Why is 1/0 infinity but 0/0 undefined?

October 10, 2014 at 9:34 pm | Posted in interesting maths, Mathematics | 1 Comment
Tags: , , , , , , , , , , , , ,

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:

x divided by infinity

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?

y=x

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

Keep reading. Keep sharing.

– Himanshu

Advertisements

Converting Miles to Kilometers – Fibonacci Numbers to the Rescue

October 9, 2014 at 12:45 am | Posted in Mathematics | Leave a comment
Tags: , , , , , , , , ,

MilestoneA lot of times we encounter a situation where we want to convert miles into kilometers. If we go by the formula where 1 mile = 1.609 kms, we will probably reach the actual conversion, but multiplying a number with 1.609 is not which everyone wants to do.

This is where Fibonacci series comes to the rescue. A Fibonacci series is a mathematical number series in which every number is a sum of two preceding numbers. For example:

1,1,2,3,5,8,13,21,34,55,89…….

In the above series, every number is the sum of preceding 2 numbers. We can assume an invisible zero before the first 1.

If we calculate the ratio between two consecutive terms, we find out that the ratio actually converges to 1.61803 after the first 14 terms.

Surprisingly this number is very close to the conversion ratio for miles to km conversion. If we consider anything between 1.6 – 1.65 as a decent conversion ratio for a back of the envelope conversion, we can safely say that once the series reaches 8, we consistently get the ratio between two consecutive terms to be well within the range.

What does that signify?

We have a very simple way of calculating the number of kilometers from miles (for kms > 5). Simply take the next value in Fibonacci series and that will be a good approximation in kilometers. 

For example if we need to convert 13 miles to kms, simply take the next number in the series i.e. 21. In actual, 13 miles = 20.917 kms. Not bad.

Let us take another example:

55 miles = ? kms

Simple, from Fibonacci series, it should be close to 89 kms. By actual formula, it is 88.495 kms. Less than 1% error again. Superb.

What if the number is not a part of the Fibonacci series. Simple again. We all know about distributive law in Mathematics:

a x (b + c) = a x b + a x c

This essentially means we can add different Fibonacci numbers to reach the number we want to convert. For example, if we want to convert 26 to Fibonacci series, 26 = 21 + 5 so we take the next numbers from the Fibonacci series for 5 & 21 and add them:

Kms (26 miles) = Number next to 5 + Number next to 21

Kms (26 miles) = 8 + 34 = 42

If we use the formula, we get 41.834. Again percentage error is less than 0.5%.

Lastly, we take a big number say 145 and see how can we apply our analysis there:

141 = 144 – 3

Kms(141 miles) = Number next to 144 – Number next to 3

Kms(141 miles) = 233 – 5 = 228

By formula; its 226.869

Isn’t it Cool.

Thanks for reading – Himanshu Joshi.

Create a free website or blog at WordPress.com.
Entries and comments feeds.

%d bloggers like this: