my teacher asked this but i dont know why you cant divide by zero
why cant you divide by zero, wont it just be zero?
1)Let us try to use a nonzero number divided by 0 . For example, 5 divided by 0: 5/0=? so 0*?=5 Since any number multiplied by 0 equals to 0. Hence, there is no number that solves the equation. Hence, the value of a nonzero number divided by 0 remains undefined.
2)Let us try to do 0 divided by 0 . 0/0=? so 0*?=0, Any number multiplied by 0 equals to 0, so ? can be any number. Hence, the value of 0 divided by 0 is still undefined.
It is known that 0 times anything equals 0:
0 x 6 = 0
0 x 2 = 0
0 x 9999 = 0
With that said, for the sake of argument, let's assume we can divide each side by 0, and we get:
6 = 0/0
2 = 0/0
9999 = 0/0
So which is it?
Since dividing a constant by zero does not give you a single answer each time, it is accepted that you cannot divide by zero, and the answer is undefined.
Well, I explain it with money. Take a dollar. Divide it into piles of 50 cents, how many piles do you get?
So 1/0.5 = 2
Now, divide it into quarters, how many? Good, 4. Now we are getting somewhere.
1/0.25 = 4
Yes, let's speed this up a little bit. Divide it into pennies, what do you get? Yes, I know you aren't stupid, sorry if this line of questioning is a little condescending.
1/0.01 = 100
Now do you agree that if we had a coin that was really small, and you had to collect 100 of them to buy 1 penny, then you could make even more piles with that doller? And let's call that thing a puny. So 100 punies = 1 penny, and that 10,000 punies equals 1 dollar. Sooooooooo
1/0.0001 = 10,000
And we could keep going on, and dividing that number 1 by smaller and smaller numbers, and then I'm going to ask you to divide that dollar up into piles of zero.
How many piles do you get?
Without going to the unnecessary point of considering advanced mathematics (calculus and beyond), division is just asking the question "how many times do you have to add the divisor (the number in the denominator) to get the dividend (the number in the numerator)??"
For example, 6 ÷ 3 = 2 because you have to add 3 to itself two times in order to get 6.
Now consider 6 ÷ 0.... (or any other non-zero number divided by 0). How many times do you have to ADD zero to get 6? There is no valid answer - not even "infinity" works, because if you add zero to itself "infinity times" then you still end up with zero, not the number you are trying to get to.
Note that this is a very simplified answer that is intended to give a simple conceptual view, not a strict mathematical view. Specifically, it is NOT true that 0/0 = 1, despite the fact that you could add 0 to itself 1 time to get 0. You could argue that any answer is possible - that is, I can add 0 to itself 10 times and still get zero. Thus, 0/0 could be 1, or it could be 10, or pretty much any other number. In fact, that is exactly what happens, and you need more advanced math to determine the true answer in any particular case (it can be a different answer for different problems.)
To answer this, first let's consider some other cases.
6 divided by 6 = 1.
6 divided by 3 = 2.
6 divided by 2 = 3.
6 divided by 1 = 6.
Let us pretend these are apples. So in the first case, I am asking to divide 6 apples into 6 groups. How many apples are in each group? Well, only 1. If I put 1 apple into 6 different spots on my kitchen table, I will have 6 groups of apples.
I then take the 6 apples and divide them into 3 groups. When I divide them out over the kitchen table, I then find I have 2 apples in each group. So I have 3 groups of 2 apples.
For the third option and fourth option above, I am separating the apples into 2 groups of 3 and then just 1 group. So all 6 apples end up in just 1 group.
We then ask ourselves what is 6 divided by 0. I want to take my six apples and divide them into zero groups. Well... I can't just make the apples disappear. I am still stuck with 6 apples. But if even if I have all six apples in one spot on the table, there would still be 1 group of apples, not 0 groups of apples.
There is no way to evenly divide something up into 0 parts. You could say, I'll form parts that are smaller than "1"... say... 0.5 ... or 0.25 or 0.125 ... getting smaller and smaller like on a ruler when you are dividing up 1 inch into pieces. However, you will always be dividing it up into SOMETHING..
Of course you can divide by zero, and I'm glad you're willing to consider this question. The issue is in what context could it ever mean anything? Given the many answers included here, there are many examples where doing so doesn't make sense, but does this mean you can't 'ever' meaningfully divide by zero? No. I just don't know of any case where it would make any sense to do so. That's why mathematicians call it undefined.
And no it doesn't equal zero to divide by zero, I couldn't imagine in any possibly conceivable sense that dividing by something 'unrelated' would guarantee something completely 'unrelated'.... (sounds like magic actually, Haha).... Just suffice it to say if zero means "no amount", then what does dividing something by "no amount" mean? that's all it is. Later in mathematics it represents something much more interesting though.
This is a great way to point out that math is actually a language: numbers and groupings of numbers represent something, something conceivable, even at their most abstract.
