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

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

Keep reading. Keep sharing.

– Himanshu

## Converting Miles to Kilometers – Fibonacci Numbers to the Rescue

October 9, 2014 at 12:45 am | Posted in Mathematics | Leave a commentTags: fibonacci, fibonacci series, kilometers, kms, mathematics, maths, maths tricks, maths trics, miles, miles to kilometers

A 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.**

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