Dividing by Zero

Division:

Division is a method of dividing a group of things into equal parts. It is one of the four basic arithmetic operations that give a fair result of sharing. The division's main aim is to see how many equal groups or how many in each group share fairly.


Example: In division, if 12 is divided into 3 equal groups, it will give 4 in each group.

Some facts about division:

  • When we divide something by 1, the result will always be the same number. This means that if the divisor is 1, then the quotient will be equal to the dividend.

  • Division by zero is considered undefined. (We'll discuss this in detail)

  • If the dividend and divisor are the same in the division, then the result will always be 1.  

For example: 5 ÷ 5 = 1. 


Zero:

Zero is an integer number just before 1. It's an even number that is neither positive nor negative. While zero is considered to be the whole number, it is not a counting number. The value of the zero number is nothing.


Zero Divided by a Number:

Dividing 0 by any number will give us a zero. Zero will never change when you multiply or divide any number by it.

⇒ \[\frac{0}{x}\] = 0


A Number Divided by Zero:

Never divide any number by zero. We've all been taught this at school, and it's good advice. It's rarely meaningful to divide anything by zero. Dividing by zero does not make sense, because in arithmetic, dividing by zero can also be interpreted as multiplying by zero. Suppose we got an equation, 5/0=X. This also interprets the same equation as 0*X=5. Here, there's no number that could accommodate X to make the equation work. 

With reference to the example given above, if we consider 0 by 0 to X i.e., 0/0=X, it can also be rewritten as 0*X=0, and the problem is that every number works. X could be anything, so, this equation is not useful at all. Hence, If we divide by zero, it is considered as "Undefined."

For example, if we have 20 bananas and we want to distribute them evenly to 4 people, then by definition of the division each student would receive 20/4 bananas, i.e. 5 bananas each. If we use the same logic, x/0 means distributing x bananas equally among 0 people. It's completely pointless; there's no rational way of distributing a group of items to 0 people, so we can say it's undefined.


What is Undefined? 

Sometimes, when you see "undefined" in the math class, it seems very strange. Mathematicians have never defined what it means to divide by zero. What's the value of that? They didn't do that because they couldn't come up with a good answer. There is no good answer, no good definition. And because of that, any non-zero number, divided by zero, is left "undefined."


Solved Example: 

If you have 5 Apples and 5 Friends at your home, how many Apples does each friend get, in a fair share? Everyone will get an Apple each, right?

If you have the same number of apples and no friends at your home, you're partitioning Apples among no people? How can we make sense out of this? This doesn't make sense, and that's what is called undefined.  


Did you know?

  • Zero is a real number, integer, Rational, as well as the whole number.  

  • Zero is always neutral, meaning that there is no such thing as -0 or +0.

  • The power of any number that is raised by zero is always one. 

FAQs (Frequently Asked Questions)

1. Can You Divide a Number by Zero?

Ans: Dividing any number by zero does not make sense, because in maths, dividing by zero can be interpreted as multiplying by zero. There's no number that you can multiply by zero to get a non-zero number. There's no solution, so any non-zero number divided by 0 is undefined.

2. Is 0 Divided by 5 Defined or Not?

Ans: That is perfectly defined. 

0 ÷ 5 = 0 

For example, if you have zero apples at your home and there are five people to share, each person will get 0 apples.

3. What is 0/0?

Ans: One might argue that 0/0 is 0 because 0 divided by 0 is 0. Another may argue that 0/0 is 1 because anything divided by itself is 1. And that's the problem! Whatever we say 0/0 is equal to, we contradict one or the other key property of numbers. To avoid "breaking math," we simply say 0/0 is undefined.

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