Kids learn division in their primary school and find it quite easy and doable. But when they encounter a zero in division, either in the numerator or denominator or both, it gets a bit tricky.
Here is my humble attempt to develop the basic intuition ( for Primary/Lower secondary students) for these cases with the help of a few simple stories.
- 0 / N (Zero divided by any number)
- N / 0 (Any number divide by zero)
- 0 / 0 (Zero divided by zero)
Zero divided by any number (0/N)
Today, one of my grade 5 students asked me what will 0/2015 be.
I didn’t want to answer his question right away. I was thinking of a better way to discuss so that he could discover this concept himself.
Student – Will it be zero?
I did not find conviction in his voice. I ignored his question and started a chain of questions in my Guided Discovery Way
Me – What is division?
Student – Division means dividing something into ‘equal parts’.
Me – Ok, give me an example.
Student – If I have 10 mangoes, and I need to share it with 4 friends, then everybody will get 2 mangoes. 10/5 =2.
Me – Cool, If you have no mangoes and you need to share it with your 4 friends then how much everybody will get?
Student – I have no mangoes, so actually I don’t have anything to share….umm…(pause)….so everybody will get nothing.
Me – What do you mean by nothing?
Student – Nothing means zero.
Me – So, everybody will get 0 mangoes. That means 0/5 = 0?
Student – Yes, 0/5 = 0.
Me – Ok, if you have no mangoes and you need to share it with your 9 friends then how much would everybody get?
Student – Yes, got it. 0/10 = 0.
Me – Ok, so 0/2015 ?
Student – Zero:) and the smile told me that he had got it which in turn brought a smile on my face 🙂
Me – Now, the final question for this discussion. What will be 0 divided by any number ? 0/ N = ?
Student – Zero 🙂
If I had told directly that zero divided by any number is zero, the student might have just accepted it without even understanding it. Math is not to be accepted, it is to be understood.
Any number divided by zero (N/0)
Lets try to develop this intuition with the help of simple numbers, 10 & 1. Me : 10/1 ? Student: 10 Me: 10/0.1? Student : 100 Me : 10/0.01 ? Student: 1000 Me: 10/0.001? Student : 10000 Me: 10/0.0001? Student : 100000 Let’s see the pattern, how it looks like
As the denominator keeps decreasing -> the value of the fraction keeps increasing.
So, if the value of the denominator is so small that it diminishes to zero, then the value of the fraction has to be so large, that it approaches infinity. So, N/0 —> infinity (N/0 approaches to infinity).
Actually, N/0 will be undefined. Trying to build a simple explanation (for primary level students) with few simple stories. Will upload it soon.
Zero divided by zero (0/0)
0/0 is neither of the form 0/N nor of the form N/0.
It is clearly neither zero nor infinity.
Actually, the result of 0/0 is indeterminate (the set of numbers whose value can not be determined by mathematical logic and rules)
Let’s look at this expression in story way
If I have 2 chocolates which I decide to equally distribute among 2 cute kids. Both will be happy. Both will get 1.
But if I don’t have any chocolates and then I decide to do the same distribution, but this time among zero kids! Look at the imaginary unhappy faces of those kids.
I am trying to distribute nothing among no one. Clearly, I can do it in many ways, but all will be non-realistic. That is, there really is no way to do this distribution. Hence no one can determine what 0/0 would look like. (I had read this story on Quora.)
Let’s look at this expression in logical way but from the reverse direction
Let us consider a universe where division by zero is actually defined.
As per the basic definition of division
=> 3=0/0 => 6=0/0
It is clear that zero divided by zero can take any value. Thus, zero divided by zero is indeterminate.