Why is dividing by zero = undefine.

Question

I know that $ a\cdot\left(\frac1a\right)=1$ as long as $a$ is not $= 0$,

but when we divide $(\frac a0)$ is $=$ we say it's undefined.

why is it really undefined? what's the big conspiracy here.

Answer 1

Here is a philosophical approach. When you want to work for a stereo worth 100 dollar , and you make 10 dollar an hour, you work ten hours, right? Now if you would work for 5 hours, you would have to sweat for 20 hours, ok? So now you are going to work for free. How many hours do you have to work to purchase that stereo?

Answer 2

$$ \frac{481}{13} = 37,\text{ since }13\cdot37=481. $$ $$ \frac50 = x\text{ since }x\cdot 0 = 5. $$

The point is that there's nothing you can multiply $0$ by to get $5$.

Answer 3

There is no conspiracy. Mathematics is purely axiomatic, which means that all statements follow from axioms which are assumed to be true. For example, if I take an axiom in my system to be: "The sun always shines," I will derive truths from this assumption (e.g. "I will never need to carry an umbrella") without doting on whether or not my assumptions are "justified" (in some practical sense).

That being said, our construction of the real numbers is very justified! Read below.

Let $b$ be a number such that $ba=1=ab$. As long as $a\neq0$ we can show that this number exists and is unique. This (along with the fact that no such $b$ exists for $a=0$) follows from extending the arithmetic operations $+$ and $\times$ from the rational numbers (all numbers of the form $a/b$ where $a$ and $b$ are integers) to the real numbers (the numbers you are used to; like $\pi$, $17$, $2/5$, etc.) when building the real numbers using, say, Dedekind cuts. For a reference, see an introduction to analysis book, such as Principles of Mathematical Analysis (http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X). We write $b\equiv\frac{1}{a}$ to denote this number, called the inverse of $a$.