Divide by zero!

Okay, time to be the one who destroys the conversation.

Division isn't really an operation that is defined on the real numbers normally (the real numbers are the numbers we use every day, like -80 or pi).
There are two main operations that are defined on the real numbers- Addition and Multiplication. Those two operations make the real numbers into a big, beautiful mathematicial object, called a "field" (That's why Wikipedia might refer it to "The Real Number Field").
Apparently, addition and multiplication aren't random- They have special properties, that make the real numbers into a field.
One of the properties is the existence of an inverse.

We all know that every number has an additive inverse- his "negative form". For 5, there's -5, for -2, there's 2. What's special about the inverse, is that every number plus its inverse always equals 0.
So, for subtraction, a-b is defined as a plus the inverse of b. For an example, 5-(-3) = 5 + 3 = 8 (because 3 is the additive inverse of -3).

Now, whenever there's an additive inverse, there's a multiplicative inverse- Every number times its multiplicative inverse equals 1.

Division, a/b, is defined by multiplying a times the inverse of b. For an example, 8/(7/3) = 8*(3/7) = 24/7, because the multiplicative inverse of 7/3 is 3/7.

But.. There's a problem.

We can't divide by zero.

Why? That's simple- When we try to divide 5 by 0 (5/0), we want to multiply 5 by the multiplicative inverse of 0.
But, there's no number, such that the number times 0 equals 1. Why that? Because every number times 0 equals 0. You can actually prove it.

The real numbers are not the only field in mathematics- There are much larger, much more complex ones, that have a very different meaning of "addition" and "multiplication". And, for EVERY single one of those, you can't divide by zero.


That was nerdy enough. I think I'll go now.
 
Ohad said:
Okay, time to be the one who destroys the conversation.

Division isn't really an operation that is defined on the real numbers normally (the real numbers are the numbers we use every day, like -80 or pi).
There are two main operations that are defined on the real numbers- Addition and Multiplication. Those two operations make the real numbers into a big, beautiful mathematicial object, called a "field" (That's why Wikipedia might refer it to "The Real Number Field").
Apparently, addition and multiplication aren't random- They have special properties, that make the real numbers into a field.
One of the properties is the existence of an inverse.

We all know that every number has an additive inverse- his "negative form". For 5, there's -5, for -2, there's 2. What's special about the inverse, is that every number plus its inverse always equals 0.
So, for subtraction, a-b is defined as a plus the inverse of b. For an example, 5-(-3) = 5 + 3 = 8 (because 3 is the additive inverse of -3).

Now, whenever there's an additive inverse, there's a multiplicative inverse- Every number times its multiplicative inverse equals 1.

Division, a/b, is defined by multiplying a times the inverse of b. For an example, 8/(7/3) = 8*(3/7) = 24/7, because the multiplicative inverse of 7/3 is 3/7.

But.. There's a problem.

We can't divide by zero.

Why? That's simple- When we try to divide 5 by 0 (5/0), we want to multiply 5 by the multiplicative inverse of 0.
But, there's no number, such that the number times 0 equals 1. Why that? Because every number times 0 equals 0. You can actually prove it.

The real numbers are not the only field in mathematics- There are much larger, much more complex ones, that have a very different meaning of "addition" and "multiplication". And, for EVERY single one of those, you can't divide by zero.


That was nerdy enough. I think I'll go now.
Click to expand...
So I'm not the only one who knew all of this.
I'm surprised more haven't picked up that subtraction is just "inverse addition" . :P
 
As far as I'm concerned, a number divided by zero is an infinite number.
 
Halibut said:
As far as I'm concerned, a number divided by zero is an infinite number.
Click to expand...
Well, 0 can go into 5 an infinite amount of times, so that logic makes sense.
However, to use Ohad's logic


2 x 1/2 = 1
3 x 1/3 = 1
1 x 1/1= 1 (obviously)
4 x 1/4 = 1
1/(1/2) x (1/2)/1 = 1
everything bolded is simply a reciprocal.
HOWEVER, for 0, there is more than one reciprocal
1/0, 2/0, 3/0, 4/0, 5/0, 6/0, etc.
and when it's done

0(6/0)
0 and 0 can't cancel out to be 1
so division by 0 is impossible
 
Smitty Werben Man Jensen said:
And the limit of sin(x)/x as x approaches 0 is 1.

Who would have thought?
Click to expand...
Graphs are a different story though. :P
I'm talking basic numbers my friend.
 
One of my teachers always taught it like this:

0/x = not doable or OFF
x/0 = No (n/0). 0 will work as the denominator but never numerator.

It's easier to understand if you write it down on paper.
 
Another way to think about it is-

When we try to define functions on numbers, we usually want them to "work well". The formal property is called being a continuous function, and you can think of a continuous function as a line that got bended down and up, and stretched to the sides, but didn't get "ripped" or broken.

For an example-

http://www.bbc.co.uk/schools/gcsebitesize/maths/images/graph_24.gif

(Sorry, it seems like I can't upload the pictures in here, so I'll post their links)
The graph of this function- x2- is like a straight line that someone pushed up in both sides, right? So, x2 is continuous.

https://encrypted-tbn3.gstatic.com/images?q=tbn:ANd9GcQNxT5a2iz0Pao8fDSgGkmtjFDjEVfC8MM4esddcT_hyP_vQ38fXQ


This function- sin(x)- Is a line that sometimes gets up, and sometimes down, like a wave. It's also continuous.

But, take a look at the function f(x)=1/x-

http://www.sagemath.org/calctut/pix-calctut/onesided05.png


This one.. Just doesn't seem as simple, is it?
Why is it? That's because 1/x is undefined at x=0, which means, 1/0 is undefined.
When we get close to zero from the positive side, the function gets bigger and bigger- But, when we get close to zero from the negative side, the function gets smaller and smaller.
There's no way to match a value for 1/0, such that the function will be simple and continuous. There's always going to be this strange disconnection in the middle.
Mathematicians have then decided to simply not give any value to 1/0, because choosing a value will make no sense when looking at the function. We can say "Hey, let's define 1/0 to be 5!" but it ruins the shape of the graph, and it ruins the properties of the function.


(By the way, I think that 1/x is the simplest function that still has very interesting and not trivial properties.)
 
I really don't think there are any exceptions. But I am questioning why I can't do 0/0, as 0x0=0.
 
More_Spongebob said:
I really don't think there are any exceptions. But I am questioning why I can't do 0/0, as 0x0=0.
Click to expand...
Because there is no reciprocal of 0, period. Division "technically" doesn't exist, rather, it's simply inverse multiplication. AKA, multiplying a number with a reciprocal.
For example, 10 divided by 2 is ACTUALLY 10 times 1 half.
There is no way to reciprocal the number 0 as there is no way to get 0*x to equal to 1 with any value of x. It is impossible, as any reciprocal is equal to one when multiplied with the other reciprocal.
Besides, with mathematical logic, 0x0 would technically be 1, but, 1x0 isn't the only way to get 0.
 
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