Is division by Zero allowed?

Why can’t you divide by zero? Who forbade it? School stubbornly forbids us from dividing by 0, but as soon as you step into university, you get an indulgence. What was considered forbidden in school is now possible.

You can divide by zero and get infinity. Higher mathematics… Well, almost. It can be explained more simply. So why can’t you divide by zero, but you can multiply by it?

History and philosophy of zero

In fact, the history of dividing by zero troubled its inventors (and zero was invented in India). But Indians are philosophers accustomed to abstract problems. What does it mean to divide by nothing? For Europeans at that time, such a question didn’t even exist, as they knew nothing about zero or negative numbers (which are to the left of zero on the number line).

In India, subtracting a larger number from a smaller one to get a negative number was not a problem. After all, what does 3 – 5 = -2 mean in everyday life? It means that someone owes someone 2. Negative numbers were called debts.

Now let’s simply figure out the issue of dividing by zero. In the distant year 598 AD (just think how long ago, over 1400 years!), a mathematician named Brahmagupta was born in India, who also pondered the question of dividing by zero.

He suggested that if you take a lemon and start dividing it into parts, eventually the slices will become very small. In our imagination, we can reach the point where the slices become equal to zero.

So, the question is, if you divide a lemon not into 2, 4, or 10 parts, but into a number of parts approaching infinity—what size will the slices be? You will get an infinite number of “zero slices.” It’s quite simple: if you cut the lemon very finely, you get a puddle with an infinite number of parts—lemon juice.

Just ask yourself the question:

If dividing by infinity gives zero, then dividing by zero should give infinity.

x / ∞ = 0, so x / 0 = ∞

What happens if you divide by zero?

But if we turn to mathematics, it seems illogical:

a * 0 = 0? And if b * 0 = 0? Then: a * 0 = b * 0

And from this: a = b

That is, any number is equal to any other number. This is the first inconsistency of dividing by zero; let’s continue. In mathematics, division is considered the inverse operation of multiplication. This means that if we divide 4 by 2, we need to find a number that, when multiplied by 2, gives 4.

Dividing 4 by zero—we need to find a number that, when multiplied by zero, gives 4. That is, x * 0 = 4? But x * 0 = 0! Another problem. It turns out we’re asking: “How many zeros do you need to take to get 4?” Infinity? An infinite number of zeros will still sum to zero.

And dividing 0 by 0 gives an indeterminate form, because 0 * x = 0, where x can be anything. That is, there are infinitely many solutions. So what do we get in the end?

A simple explanation from life

Here’s a problem from physics and real life. Suppose we want to calculate how long it will take to walk 10 kilometers. So, Speed * time = distance (S = Vt). To find time, we divide distance by speed (t = S / V). But what if the speed is 0? t = 10 / 0. It will be infinity!

Standing still, speed is zero, and at that speed, we will take forever to reach the 10 km mark. So time will be… t = ∞. There we have infinity!

In this example, dividing by zero is possible, and life experience allows it. It’s a pity that school teachers can’t explain such things as simply.

Another explanation

Let’s define what division is. For example, 8 / 4 means the question “how many fours can fit into eight?” Answer: “two fours,” that is, mathematically 8 / 4 = 2.

And if we ask ourselves 5 / 0 = ? How many zeros can fit inside five? As many as you want. An infinite number. We divide by zero and get… infinity again.

But if instead of abstract numbers we take material things, like apples. 6 / 3 — “if you put 6 apples into boxes with 3 each, how many boxes do you need?” Answer: “2 boxes.” Let’s go further: 4 / 0 — “if you put 4 apples into boxes with zero each, how many…” It turns out that boxes aren’t needed; we’re not putting anything anywhere!

A very simple explanation

Very simply, “on fingers”

10 / 2 = 5; 10 / 4 = 2.5; 10 / 8 = 1.25… The larger the number in the denominator, the smaller the result.

10 / 2 = 5; 10 / 1 = 10; 10 / 0.5 = 20… The smaller the number in the denominator, the larger the result. And if we take a very small number? For example, 0.0000001, we get 100,000,000. And if we go further in our reasoning and reduce the denominator to zero? In the end, we get something so large that it’s called “infinity.”

So, can you divide by zero?

It all depends on why you need it and within what rules you decide to “divide.” If it’s algebra, then it’s simple — “you can’t divide by zero” because there’s no such thing as “infinity” (it’s not even a number), and it’s unclear what should result.

Dividing by zero and higher mathematics

Can you divide by zero in higher mathematics? Yes, please. After all, zero can be represented as the digit zero (the digit means the number with the value “0,” that is, absolutely nothing), or as some infinitesimally small quantity (that is, approaching zero, almost nothing, but still not nothing). Then there’s nothing stopping you from calmly dividing by “infinitesimally small.”

The illogicality and abstractness of operations with zero are not allowed within the narrow confines of algebra; more precisely, it’s an undefined operation. It requires a more serious apparatus—higher mathematics. So, in a way, you can’t divide by zero, but if you really want to, you can… But you need to be ready to understand things like the Dirac delta function and other hard-to-grasp concepts.

Divide to your heart’s content if you’re not afraid of infinity in the result.