Dividing by Zero

Dhristi JEE 2022-24

What is 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. It can also be said that division is the inverse operation of multiplication. 


Example:

In division, if 12 is divided into 3 equal groups, it will give 4 in each group. This in a mathematical sense can be written as 12/ 3= 4. 


Some facts about division:

  • In division, the number which gets divided is called the dividend. The number which it gets divided is called the divisor. The number obtained from dividing is called the quotient while the number left is known as the remainder. 

  • For example, if we divide 17 with 2, we get 8. Here, 17 is the dividend, 2 is the divisor, 8 is the quotient while 1 is the remainder. 

  • The product of the quotient and the divisor added to the remainder is always equal to the dividend. This can be written as (Divisor × Quotient) + Remainder = Dividend or (d × Q) + R = D

  • For example, if we divide 23 with 2, we get 11. Here, 23 is the dividend, 2 is the divisor, 11 is the quotient while 1 is the remainder. If we follow the above rule then by solving (2 x 11) + 1 we will get 23 which is the dividend. Therefore, the property given above holds true. 

  • 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.

For example: 10 ÷ 1= 1 

  • In division, the remainder is always smaller than the divisor. 

  • 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.

⇒0/x = 0


For example, a  person has zero toffees which are to be divided among 7 ( let’s say) children. This means that there is nothing to be shared or distributed among 7 children. If nothing is shared, then no one will get any toffees. Hence, 0 divided by any number gives 0 as the quotient.

Therefore, 0/1 = 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 maths 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?

  • The concept of the number zero came in the 7th century much after the invention of other natural numbers. Brahmagupta was the mathematician and astronomer who found the concept of Zero. He also gave rules for addition and subtraction  with zero. 

  • 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.

  • Zero is neither a prime number nor a composite number. 

  • If zero is added or subtracted to any number, then the number remains the same. But if zero is multiplied with any number then the product is zero. 

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


Conclusion

Highly qualified subject matter experts, curate the study material with utmost care for the needs of a student. The articles, solutions, and sample questions are a great way to conceptual learning. As it is known that mathematics requires practice, students are expected to refer to the  material provided is important for better learning. To understand this key requirement and work upon the curriculum keeping that in mind is an important aspect to success. You can easily download the free PDFs and learn constructively. 

FAQs on Dividing by Zero

1. Can You Divide a Number by Zero?

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?

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?

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 maths," we simply say 0/0 is undefined.

4. How is division the inverse of multiplication?

To understand how Division is the inverse of multiplication, consider the following example. Divide a number ‘y’ by 8. According to the inverse rule, multiplication of the result y/8 with the divisor, 8 will give the dividend, y as the result. In mathematical sense, 

(y/8) x  8 = y

Therefore, if a number (say y ) is divided by any other number (say 8) , multiplying the quotient by 8 undoes division by 8. This is applicable for any number which is not equal to zero. 


Example- Divide 32 by 2. 

= 32/2

= 16 

Now, multiply the quotient that is 16 with the divisor that is 2.

= 16 x 2 

= 32 which is the dividend. 

Therefore, we can say that division is the inverse of multiplication. 

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