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

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

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Why do we break traffic lights? – A Quantitative Analysis

January 4, 2014 at 12:05 am | Posted in india | Leave a comment
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India is my country and all Indians are my brothers and sisters. So this analysis believes in right to equality and assumes each of us to be equally prone to break a traffic signal. I am sure most of us would have broken a traffic signal atleast once in your lifetime or have seen someone breaking it.

Pb = Probability of breaking a traffic signal by you

Pc = Probability of getting caught when you break a signal

Pp = Probability that a policeman is standing on the traffic light to catch you

This analysis uses some basic approaches of game theory and Bayesian probability models. Please drop a comment if you have doubts in the analysis.

The Decision Tree

To start any analysis that involves probability, we should start with creation of the decision tree for various outcomes that can happen. If we draw a decision tree for a normal scenario:

Image

 

Here we assume some more things:

  • The payoffs for breaking a traffic signal are definitely more than those for not breaking a signal.
  • When a rider does not break a signal, he loses some time and we can define the losses in financial terms.
  • When a rider breaks a signal, he saves time thus he does not lose any money (arbitrary)
  • When he gets caught, he has to pay some fine and some more extra money to the policeman to get away.

 

Assigning Values to Variables:

We do not know how much the loss is, in terms of money, for the rider when he does not break a signal. We assume it to be Rs. 100 to start and then proceed with our calculations. Similarly the gains that he gets by breaking the signal is equal to Rs. 100/-

Number of vehicles at any signal: 10

Number of vehicles breaking signal: 2

Number of vehicles getting caught: 1

Probability of breaking a signal, Pb : 2/10 = 0.2

Probability of getting caught, Pc: 1/2 = 0.5

Probability that a policeman stands (we can safely assume that in India, we see a policeman 1 in 10 times), Pp : 1/10 = 0.1

Fine when the rider is caught: Rs. 500

 

Updating the Decision Tree:

Image

Next we add the payoffs / losses at various options and try to calculate the average payoffs at the various option nodes.

Image

 

All we are left to is to calculate the payoffs/losses at the nodes C, B & A and try to see if breaking a signal sounds more logical.

At node C:

Nett payoff = 0.5 * 100 + 0.5 * -500 = 50 – 250 = Rs. – 200/-

At node B:

Nett payoff = 0.1 * (-200) + 0.9*100 = -20 + 90 = Rs. 70/-

At node A:

Nett payoff = 0.2 * 70 + 0.8*(-100) = 14 – 80 = Rs. -66/-

This essentially means that if there is a policeman, it is more logical for the rider not to break the rule because average expected payoff (Rs. 100 * 0.5 = Rs. 50) is less than the average expected loss (Rs. 500 * 0.5 = Rs. 250). However, if there is no policeman, the payoffs for the user to break the red light are much higher (0.9 * 100 = 90). So when there is no policeman, it sounds better to him to break the rule because he is atleast getting Rs. 90/- out of his payoff of Rs. 100/- on an average.

Overall at node A, there is only one option through which he gets a positive payoff and that is by breaking the traffic signal. So it sounds more logical to the rider to break the rule. 

Next Steps:

If someone is interested in further analysing the behaviour using Game Theory of economics, he can approach the problem as under:

1. Change the payoffs and probabilities with real ones.

2. Calculate the threshold value of fine below which the rider will be tempted to break the law.

3. Calculate the optimum probability of a policeman standing on a traffic signal. 

Please contact me in case you need any suggestions on game theory or Bayesian probablistic models.

 

मज़ा लिया जाए – Ghazal

December 12, 2012 at 11:42 am | Posted in india | Leave a comment
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दिन है इतवार का मज़ा लिया जाए
थोडा इंतज़ार का मज़ा लिया जाए

जवानी का पतझड़ कब का गुज़र गया गया
बुढापे  की बहार बहार का मज़ा लिया जाए

कलम दवात मेरी पूरी ज़िन्दगी को खा गए
खंजरों की धार का मज़ा लिया जाए

बेईमान हो गए तो उनकी हैसियत बदल गई
फिर से इमानदार का मज़ा लिया जाए

पहली नज़र में ही उसे दिल दिया मैंने
मोहब्बत की इस हार का मज़ा लिया जाए

कभी कभी बच्चे भी बनके देख लो यारों
माँ के उस दुलार का मज़ा लिया जाए

घटनाएं रोज़ ही होती रहेंगी नया कुछ नहीं है
चलो पुराने अखबार का मज़ा लिया जाए

Pyar ho to aisa

June 6, 2010 at 6:17 pm | Posted in india, knowledge, poem | Leave a comment
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काली प्लेट पर सफ़ेद रसगुल्ला

प्यार हो तो ऐसा पूरा खुल्लम खुल्ला

गालों की लाली अब और लाल होगी

किसी से मिलने को वो बेहाल होगी

जो परछाई उनकी भी उनकी न होगी

तो बातें तो होंगी होगा हल्ला गुल्ला

प्यार हो तो ऐसा पूरा खुल्लम खुल्ला

सहेली से मिलने को बेचैन होगी

राहों में रखे हुए नैन होगी

जो यारों ने उनके की हंसी ठिठोली

तो भागेगी दांतों में दबा कर के पल्ला

प्यार हो तो ऐसे पूरा खुल्लम खुल्ला

ऐ ज़िन्दगी – Ae Zindagi

May 7, 2010 at 7:16 pm | Posted in india, poem | 1 Comment
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ऐ ज़िन्दगी

कभी कटती है कभी कटती ही नहीं,

ज़िन्दगी से मेरी पटती ही नहीं |

हर किसी के लिए खुलती सी जाती है,

सिर्फ अपने लिए सिमटती ही नहीं |

रोज़ कितनी बार मुझको मार देती है

कभी मेरे हाथों पिटती ही नहीं |

दिन रात इसकी बोली मैं घोंटता रहता हूँ

कभी मेरी बोली ये रटती ही नहीं |

जो जानता नहीं उसे ये मिल नहीं पाती

जीने वालों के लिए कभी घटती ही नहीं |

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