11/06/2015
Tutorial Subject – Mathematics
Tutorial number #01
Infinity and related concepts
What actually is infinity? Often come across the term infinity (∞) in mathematics.
For example: tan 90 = ∞
This brings us to a very important question. What does infinity mean?
Infinity is not a real number. It is the concept of an unimaginably large number. Take the following example:
Let x = a/b, where ‘a’ and ‘b’ are real numbers.
Now, as ‘b’ grows smaller and smaller, the value of ‘x’ becomes larger and larger. Let us consider the value of ‘x’ as ‘b’ tends towards 0. The value of ‘x’ tends towards an infinitely large number we term as infinity. What we usually term to be ‘infinity’ is actually either ‘undefined’ or ‘indeterminate’.
Why division by zero is not allowed: The difference between undefined and indeterminate.
Let us assume for the sake of argument that there exist two real numbers ‘a’ and ‘b’ such that:
a/0 = b, a ≠ 0.
=> a = 0.b
=> a = 0 for any real number b, which is a contradiction. Hence, the form a/0 is undefined when a≠ 0.
Now, we assume that a = 0. Then this implies that b.0 = 0, which holds identically for any real ‘b’ (1.0 = 0, 2.0 = 0 … and so on). As a result, the value of ‘b’ cannot be determined. And thus, division by zero is not allowed.
As such, the form “x = 0/0” is ‘indeterminate.’
A few more examples:
Division by zero is an operation for which you cannot find an answer, so it is disallowed. You can understand why if you think about how division and multiplication are related.
12 divided by 6 is 2 because
6 times 2 is 12
12 divided by 0 is x would mean that
0 times x = 12
But no value would work for ‘x’ because 0 times any number is 0. So division by zero doesn't work.
Another way to look at it is to imagine that there is a magic coin that takes up no space at all. How many such coins can you put into your coin box? Some people may say that the answer is “infinite”. But the real answer is – THERE IS NO ANSWER!
Now we consider the tan 90 problem.
tan 90 = sin 90 / cos 90 = 1/0, which is undefined.
Hope you enjoyed it, more are on their way!
