# What is 0 divided by 0?

This is part of a series on common misconceptions.

What is \(0/0?\)

**Why some people say it's 0:** Zero divided by any number is 0.

**Why some people say it's 1:** A number divided by itself is 1.

**Why some people say it's undefined:** Dividing by zero is undefined.

The statement is \( \color{red}{\textbf{indeterminate}}\).

Proof:Let \( \dfrac{0}{0} = a \). Then \( a\) is "the" number satisfying the equation \( 0 = 0 \cdot a \). But

everyreal number \(a\) satisfies this equation. The quotient \( \dfrac{x}{y}\) is defined and unique as long as \( y \ne 0,\) but it is no longer well-defined when \( x=y=0.\)

Rebuttal: Any number divided by itself is \( 1.\)

Reply: This is true for any nonzero number, but dividing by \(0\) is not allowed.

Rebuttal: \( 0\) divided by any number is \( 0.\)

Reply: This is true for any nonzero denominator, but dividing by \( 0\) is not allowed no matter what the numerator is.

Rebuttal: Any number divided by \( 0\) is \( \infty.\)

Reply: Even for nonzero \( y,\) writing \( \dfrac{y}{0}=\infty\) is not entirely accurate: see 1/0 for a discussion. But this reasoning only makes sense for a nonzero numerator.

Rebuttal: If we choose to set \( 0/0=1,\) or \( 0,\) it is not inconsistent with other laws of arithmetic, and it makes one of the rules in the above rebuttals true in all cases.

Reply: This is a combination of the first two rebuttals, so here is a "big-picture" reply. Any specific choice of value for \( 0/0\) will allow some function to be extended continuously. For instance, if we mandate \( 0/0=1,\) then the function \( f(x) = x/x\) becomes continuous at \( x=0.\) If \(0/0=0,\) the the function \( f(x)=0/x\) becomes continuous at \( x=0.\)But this is not satisfactory in all cases, and the arbitrariness of the choice will break other laws of arithmetic. For instance, \[ \begin{align} \frac00 + \frac11 &= \frac{0\cdot 1+1\cdot 0}{0\cdot 1} \\ &= \frac00, \end{align} \] which doesn't make any sense for any (finite) choice of \( \frac00.\)

Here is the proof of \(\dfrac 00=2\):

**Step 1:**
\[\dfrac 00 =\dfrac{100-100}{100-100}\]

**Step 2:**
\[ =\dfrac{10^2-10^2}{10(10-10)}\]

**Step 3:**
\[ =\dfrac{(10+10)(10-10)}{10(10-10)}\]

**Step 4:**
\[= \dfrac{10+10}{10}\]

**Step 5:**
\[=2.\]

Since 0/0 is an indeterminate form, we can't equate it to any real number. Then which step of the above is faulty?

**See Also**

**Cite as:**What is 0 divided by 0?.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/what-is-0-0/