Why Dividing by Zero is Undefined?

Students tend to hear one thing copiously, which is that “you can never divide a number by zero.” And once they enter high school, they hear the phrase “a number divided by zero yields an undefined result.” almost daily.

5 min readOct 7, 2021

In the world of mathematics, every operation and its result has a very rational explanation because mathematics is built upon a foundation of very simple and essentially bulletproof principles. If we change some of these principles, we end up with an outcome where the rock-solid foundation of mathematics undergoes intriguing metamorphoses.

Many students relearn the same principles over and over again during their mathematical education. However, they tend to hear one thing copiously, which is that “you can never divide a number by zero.” And once they enter high school, they hear the phrase “a number divided by zero yields an undefined result.” almost daily.

Though dividing a number by zero might seem very basic at the onset, it does not quite work that way. If we are to ponder over the concept for a moment, we notice that it becomes increasingly complicated to wrap one’s mind around it. Because mathematicians have yet to define the outcome of dividing a number by zero effectively, it is still undefined. And for many, the lack of a definition for the outcome of dividing a number by zero is still quite difficult to process.

But how can a rational concept such as mathematics not define an operation as simple as this?

First of all, before we divide a number by zero, we need to assume that we know how to compute addition and subtraction. To better understand the division of any mathematical number, we can approach the matter in the following manner:

For example, let’s divide the number 1 by numbers very close to zero. 1/0.1 equals 10. 
1/0.01 equals 100.
1/0.001 equals 1000.
1/0.0001 equals 10000.
1/0.00001 equals 100000.
1/0.000001 equals 1000000.
1/0.0000001 equals 10000000.
1/0.00000001 equals 100000000.
1/0.000000001 equals 1000000000.
1/0.0000000001 equals 10000000000.
1/0.00000000001 equals 100000000000.
1/0.000000000001 equals 1000000000000.
If we go further we get even closer to zero, such as dividing 1 by 0.0000000000000000000001, we get 10000000000000000000000 (Septillion). So the closer to zero that the number we use to divide 1 is, the greater the result, and it’s a clear pattern that goes on.

Then if we were to adopt a logical approach, shouldn’t we get a result of infinity if we divide one by zero? Though it might seem rationally straightforward at first, the result approaching infinity as the divisor approaches zero is not the same as simply assuming one divided by zero is infinity. In one, we approach; in the other, we assertively claim so, but why?

I personally love it when people get into these sorts of inquiries.

To understand this problem, we first need to know what it means to divide. If we are to give a simple example, the number 10 divided by the number 5 indicates how many number 5’s need to be added to reach a number 10. So, in other words, five multiplied by what gives us 10?

If we examine this example in detail, we notice something; dividing a number means doing the inverse of multiplication. So that means if we can multiply any given number by another number, “x,” that means we can multiply the result which we end up with to get a result that is the original number. If we can find such a number, it is called the multiplicative inverse of that number.

An example of this is when you multiply 2 and 3, you get 6. That means to get three again, you need to multiply six by ½. So the multiplicative inverse of 2 is ½. Using these values, we can get another interesting result.A number multiplied by its multiplicative inverse is always equal to 1.

That means if we are dividing a number by zero, zero also needs to have a multiplicative inverse, which would need to be 1/0. Furthermore, when we multiply 1/0 by 0, we should get 1. However, by definition, any number multiplied by zero is always zero. So zero does not have a multiplicative inverse.

So does the answer we get take care of things? If we go to the past, we can find some cases where mathematicians have changed the rules. A long time ago, negative numbers didn’t have square roots. Then mathematicians defined it as an imaginary number “i” under the name of complex numbers,” leading to a whole new world in the field of mathematics.

Well, can’t mathematicians change the rules again and invent a new one where they define 1/0 as infinity? Let’s try it.

Assume that;1/0 = ∞

By definition, a number multiplied by its multiplicative inverse must be equal to 1. So infinity times zero must be 1.

0 x ∞ = 1

So zero times infinity plus zero times infinity must be equal to 2.

(0 x ∞) + (0 x ∞) = 2

If we use the distributive property, zero plus zero multiplied by infinity would be 2. Right?

(0+0) x ∞ = 2

Because zero plus zero is equal to zero, its equivalent of zero multiplied by infinity can be written as such.

0 x ∞ = 2Yet earlier, we had assumed that zero multiplied by infinity was equal to one.

So now we end up with one equalling 2. We didn’t make any errors in our operations; we merely got an incorrect result. If we were to accept this result as correct, we would accept that the number 1 is equal to all other numbers mathematically. That would mean the end of all Algebra as we know it.

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