Division by zero

Division by Zero is an illegal operation, regardless of the value of the dividend. It has an undefined value, and therefore has no meaningful value. No conclusions can be drawn, and usually the operation if otherwise unnoticed will produce a contradictory or illogical result. No numeric or abstract concept exists that explains or justifies a division by zero, and therefore, if such an expression is reached, we can conclude that a mathematical error or false premise is to blame. Most calculators return an error comment, sometimes a small capitalized E.

Example of the fallacy

Given the obviously true statement:

$0 = 0$

Each side may be multiplied by an arbitrary value:

$0 \cdot 4 = 0 \cdot 3$

The above is still true, given the fact that zero multiplied by any number is still zero. This is known as the zero property of multiplication.

If division by zero were permissible:

$(0 \cdot 4) / 0 = (0 \cdot 3) / 0$

$4 = 3$

Simplifying by dropping out common factors, an obvious fallacy results. Any number can be made to equal any other number or expression via this error in algebra.

In math culture

Either as a joke or as a true misconception, some believe in the proof that 1=2 based on a trickery of algebraic manipulations

Define (as a premise):

$a = b = 1$

Therefore:

$a = b$

$a^2 = ab$

$a^2 - b^2 = ab - b^2$

$(a + b)\cdot(a - b) = (a - b)\cdot b$

$a + b = b$

$1 + 1 = 1$

$2 = 1$

(This statement is false, as proven below)

This, and proofs like it, all share the common algebraic error of a division by zero. As in the first example, above, zero equals zero, and from it any number can be made to equal any other number due to division by zero.

The transition from step four to step five involves the division by the term $(a - b)$.

$(a + b)\cdot(a - b) = (a - b)\cdot b$

$a + b = b$

The trick in the math lies here, as a division by zero is obscured with a seemingly innocent algebraic expression. If the first line of the proof, $a = b = 1$, is valid then $(a-b) = (1-1) = 0$. And so, division by $(a - b)$ IS a division by zero.

Reasoning for being undefined

Division can be thought of as being a multiplication of one input value by the multiplicative inverse of the other. That is, $\frac{x}{y}=x\cdot y^{-1}$. If zero had a multiplicative inverse (a number by which we can multiply by our given number to obtain one), then division by zero would be possible. However, we know that $0x = 0$ for all $x$, so zero does not have a multiplicative inverse, and thus $x\cdot 0^{-1}$ is meaningless.

We could define some constant to be 1/0 in the same way that mathematicians defined i to be sqrt(-1) but even that wont work because $1 = K_{1/0} \cdot 0 = K_{1/0} \cdot (0 + 0) = K_{1/0} \cdot 0 + K_{1/0} \cdot 0 = 1 + 1$ which is a contradiction. A contradiction is inevitable as long as we accept that 0+0=0. Zero is unique in that doubling it does not result in a number twice as big.