Dividing by zero would mean multiplying by the reciprocal of 0.
But 0 has no reciprocal (because 0 times any number is 0, not 1.)
Therefore, division by 0 has no meaning in the set of real numbers.
Multiplicative property of 0
If a is any real number,
then a0 = 0 and 0a = 0
Statement __________ Reason
0 = 0 + 0 ________ 1. Identity property of addition
a0 = a(0 + 0) _____ 2. Multiplication property of equality
a0 = a0 + a0 _____ 3. Distributive property of mult. with respect to add.
But a0 = a0 + 0 ___ 4. Identity property of addition
a0 + a0 = a0 + 0 __ 5. Transitive property of equality
a0 = 0 _________6. Subtraction property of equality
0a = 0 _________7. Commutative property of multiplication
Therefore, 0 times any number is 0, not 1.
(Source: Algebra: Structure and Method Book 1)
The two cases are:
Dividing a nonzero number by zero, violates the multiplicative property of zero and therefore the properties of the real numbers upon which it is proven, as shown above.
Dividing zero by zero, which does not violate the multiplicative property of zero, but multiplication by zero is an operation that can not be "undone."
The following argument is presented in an older edition of the above source:
If a/0 = c, then a = 0*c.
But 0*c = 0.
Hence, if a is not equal to 0,
no value of c can make the statement a = 0*c true,
while if a = 0,
every value of c will make the statement true.
Thus, a/0 either has no value or is indefinite in value.
But, there must be "purely formal" steps which are assumed to be valid in the above argument; namely, if given a/0 = c is an algebraic statement, how can you then derive a = 0*c?
a/0 = c
0*(a/0) = 0*c But in transforming an equation, we should never use zero as a multiplier since we know that the final equation will be satisfied by any real number.
Furthermore, to manipulate the left hand side: 0*(a/0),
implies you are writing 0*(a/0) as 0*(1/0)*a, this must be one of the "formal" steps
and finally asserting 0*(1/0) = 1.
But 0 has no reciprocal because 0 times any number is 0, not 1, this must be another "formal" step.
1*a = 0*c
Thus, arriving at the statement: If a/0 = c, then a = 0*c.
Here is a proof by contradiction:
If (1/0) = x, and x is the multiplicative inverse of 0,
0 * (1/0) = 0 * x (multiply both sides by 0. (if two numbers are equal, then to multiply them by the same number, they should still be equal)
1 = 0. (since 0 * 1/0 = 1 (left hand side) and 0 * any number = 0 (right hand side))
1=0 is absurd. So the assumption that (1/0) exists and is a real number, must have been wrong (proof by contradiction).
Division is not always possible in the system of numbers consisting of the integers (6 is divisible by 2 and 3 but not by 5), but in those cases where it is, the result is always uniquely determined.
In the system of all rational numbers (that is, the integers and fractions) division is not only unique but is always realizable with one exception—division by zero. On the basis of the definition of division given above, it is apparent that it is not possible to divide a number different from zero by zero.
The result of dividing zero by zero, according to the definition, can be any number since c*0 = 0 in all cases.
It is usually preferable in algebra (in order not to violate the uniqueness of division) to consider division by zero to be impossible for all cases. Thus division by zero is defined to be undefined.