Here is given $A(x_1,y_1), B(x_1,y_2), C(x_2,y_3)$ and $D(x_3,y_3)$. I have recently read that, multiplication of two perpendicular lines is always $-1$.
From the above graph, the slope of $AB, m_1 = \frac{y_2-y_1}{x_1-x_1} = \frac{y_2-y_1}{0} = \infty$
and the slope of $CD, m_2 = \frac{y_3-y_3}{x_3-x_2} = \frac{0}{x_3-x_2} = 0$
And then we get $m_1m_2 = -1$
$\infty*0 = -1$
Here it is clearified that multiplication of infinity and zero always leads to $-1$ from the above formula. Then in which fact, do we strongly put emphasis on? $\infty*0 = -1$ or $\infty = \frac{-1}{0}$
If it indicates the 1st condition, then why $\infty*0$ would be only equal to $-1$. It could have been resulted in any real number. And how we define the infinity in this case?
Any kind of conception would be greatly helpful for me to remove my ignorance. Thanks in advance.
Best Answer
Good question! The trick is: infinity is not a number (in most frameworks).
When we write ∞, we're using it as a convenient shorthand for "what happens when this variable gets arbitrarily large". And if you try to use it as a number, things go horribly wrong.
When you ask about something like $0 \times \infty$, what you're usually asking is: "what happens to $0 \times k$, as $k$ gets arbitrarily large?" Using limit notation, we'd say "what is $\lim_{k \rightarrow \infty} 0 \times k$?"
And sometimes this can be answered: we can see that, no matter how big $k$ gets, $0 \times k$ is always zero. So $\lim_{k \rightarrow \infty} 0 \times k = 0$. Sometimes it can't. Suppose we instead asked about $\frac{1}{k} \times k^2$: the first part is going to zero, the second part is going to infinity. But this time, as $k$ gets bigger and bigger, the product just keeps getting bigger and bigger too. So in this case, our "$0 \times \infty$" ends up being infinite.
So the answer to $0 \times \infty$ is: "it depends how you get there". Infinity isn't a number, and it doesn't act like one: rather, it's a shorthand for "let's see what happens when we let this variable get arbitrarily large".