Mathematics

# Division by zero is it really Impossible – Yes

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As an educator specializing in mathematics I often hear the question about whether division by zero is possible. Some of the comments about division by zero is, "the world blows up when trying to divide by zero". That is obviously absurd but dividing by zero is an absurd idea in and of itself.

Division is the opposite of multiplication just as subtraction is the opposite of addition. What I mean by opposite is that multiplcation will "undo" division and addition "undoes" subtraction. For example, 5+1=6, to "undo" the addition of 1, subtract 1 from 6 to get 5. For division and multiplication the same idea holds true. Take 15 and divide it by 3 to get 5. To "undo" the division take 5 and multiply by 3 to get the original 15. So I'll use the same concept to try and define a division by zero. Let's take 6 times 0, which is obviously 0. To "undo" the multiplication problem we should be able to take the 0 we got in the answer and divide it by 0 and get 6. Does 0 divided by 0 equal 6? Obviously not, in fact it is said to be undefined. If you do that problem on a calculator you get and "error" message.

Think of a pie and you want to divide the whole pie into 4 equal parts. Now 6 parts, 8 parts, etc. Each piece of pie gets smaller but no matter how many pieces you want to divide the pie into, technically that can always be done, albeit very difficult as the number of pieces gets larger. This is the same concept of the limit. As the number of pieces gets larger, each slice gets smaller. So for 1/n (1 representing the whole pie and n being the number of slices), as n gets larger each piece of pie gets smaller. Take the problem 1/n, as n gets smaller now and approaches 0, 1/n gets larger! Yes this is true. Say n is 1, then 1/1 = 1. When n= 1/2, 1/n = 2. When n= 1/100, 1/n = 100. When n= 1/10000, 1/n = 10000. What is happening here? 1/n is getting larger and larger as n approaches 0, so the limit as n approaches zero of 1/n is infinity. That would seem to indicate that 1 divided by 0 would be infinity as well, but that defies the rules of division and multiplication I mentioned above. The fact is, that n can NEVER actually EQUAL zero in the problem 1/n, it can just approach zero, get so infinitesmally small but never EQUAL zero. Division by zero is simply impossible.

Finally, think of this when considering division in general and division by zero. Take a pie and divide it into 1 part. That is obviously the entire pie. If someone says, "divide that pie into 0 parts". What would you do? You'd give the person a baffled look and say that a pie cannot be divided into zero parts, which is the same idea as dividing any number by zero. No the world wont "blow up when dividing by zero", nor can any number be divided by zero. It's simply impossible.

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