Gady, you teacher asks a good question!
Imagine, if you decide to give something that you can separate easily. Got that? Now let's say that it can be separated easily, where each person can get one portion. This makes sense when you are dealing with whole persons right. But it does not make sense when there are zero persons. This is what Math teachers will call undefined. Take for example:
- if it is for two people, you divide that in half,
- if one person, you give the whole thing,
- so if for zero people, what do you do?
Well you can't give any portion, if you don't know what portion to give by dividing by zero and we know it can't be the whole thing or a portion of it. So they (the Math people of long ago) determined to agree this is undefined (not defined).
So, if anyone asks, what do you get when you divide by zero, just say the Math people said it is undefined or not defined. They may most likely say back, that is good or I could never do it either. That is when you can smile and laugh and breathe till they ask another question that stumps you.
Have a great Math day!!!
From Tai W (the Math guy from Modesto, California)
PS: disregard my comment above. I should have taken my time but thought there would have been an edit button after I clicked add comment. Did revise, herein, so here we go again as a Mentor note to myself and others (live a little and learn a little, hope I live a lot and learn a lot to share!). Learning, Math and Life should be fun and personal because it makes it easier to remember in a personal universe!
Another way to see why this is true: try dividing by a very small number. For example: 2/0.001 = 2000. Now make that denominator even smaller: 2/0.0001 = 20,000. And smaller: 2/0.0000001 = 20,000,000.
As you can probably tell by now, as the denominator gets smaller and shrinks towards zero, the quotient (the answer) grows larger, towards infinity. So technically speaking, you can say that dividing anything by zero = infinity, and that's "undefined" (can't put a number on it) in math.
You cannot divide by zero because zero has no multiplicative inverse: No real number multiplied by zero equals any number other than zero. This is clearer when you realize that any number multiplied by zero must be zero; therefore, you cannot divide by zero (it is undefined).
Here is a graphical way to think of it.
Suppose you wanted to divide the number 1 by 0. In other words, you want to find y = 1/x when x = 0. Well, try graphing the function y = 1/x. You will notice that to the left of x = 0, the value of y approaches negative infinity. Meanwhile to the right of x = 0, the value of y approaches positive infinity. Since 1/0 can't be both negative infinity and positive infinity, it is undefined.
Let's say that you have 20 apples, but you can't find anyone who wants them. Mathematically, the answer would 20/0. Dividing by zero is like saying that no one cares about the thing you are trying to divide. If no one cares, why bother?
Picking up where Kevin S. left off:
What he demonstrated was 0/0 is indeterminate (all real numbers are equally good values for this expression) so we call it indeterminate.
Dividing anything else like 2 or -7 by zero makes things worse and we call the answer "undefined". Why?
Assume there is a real number x such that x=a/0 where a is not zero. By definition of division, we have a=0x=0, a contradiction. Hence there is no such x.
You cannot divide by 0 because there is nothing to divide. Zero is nothing. Think logically - if there are 12 pieces in a Hershey bar and you are divying it up between six kids, then each will have 2 pieces. Thereby leaving nothing (0) left should another kid come and want some.
There are branches of mathematics where division by zero is defined as infinity. Those branches have severe limitations in solving most real world problems.
Let us define division by zero and see what happens.
X/0 = Y
Now if we accept that any number times 0 is 0, we have problem.
Therefore (5)0=(4)0, right?
But if dividing by zero is allowed, then
This is not a very useful result. To make consistent set of mathematical laws while defining division by zero, we have to throw out a ton of useful laws, such as defining multiplication by zero and the identity property of division: A/A=1. The benefit of defining zero does not outweigh the costs.
Depending on what level of mathematics you are using, the proper answer to the question would be to use limits. The idea of Limits is the precursor to Calculus.
The limit of X/Y as Y=> 0 would be infinity. If you simply imagine the value of Y to be smaller, and smaller, the value of X/Y becomes larger and larger, until when Y approaches 0, X/Y approaches infinity.
I think this is best approached this way:
If zero is the divisor then what dividend or value produces a result or quotient which is the number of times zero can be divided by it? Suppose the divisor is 1 then then quotient is equal to the dividend but for zero we cannot determine how many times a dividend can be divided by it so it is undefined.
Because it is not a divisor
Think of it tihs way:
If you have 15 / 3, you have 15 things and you are putting 3 things in a group. If you divide it this way, you will get 5 groups.
If you have 15 / 1, you have 15 things and you are putting only 1 thing in a group. If you divide it this way, you will get 15 groups.
If you have 15 / 0, you have 15 things, but you will HAVE to make at least 1 group because you already have stuff. It is impossible to divide something by 0.
You CAN do 0 / 0, because you have nothing.
You CAN do 0 / 15, because you have NOTHING right now, but IF you had something, you would put 15 things in the groups.
I hope that helps in the concetual thinking of not being able to divide by 0.