Please explain why division by zero is undefined and yet it is possible to divide zero by a nonzero number?

Question

Can someone help me with this problem?:

Explain why division by zero is undefined and yet it is possible to divide zero by a nonzero number. (HINT: This involves the definition of a reciprocal.)

Answer 1

Think about it this way, you must be able to reverse your steps...

If you divide 6 by 3 you get 2, so to reverse your steps, 2 times 3 must equal 6 (which good for us, it does!)

Now say you divide that 6 by 0 then no matter what answer you get, to reverse your steps you have to multiply that number by 0 and get the 6 back. But, no matter what number you try, you multiply the 0 and get 0 and can never get the 6 back. This is to say no number exists for 6 divided by 0 (or any number you choose to begin with though 0 is a bit trickier to explain) since no number times 0 equals 6.

The other way does work because if you divide 0 by any nonzero number, you get 0. You multiply any nonzero by 0 and you get 0.

If you are wondering what is wrong with 0/0 = 0, the issue is uniqueness. Even though you can reverse your steps there, there is nothing wrong with saying 0/0 = 1, 0/0 = 17, 0/0 = pi... all of these work for the reversing your steps part, but since no single solution exists for 0/0 it is undefined.

Hope it helped

Answer 2

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RE:
Please explain why division by zero is undefined and yet it is possible to divide zero by a nonzero number?
Can someone help me with this problem?:

Explain why division by zero is undefined and yet it is possible to divide zero by a nonzero number. (HINT: This involves the definition of a reciprocal.)

Answer 3

First define division.

Division is a process which gives you an answer which when multiplied with the divisor gives the divident.

So you can easily see that division of zero by a non zero number is zero, because multiplication of any number (divisor) by zero (result of division) will yield zero (divident).

If you cant digest how multiplying a number by zero yields a zero, imagine you are multiplying a number by fractions like 1/2, 1/4, 1/100, 1/1000 etc. As the fraction goes on decreasing, your result goes on decreasing and as your fraction tends to zero you finally get a zero.

Next consider the case of division. Imagine you are dividing a number by fractions like 1/10, 1/100, 1/1000 etc. You find that your result goes enormously large as fraction goes on decreasing, and you are not tending to any finite value as you near to zero, and what you get is infinity, which is not a defined value. In other words multiplying infinity with your divisor, zero, doesnt give your divident. Hence division by zero is not defined.

Even though division by zero is not defined, division by quantities that tend to zero are quite useful and widely employed in mathematics.

For example, Consider this:

The quantity 2/(1/x) as x tends to zero yields a zero.

It is because 1/x tends to infinity as x tends to zero as we have seen above, and hence 2/infinity is too small and tends to zero, and is hence zero.

Answer 4

division (formally) is multiplication by the reciprocal, and reciprocals always have a product of 1. But 0 times anything is always 0, so it cannot have a reciprocal, so we cannot divide by it.

More intuitively, if I have 6 pennies and divide them equally into 3 piles, I get 2 in each pile [ 6 ÷ 3 = 2]. If I have 0 pennies and divide them into 3 piles, each pile will have 0 pennies. [ 0 ÷ 3 = 0] It isn't quite nonsense. But asking how many pennies end up in each pile if I divide 6 pennies up into 0 piles is clearly nonsense. No number of pennies in each pile makes any sense if I don't have any piles, you know?

Answer 5

division by zero is like saying... how many times can i fit zero into 5??!? the answer is an INFINITE or UNDEFINED amount of times... whereas if i say, how many times does 5 go into zero? the answer is simply it DOESNT. thus ZERO times. hope this helps

Answer 6

Division of or by Zero will always be Zero. ~