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 Ā (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 āequal partsā.
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 donāt have anything to shareā¦.ummā¦(pause)ā¦.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 š
Me – Now, the final question for this discussion. What will be 0 divided by any number ? 0/ N = ?
Student – Ā Zero š
If I had told directly that zero divided by any number is zero, the student might have just accepted it without even understanding it. Ā 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 ? Ā Ā Ā Ā Ā Ā Ā Ā Student: 10
Me: 10/0.1? Ā Ā Ā Ā Ā Ā Ā Student : 100
Me : 10/0.01 ? Ā Ā Ā Ā Ā Student: 1000
Me: 10/0.001? Ā Ā Ā Ā Ā Student : 10000
Me: 10/0.0001? Ā Ā Ā Ā Student : 100000
Letās see the pattern, how it looks like
Ā Ā Ā Ā Ā Ā
Ā As the denominator keeps decreasing ->Ā 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, Ā N/0 —> infinity (N/0 approaches to infinity).Ā Ā
Actually, N/0 will be undefined. Trying to build a simple explanation (for primary level students) with few simple stories. Will upload it soon.
Why ādivision by zeroā is undefined?
Zero divided by zero (0/0)
Ā Ā 0/0 Ā is neither of the form 0/N nor of the form N/0.
Ā Ā It is clearly neither zero nor infinity.
Ā Ā Actually, the result of 0/0 is indeterminate (the set of numbers whose value can not be Ā Ā Ā determined by mathematical logic and rules)
Letās 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.Ā (I had read this story on Quora.)
Letās 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
Ā Ā Ā Ā Ā 0ā3=0 Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā 0ā6=0
=> Ā 3=0/0 Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā => Ā 6=0/0
It is clear that zero divided by zero can take any value.Ā Thus, zero divided by zero is indeterminate.
Maths2Art 
Here a simple question
If N/0 = infinity
N = infinity Ć 0
N = 0
How???
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Thank you for raising such a beautiful question. Let me try to attend it in 2 parts
1. N/0 = infinity
here are the few lines from the article,
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.
N/0—> infinity. Actually, I should write Lim (N/0) = infinity. But, I was writing this article for upper primary level so didnāt use the terminology of limit. I am trying to keep the essence using plain English (N/0 approaches infinity)
I will make a correction in my article and update ‘N/0 = infinity’ to ‘N/0 —> infinity’
2. infinity x 0
Infinity is not a number,it is a concept.
The problem is that the laws of addition, multiplication (Arithmetical Operations) we are using hold for numbers, but infinity is not a number, so these laws do not apply to infinity.
E.g., if we say “ā+1=ā”, this seems to imply 1 = 0.
In summary, the expression āĆ0 using multiplication defined for the numbers does not have any meaning, so it cannot be said to be equal to 0. It is Indeterminate.
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This is really a good article. I would like to suggest some improvement in the article. You can simultaneously explain each division in algebraic way also by asking the following questions: 1. Why should 0/N be only zero and not any other integer, i.e. what happens if we assume 0/N equal to some some other integer (Like it has been explained for 0/0 )? 2. Is there any largest number which equals N/0 ? Or, why can we not assign a particular integer or real value for this quantity ?
Finally, there is typo “share it with 4 friends.” Replace 4 by 5.
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Thank you very much for your suggestions. Will keep it in mind.
Very good observation on ” share it with 4 friends “. I was waiting for someone to point it out. Actually, we were trying to make it bit interesting in our classroom so thought of writing – “I have to share with my 4 friends” meaning “sharing need to be done among 5 people”.
If I had a prize for the most observant blog reader, I should have given it to you.
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The concept of zero first appeared in India around A.D. 458, according to the book āThe Crest of the Peacock; Non-European Roots of Mathematics,ā by Dr. George Gheverghese Joseph. Joseph suggests that the Sanskrit word for zero, ÅÅ«nya, which meant āvoidā or āemptyā and derived from the word for growth, combined with the early definition found in the Rig-veda of ālackā or ādeficiency.ā The derivative of the two definitions is ÅÅ«nyata, a Buddhist doctrine of āemptiness,ā or emptying oneās mind from impressions and thoughts.
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Thank you for this informative comment on the article. I have added this book in my ‘to be read’ list. This should surely help me in knowing more about Maths history.
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Replace 4 with Five. Otherwise you have to cut it into two.
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Very good observation on ā share it with 4 friends ā.Actually, we were trying to make it bit interesting/tricky in our classroom so thought of writing ā āI have to share with my 4 friendsā meaning āsharing need to be done among 5 peopleā.
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replace 4 with 5 or say that your sharing the mangoes among you and your friends.
Reyansh
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Good observation Reyansh !
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