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Role of Zero in division

© Vidya Narayan Wadadekar

The division operation involving zero is a very difficult to understand, hence considered separately in this article.

Zero as a dividend

When I write a problem '0 ÷ 5 =?' children usually come out with the correct answer 'zero'. But, if I ask for the reason, they look confused. It means at intuitive level the children feel that the answer should be zero, but lack reasoning. This situation thus demands some concrete work which helps understanding.

I invite a child and tell him I do not have any marbles here. However, I want you to distribute them equally among 4 children. I invite four more children and make them stand before him. Since, the child has no marbles to distribute he does not distribute anything to the four children. Thus, the children receive nothing. Mathematically we state this as '0 ÷ 5 = 0'.

I also remind them that 5 times 0 = 0 or 0 times 5 = 0 Thus, 0 x 5 = 0. Since division is an inverse multiplication 0 ÷ 5 = 0

I provide some more examples of following type to solve:

• 0 ÷ 10 = ?
• 0 ÷ 50 = ?
• 0 ÷ 100 = ?
• 0 ÷ 150 = ?
• 0 ÷ any number greater than zero = ?

  After solving these examples they will be in a position to conclude: Zero divided by any number, greater than zero, gives a zero quotient.

  Zero as a divisor

  This is a bit difficult for the children to understand. Even adults are confused with the following types of problems:

  • 5 ÷ 0 = ?
  • 50 ÷ 0 = ?
  • 500 ÷ 0 = ?
  • 5000 ÷ 0 = ?
  I give a demonstration in my class to solve this problem. They think that the answer is zero. I then ask them that, 'we have already seen that '0 ÷ 5 = 0' Then, do you think that '5 ÷ 0' will also give a zero quotient?' They start thinking. I give another example: If 20 ÷ 5 = 4 then 5 ÷ 20 = ?. I make it more clear by constructing a situation.

  If 20 balls are distributed equally among five children, how many balls each child will have? They know the answer, which is 4.

  If 5 balls are distributed among 20 children how many balls each child will have? They know that the answer is, not even one ball.

  Again I bring them to the problem '5 ÷ 0 = ?' They agree that the answer cannot be 'zero'. I help them to find out the answer by giving a concrete activity.

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