# Why is division by zero undefined or no definition?

Division by 0 is NOT ALLOWED in mathematics. Reason :-

Let's see the following scenario :-

5/0=?

Thus 5/0=x , = 5=0×x, = 5=0.

See! What we proved! 5=0 ! Impossible.

Here the value of the answer (x) is also not defined because there is no such number which gives the answer 5 when multiplied by 0 (0×x=5; in this case x doesn't exist).

Thus, division by 0 is not allowed and undefined.

Hope you understand.

## 4 Replies to “Why is division by zero undefined or no definition?”

1. Jonathan Lang says:

Thanks for the A2A!

Well, you can define it, but when you do, you lose some important features. For example, there was a time when complex numbers didn’t exist. Well, people decided to define $\sqrt{-1}$, and they gained lots of important things, but they also lost things, like the fact that $\sqrt{ab}=\sqrt{a}\sqrt{b}$ for all numbers and that numbers can be ordered, but the pros outweighed the cons. This is why complex numbers are taught in school curriculums. On the other hand, people have defined division by $0$, but in that case, the cons outweighed the pros, so division by $0$ isn’t usually taught, unless you get to higher math.

2. Jonathan Lang says:

$1 \div 1 = 1, 1 \div 2 = 0.5, 1 \div 3 = 0.\overline{3}, 1 \div 4 = 0.25$

$1 div 5 = 0.2, 1 \div 6 = 0.1\overline{6}, 1 \div 7 = 0.\overline{142857}$

$1 \div 8 = 0.125, 1 \div 9 = 0.\overline{1}, 1 \div 10 = 0.1, ….$

$\frac{1}{100} = 0.01, \frac{1}{1000} = 0.001, \frac{1}{10000} = 0.0001….$

Now, think of the other cases:

$0.1 \div 1 = 10, 0.01 \div 1 = 100, 0.001 \div 1 = 1000$,

$0.0001 \div 1 = 10000, 0.00001 \div 1 = 100000$….

As the value of grows, $\frac{1}{0} = \infty$

3. Jonathan Lang says:

Use the relative properties to define the properties of the number Zero.

1. On a deduction; If you deduct a factor of a number from the number itself, the resulting answer will also be divisible by the factor. (ie . 65 is divisible by 13, if you deduct 13 from 65, the resulting 52 is also divisible by 13).
1. You can reach zero by deducting any number from the number itself, Hence Zero should be the only number up to infinity which can be divided by all the numbers.
4. On a division
1. If you deduct the sum of digits of a number from the number itself (ie 151 – 7 = 144)
2. or if you deduct one number from another with same sum of digit (ie 25 & 16);

the resulting number would be divisible by 9.

Based on the above, Two properties of Zero can be defined.

1. Zero is the only number divisible by all the other numbers up to infinity.
2. since sum of digit of both numbers in a formula resulting in zero are equal, the resulting answer (zero) would be divisible by 9 and the sum of digit of zero should be 9.
5. Dev Suthar says:

The easiest way to answer this is to turn it around:

If $\dfrac{1}{0}=x$ then $1=0x$.

For what value of $x$ can you multiply it by $0$ and get $1$? You say that there are no such values? There you go; that's why division by zero is usually undefined.

There is one exception: if the numerator is also zero.

If $\dfrac{0}{0}=x$ then $0=0x$.

For what value of $x$ can you multiply it by $0$ and get $0$? All of them. $\frac{0}{0}$ isn't undefined (no valid answers); it's indeterminate (everything is a valid answer).