Division by Zero
In mathematics, there's a fundamental rule: no number can be divided by zero. However, there are three distinct cases to consider:
- The division of a number $ n:0 $ with $ n \ne 0 $ is impossible
- The division $ 0:0 $ is undefined
- The division $ 0:n = 0 $ with $ n \ne 0 $ is defined, and the result is always zero
Simply put, no number can be divided by zero, but zero can be divided by any non-zero number.
But why is dividing by zero impossible? What’s the difference between an impossible operation and an undefined one? Let's delve into this puzzle.
We'll now explain each case in detail.
Dividing a Number \( n \) by Zero with \( n \neq 0 \)
This operation is impossible. Division by zero isn’t defined because there is no number that, when multiplied by zero, gives a result other than zero.
In arithmetic, division is the inverse of multiplication.
So, when you say "10 divided by 2 is 5", it means that multiplying the quotient 5 by 2 gives you 10.
$$ 10:2 = 5 \Leftrightarrow 5 \cdot 2 = 10 $$
This principle works for all numbers, except when zero is the divisor.
Let’s consider a practical example.
$$ 10:0 = x \Leftrightarrow x \cdot 0 = 10 $$
To find the result of "10 divided by zero", you would need a number x that, when multiplied by zero, equals 10.
Such a number doesn’t exist because zero is the absorbing element in multiplication: the product of any number and zero is always zero.
Therefore, no number multiplied by zero can yield 10 or any other non-zero value.
This is why division by zero is impossible (undefined).
The same logic applies to all non-zero numbers.
For example, "15 divided by zero" is also undefined because no number multiplied by zero equals fifteen.
$$ 15:0 = x \Leftrightarrow x \cdot 0 = 15 $$
And so on for any other non-zero number.
You might assume that, consequently, the operation "zero divided by zero" should be possible. However, this isn’t the case because zero divided by zero is "undefined", as any number multiplied by zero results in zero. I'll explain why in the next section.
Dividing Zero by Zero
Dividing zero by zero is undefined because there isn’t a single value that satisfies the equation \( x \cdot 0 = 0 \).
Indeed, any number multiplied by zero results in zero, so a specific value cannot be determined.
For example, any number x multiplied by zero equals zero (where x can be 1, 2, 3, ...)
$$ 0:0 = x \Leftrightarrow x \cdot 0 = 0 $$
This is where the issue arises: the equation "zero divided by zero" doesn’t have a unique solution; it allows for infinite possible values for x.
Therefore, dividing $ 0:0 $ is also prohibited in mathematics, but for a different reason.
Dividing Zero by a Number \( n \neq 0 \)
This operation is defined, and the result is always zero. If you divide zero by any non-zero number, the quotient is zero.
For instance, "0 divided by 2" is a valid operation, and the result is always 0.
$$ 0:2 = 0 \Leftrightarrow 0 \cdot 2 = 0 $$
The quotient is always zero due to the zero product property.
Similarly, you can divide 0 by 5:
$$ 0:5 = 0 \Leftrightarrow 0 \cdot 5 = 0 $$
This operation is also defined. And so forth.
In summary, you can divide zero by any non-zero number, and the result will always be zero.
I hope this explanation clarifies everything.
