Why Division by Zero Is Undefined

We start from a simple and true statement:

$2 + 2 = 4$

Now the question asks: "then divide it by 0". That means we are being tempted to compute

$\frac{4}{0} \;\; \text{or} \;\; \frac{2 + 2}{0}.$

1. What does division mean?

Division is defined as the inverse of multiplication:

For numbers, $a \div b = c$ means $a = b \cdot c$.

So, if we write

$\frac{4}{0} = x,$

by the definition of division this must mean

$4 = 0 \cdot x.$

But $0 \cdot x = 0$ for every real number $x$. There is no number $x$ such that $0 \cdot x = 4$.

Therefore, $4/0$ does not exist as a real (or usual) number. That’s why we say:

Division by zero is undefined.

2. Why can’t we just call it infinity?

Sometimes you might see expressions growing very large as a denominator gets close to 0, like

$\frac{4}{0.1} = 40, \quad \frac{4}{0.01} = 400, \quad \frac{4}{0.001} = 4000, \ldots$

As the denominator tends to 0 from the positive side, the value tends to grow without bound. In calculus, we might say a limit tends to infinity, but that does not mean we define

$\frac{4}{0} = \infty.$

If we tried to define $4/0 = \infty$, we would break the basic arithmetic rule

$\frac{a}{b} = c \iff a = b \cdot c.$

Because then we’d be forced to accept

$4 = 0 \cdot \infty,$

which makes no sense in standard arithmetic: $0 \cdot$ anything is still 0, not 4.

3. From $2+2=4$ to “divide by 0”

The true part:

• $2 + 2 = 4$

The problematic part:

• $\dfrac{2+2}{0}$ asks for a number $x$ such that

$$2 + 2 = 0 \cdot x,$$

which is impossible for the same reason as above.

So the correct conclusion is:

$2 + 2 = 4$ is fine.

$\dfrac{4}{0}$ or $\dfrac{2+2}{0}$ is not defined in standard arithmetic.

4. What the visualization shows

The visualization below lets you:

• Fix a numerator (like 4), and
• Let the denominator slide closer and closer to 0 from the positive and negative sides.

You will see the graph of the function

$f(x) = \frac{4}{x}$

blowing up towards large positive or negative values as $x\to 0$, but there is a vertical gap (an asymptote) at $x=0$. That visual gap represents the fact that the value at $x=0$ simply does not exist: division by zero is undefined, not a huge number and not infinity.