition and Subtraction
Lets assume that you are planning a treat for your friends. You buy 5 toffees. Earlier you had no toffees. But now you have 5 toffees. Mathematically it can be denoted as
= 0 + 5 = 5
You have 5 toffees but your friends dont like the flavor, so they didnt take any. That can be denoted as;
= 5 – 0 = 5
Taking back the same example, you have 5 friends and if you are planning to buy 2 toffees per friend, you need to buy 5 * 2 = 10 toffees. Then your mind changes, you are planning to buy a chocolate and a toffee per friend. Then you will need to buy 5 * 1 = 5 toffees only. But if you change your mind and decide that you buy 2 chocolates per friend, then you need to buy 5 * 0 = 0 toffees and 5 * 2 = 10 chocolates. This makes sense. As you have 10 chocolates, your friends get 10 / 5 = 2 chocolates. Further since you have no toffees your friends get 0 / 5 = 0 toffees.
Therefore it is safely concluded that, addition, multiplication of 0 wont change the value. Multiplication by zero makes any value 0. If any value is divided by zero then it is zero again.
So far the maths align with our intuition.
Multiplication as Addition
Multiplication is a special form of addition. For an example 3 * 5 means adding 5, three times. As multiplication is commutative 5 * 3 gives the same results but as a convention it can be taken as adding 3, five times.
5 * 3 = 5 + 5 + 5 = 15 or 3 * 5 = 3 + 3 + 3 + 3 + 3 = 15
Then 5 * 0 can be taken as adding 0, five times. 0 + 0 + 0 + 0 + 0 = 0, however according to the convention 0 * 5 becomes senseless. Adding 5 zero times, meaning nothing is being done.
Division as Subtraction
Division is also a special form of subtraction. It says how many subtractions should be performed to reach zero. For an example 6 / 2 can be denoted as 6 – 2 – 2 – 2 , so it needs 3 subtractions, hence 6 / 2 = 3
In the same manner, 0 / 5 = 0 because 0 has already reached to zero and no subtraction is required.
So far zero is behaving rationally. Now will see how wired zero can become under certain circumstances.
Division by Zero
It is a fact that you know, when a number is divided by a small number the result gets very large. When the number becomes smaller the result gets much more larger. 0 is the absolute smallest number. Therefore, if a number is divided by zero then it should become the infinity. This is quite intuitive though it is not correct.
Lets take an analogy. You have to distribute 10 chocolates among 5 friends. One friend takes 2 chocolates . Then 8 chocolates left for 4 friends. Still 8 / 4 = 2. Then two more friends takes. Now you have 4 chocolates and 2 more friends to take. Another one takes 2 chocolates now just one friend and 2 chocolates, once the final friend takes 2 remaining chocolates, you have no friends to come and no chocolates to distribute. So no one gets a chocolates, as there are no chocolates to distribute and at the same time there is no friends left to get any chocolates. Therefore, following make no sense at all, does it suggest the infinity or zero. It suggests neither of them.
(no friends) / (no toffees)
Consider division as a series of subtraction, as we checked earlier. Example: take 5 / 0. How many times 5 should be subtracted by zero so that the final answer to reach 0. It can be noted that it will never reach to 0.
5 – 0 – 0 ………………………………………………. – 0 – 0 ≠ 0
That is one of the ways to demonstrate 0 / 0 non-existence, why any number is divided by zero is not infinity. Therefore in mathematics, division by zero is not determinable, considered as an invalid operation. That is why before dividing by any number we make the assumption that the number is not equal to zero. It reaches to the infinity, but it never gets there. Calculus should be used to deal with such situations, but it is not in the scope of this article.
Multiplication zero by zero
Intutively, zero times zero would equal to zero. Because when a number is multiplied by zero, the result becomes zero. But there is an issue.
Coming back to the toffee example, it is decided to buy chocolates instead of toffees. The treat needs 10 chocolates as there are 5 friends and one gets two chocolates. But how many toffees to buy. All friends got chocolates therefore there is no friend to get toffees and at the same time you have not decided to give toffees, therefore it sounds like 0 * 0. Does it make sense, if so above example on 0 / 0 should make sense as well.
On the other hand, we noted that 5 * 0 = 0 + 0 + 0 + 0 + 0,
then, 4 * 0 = 0 + 0 + 0 + 0
and 3 * 0 = 0 + 0 + 0
2 * 0 = 0 + 0
1 * 0 = 0
but what is 0 * 0 = ?
Anyhow, if you can remember, this issue occurs on 0 * 5 as well but taking commutative property, 0 *5 considered as 5 * 0. Here it has no meaning as it becomes 0 * 0 again.
That is why both 0 * 0 and 0 / 0 are counter intuitive.
Is 0 * 0, indeterminate?
In an instance, 0 * 0 becomes indeterminate. Check the following proof.
take 3 numbers such that x * y = z
then assign x = 0;
as we already know that any number multiplied by zero is zero, when x = 0
0 * y = 0 ——————————(B)
y can be any number, then divide the equation by y.
0 = 0 / y ———————————-(A)
Now you will realize, equation A can only be true if y is not equal to 0, when y is equal to 0 equation A cannot be true. It is illegal, that means in equation B, solution y = 0 does not exist. That means, 0 * y = 0 true only if y ≠ 0 only.
When 0 * 0 Becomes 0
It is known fact that, x * x reaches to zero when x is reaching to zero. But it never gets value zero. But in most equations it is easy to take 0 * 0 as 0. Even in some instances x / 0 can be considered as the infinity. Even 0 / 0 is common in calculus, it is not taken as not defined in calculus. In fact it is the begging of calculus. The answer changes as the hat you are wearing, mathematician, engineer, physicists and etc…
Therefore where you use the expression it can get the related meaning. But based on basic mathematics, it can be seen that 0 * 0 cannot be defined. There is no much harm if it is taken as 0 * 0 = 0 as well. But it cannot be generalized. This affects indexes of 0 as well. 0^2 ≠ 0 and generally 0 ^ n ≠ 0 where n > 0. This should be discussed in another article.
How this is started ?
Recently, I found an interesting question on Quora. Two numbers are such that their sum is equal to their product. What are these two numbers?
Intuitively, it seems like 0,0 and 2,2 are the solutions but 0,0 is not (as 0 * 0 ≠ 0 + 0 but 2 * 2 = 2 + 2). That started the argument on what is 0 * 0 and this is my explanation on that.
I believe 0 * 0 is not 0. This is what I believe and the basis is explained above. This is open for discussions, please feel free to add your thoughts as well.