The Short Answer
No, dividing any number by 0 does not result in infinity. Attempting to divide a number by 0 is undefined mathematically. It does not have a numerical answer.
Explaining Division by Zero
Division is the inverse operation of multiplication. When we divide a number x by another number y, we are asking how many times y goes into x. For example:
| 6 ÷ 3 = 2 |
This means 3 goes into 6 two times. Division works smoothly when the divisor (the number we are dividing by) is not zero. But what happens when we attempt to divide by zero?
Let’s think through a few examples:
| 6 ÷ 0 = ? |
| 1 ÷ 0 = ? |
| 100 ÷ 0 = ? |
We cannot answer what 1 divided by 0 is. Or what 100 divided by 0 is. Because division asks “how many times does the divisor go into the dividend”, and zero goes into other numbers zero times, division by zero is undefined.
Some people think dividing by zero results in infinity. But infinity is not a real numerical value, it is a concept describing something without bound or end. Dividing by zero does not produce an infinitely large number, it simply has no numerical answer.
Why Division by Zero is Undefined
Mathematically, division by zero is left undefined for a few key reasons:
Divisor Cannot Be Zero
The fundamental definition of division requires the divisor to be a number that can divide into the dividend. Zero cannot divide into other numbers because it is zero itself.
Would Break Other Math Rules
If division by zero was defined as infinity, it would break other fundamental math rules like:
| a ÷ 0 = infinity |
| a × 0 = 0 (for any number a) |
This would mean infinity × 0 = 0, which is a contradiction.
Similarly, if a ÷ 0 = infinity for any a, then infinity ÷ 0 must also equal infinity. But infinity ÷ 0 cannot equal infinity × 0 = 0. This is a contradiction, so division by zero cannot equal infinity.
Losses Meaning of Division
As mentioned above, division asks how many times the divisor can go into the dividend. Defining division by zero as infinity would lose this fundamental meaning.
What Happens When You Attempt to Divide by Zero
Most calculators will display an error when you attempt to divide by zero. Some examples:
| 6 ÷ 0 on a calculator: Error |
| 1 ÷ 0 on a calculator: Error |
| 100 ÷ 0 on a calculator: Error |
In computer programs and code, attempting to divide by zero will often cause an application crash or exception. The program cannot execute the division, so it exits with an error.
In equations, any expression containing a division by zero is left undefined:
| x = 6 ÷ 0 |
| y = 3x + 2, where x = 6 ÷ 0 |
| Both x and y are undefined |
Special Cases
There are a few special cases when dividing numbers that initially appear to be “dividing by zero”, but are actually well-defined:
Dividing 0 by 0
| 0 ÷ 0 = ? |
This is also undefined, because 0 divided by anything is 0, but 0 divided by 0 has no clear value.
Limits Approaching 0
| lim x->0+ f(x) ÷ g(x) |
Dividing two functions f(x) and g(x) in a limit as x approaches 0 from the positive side is not the same as dividing by 0, and can be a finite number.
Indeterminate Forms
Expressions like:
| infinity ÷ infinity |
| 0 ÷ 0 |
| infinity – infinity |
Are called indeterminate forms. They do not immediately evaluate to a single number, but they are not necessarily undefined. Techniques like L’Hôpital’s rule can find limits for indeterminate forms.
Conclusion
Dividing by zero is undefined mathematically. While concepts like infinity are useful in things like limits, division by zero does not actually equal infinity. Attempting to divide any number by zero results in an error, not a single numerical value. Special cases like dividing zero by zero, or limits approaching division by zero, require special mathematical techniques – but they do not define what it means to divide by zero in general mathematics.