What is 1 divided by 0?
This is part of a series on common misconceptions.
True or False?
\(\frac10\) is undefined.
Why some people say it's true: Dividing by \( 0\) is not allowed.
Why some people say it's false: \(\frac10 = \infty.\)
Can you see which of these is the correct explanation?
The statement is \( \color{blue}{\textbf{true}}\).
Proof:
If \( \frac10 = r\) were a real number, then \( r\cdot 0 = 1,\) but this is impossible for any \( r.\) See division by zero for more details. \(_\square\)
There are some common responses to this logic, but they all have various flaws.
Rebuttal: In calculus, \(\frac10\) equals \( \infty.\)Reply: This statement is incorrect for two reasons. First, infinity is not a real number. The proof demonstrates that the quotient \(\frac10\) is undefined over the real numbers.
It is true that, in some situations, the indeterminate form \(\frac10\) can be interpreted as \( \infty:\) for instance, when taking limits of a quotient of functions. But even this is not always true, as the following example shows:
Consider \( \lim\limits_{x\to 0}\frac{1}{x}.\)
Approaching from the right, \( \lim\limits_{x \to 0^+} \frac{1}{x} = + \infty. \)
Approaching from the left, \( \lim\limits_{x \to 0^-} \frac{1}{x} = - \infty. \)In order for \( \frac{1}{0} \) to be consistent, the limits from both directions should be equal, which is clearly not the case here.
Rebuttal: What about on the Riemann sphere?
Reply: For certain complex functions, it is convenient and consistent to extend their domain and range to \( {\mathbb C} \cup \{\infty\}.\) This set has the geometric structure of a sphere, called the Riemann sphere. For instance, suppose \(a,b,c,d\) are complex numbers such that \( ad-bc\ne 0.\) Then the function \( f(z) = \frac{az+b}{cz+d} \) can be extended by defining \( f\left(-\frac dc\right) = \infty \) and \( f(\infty) = \frac ac \) \(\big(\)or \( f(\infty) = \infty \) when \(c=0\big).\) This makes \(f\) a bijection on the Riemann sphere, with many nice properties.
So there are situations where \(\frac10\) is defined, but they are defined in a tightly controlled way. It is still the case that \(\frac10\) can never be a real (or complex) number, so—strictly speaking—it is undefined.
See the consequences of assuming that \(\frac{1}{0}\) is defined for yourself in the following problem:
What is wrong with the following "proof"?
Let \(a = b=1\), then \(a+b=b.\)
- Step 1: \(a^2 = ab \)
- Step 2: \(a^2 - b^2 = ab - b^2 \)
- Step 3: \((a+b)(a-b) = b(a-b) \)
- Step 4: \(a+b= \dfrac{b(a-b)}{a-b} \)
- Step 5: \(a+b = b\)
Conclusion: By substituting in \( a = b = 1,\) we have \(1+1 = 1 \implies 2 = 1.\)
See Also
