Why is 1/0 infinity but 0/0 undefined?October 10, 2014 at 9:34 pm | Posted in interesting maths, Mathematics | 1 Comment
Tags: 0/0, funny, himanshu joshi, india, infinite, infinity, interesting, madsadman, mathematics, maths, number system, numberline, weird, wierd
When we were kids, we were taught that anything divided by 0 is infinity i.e. a very big number that cannot be calculated. As kids, we took the liberty of taking 0/0 as infinity as well and no one corrected us. As long as there was no finite definition of x/0, everyone was happy and so were we.
The time changed, we grew up, started using calculators & spreadsheets and suddenly came across this difference. When you divide any number other than 0 by 0, your calculator returns ‘E’ or error but when you divide 0 by 0, it returns undefined. Why is it so? Is x/0 actually infinity?
Let us dig a bit deeper into why is there a discrepancy.
Let us take a very simple equation: xy=1
i.e. y = 1/x
So when x = 1, y=1 when x = 1/2, y = 2, when x = 1/4, y=4 and so on.
So (x,y) ∈ (1,1), (0.1,10), (0.01,100), (0.00001,10000)……… (1/10^n, 10^n)
Thus we see when n increases to extremely large numbers, limit(x)->0 but limit(y)-> ∞.
Thus we can say if something is divided by 0, the output is infinity(∞). But wait.
What if in our case, x < 0?
(x,y) ∈ (-1,-1),(-0.1,-10),(-0.01,-100)…..(-1/10^n, -10^n)
The result is same but in the other direction i.e. limit(x)->0, limit(y)-> -∞.
That essentially means, y is ±∞ depending on which side of the number line we approach it. So it might be okay to assume x/0 = ∞ for basic mathematics purposes but in reality it might not be true.
If we draw a graph of y = 1/x, it looks like:
If x/0 is infinity, then why is 0/0 undefined?
We approach this problem with simple two dimensional coordinate geometry.
We take a simple straight line y = x. This line has an intercept of 0 and slope of 1.
y = x i.e. y/x = 1;
Some of the points that lie on this line: (2,2),(1,1),(0,0),(-1,-1),(-2,-2)
Do you see something? If (0,0) lies on this line, (0,0) must satisfy the equation y/x = 1; i.e. 0/0 = 1?
Take another line y=2x i.e. y/x = 2;
Some of the points that lie on this line: (2,4),(1,2),(0,0),(-1,-2),(-2,-4)
Great. (0,0) lies on this line too i.e. 0/0 =2?
We all know that we can draw infinite number of straight lines through 0, so 0/0 changes its values infinite times.
Moreover, we have innumerable other curves that pass through (0,0) y=4x^2 for example. An infinite number of curves fulfilling such equations pass through (0,0), so 0/0 has another infinite set of values.
Total number of values for 0/0 = ∞ x ∞
Which value of 0/0 to pick among the so many values of 0, when we are asked?
Best answer: Undefined
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