why is 6 divided by zero, zero and zero divided by six is undefined?​

Answer:

See below.

Step-by-step explanation:

The reason a number divided by zero is undefined is as follows.

Every operation, addition, multiplication, subtraction, and division, is carried out between two numbers and must have a single correct answer.

For example 1 + 5 = 6. The only correct answer to 1 + 5 is 6. There is no other correct answer to that addition. The same is true of any multiplication, subtraction, or division. Another example: 8/2 = 4. No other correct solution exists to the division of 8 by 2 other than 4.

Division is defined in terms of multiplication.

The definition of dividing a number by another number is:

a divided by b equals c if and only if b * c = a.

For example, 10/5 = 2 if and only if 5 * 2 = 10. 5 * 2 does equal 10, so it is correct that 10/5 = 2.

Now let's try dividing by zero and see the problem that arises.

Let's try to divide 8 by 0.

8/0 =

Since we are not used to dividing by zero, we don't really know what the answer should be, but we can try different answers.

Let's say 8/0 = 8

Then by the definition of multiplication, we need 0 * 8 to equal 8, but we know that 0 * 8 = 0, so the answer of 8/0 cannot be 8. Let's try a different answer.

8/0 = 0, then we need 0 * 0 to equal 8, but 0 * 0 = 0, not 8, so 0 also does not work.

We can generalize and try

8/0 = k, but then that requires 0 * k to equal 8. Since multiplication by 0 always has a result of 0, there is no number k that will multiply by 0 and give 8. This means that a number divided by 0 cannot give a result.

Now let's try one more thing.

What about 0 divided by 0.

Let 0/0 = 1. Then 0 * 1 = 0. It seems to work, but here there is a different problem. We could also say that 0/0 = 2 since 0 * 2 = 0; 0/0 = 3 since 0 * 3 = 0, etc. and now 0/0 can equal every number. The result of a division must be a unique answer, not all numbers.

When we try to divide a non-zero number by 0, no number works in the definition of division, so dividing a non-zero number by 0 is undefined. When we try to divide 0 by 0, every number works. Since there is no unique solution, 0/0 is also undefined. Therefore, division by zero is undefined.

Now the last part of your question.

What about 0 divided by 6?

0/6 = 0 if and only if 6 * 0 = 0. Well, 0 * 6 does equal zero. There is no other number that you can use as the quotient of 0/6 and have the corresponding multiplication equal zero, so 0/6 is indeed 0, and that operation is defined.

0 divided by any number other than zero is a defined division and equals 0.

Conclusion:

(any number)/0 is undefined

0/(any number other than zero) is defined and equals zero

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