Prove that the real number 0 does not have a reciprocal.?

Question

(Hint: Assume b=1 over 0 is the reciprocal of 0. Supply a reason for each of the following steps.

1=0 * 1 over 0 as in 1/0.

=0*b

=0

Answer 1

Any real number times its reciprocal equals 1.

Ex:

2 * 1/2 = 1

However: 0 * 1/0 = 0, and therefore, it has no reciprocal.

Answer 2

In order for 0 to have a reciprocal, there must exist a

number b such that b =1/0 and

0*b = 1

Now

let us suppose this number exists.  

then 1 = 0(1/0)=0(b) by the definition of the reciprocal.

--------------------------------------...

Definition of reciprocal of a number h

If h is a number the definition of the reciprocal of h is

the number that can be multiplied by h to give us 1, the

identity number for multiplication

--------------------------------------...

Since 1=0*b

we  have

0*b = 0   since 0 times any number equals 0.   0 is the

identity number for addition and

0*(any integer) = 0+0+0.....+0 that number of times

and that is zero since 0 is identity for addition.

Moreover 0 divided by any nonzero number is 0 so

0*(any rational number) is 0

Thus 0*b =0

We therefore have

1=0*(1/0)=0*b=0  by the transitive property of equality

Consequently 1=0.

But this says the identity number for multiplication is

equal to the identity number for addition.   That is a contradiction to the axiom that the real numbers must have a

unique multiplicative identity that is different from the additive identity.    In common sense terms, we know

1 does not equal zero and it would have to, if 0 had a reciprocal.    This contradiction provides your proof, by

contradiction that 0 does not have a reciprocal.  

  I suspect what I've written need a little cleaning up and formatting to fit the forms of the axiums and theorems you

have learned up to date but this should be the structure for your proof

Answer 3

I think you more or less got it right.

Assume 0 has a reciprocal x

Then

x * 0 = 1

0 = 1

This is a contradiction, therefore our initial assumption is wrong.

So there is no x that is a reciprocal of 0

Proof by contradiction is actually used quite frequently in math.

Answer 4

Since the reciprocal function is defined as:

f(x) = 1/x

If x = 0, you're dividing by zero, which is undefined.

Another way to look at the defintion is:

The reciprocal of x is the number that when multiplied to x is 1.

if x = 0, there is no number that can cause this to become a 1 when multiplied.