Why can you never divide by zero?

Answer 1

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.

Multiplicative property of 0

Prove: If a is any real number, then a0 = 0 and 0a = 0

Proof:

Statement _____________ Reason

1. 0 = 0 + 0 ____________ 1. Identity property of addition

2. a0 = a(0 + 0) _________ 2. Multiplication property of equality

3. a0 = a0 + a0 _________ 3. Distributive property of mult. with respect to add.

4. But a0 = a0 + 0 _______ 4. Identity property of addition

5. a0 + a0 = a0 + 0 ______ 5. Substitution principle

6. a0 = 0 ______________ 6. Subtraction property of equality

7. 0a = 0 ______________ 7. Commutative property of multiplication

Therefore, division by zero has no meaning in the set of real numbers.

(Source: Algebra: Structure and Method Book 1)

In my opinion, the answer to this question depends upon which definition of division you are using

ie. the Algebraic definition of division as the multiplication by a reciprocal

or

the Arithmetic definition of division as a/b = c because a = bc.

Any discussion about division by zero must however centre around the above proof which is based on the properties of the real numbers.

The two cases are:

1. Dividing a nonzero number by zero, which does violate the multiplicative property of zero and therefore the properties of the real numbers upon which it is proven, as shown above.

2. 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."

a/b = c is defined by a = b*c.

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 statment 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.

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.

In mathematics the art of asking questions is more valuable than solving problems.