Number Sense

Definition from Wikipedia – An intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations. It is very important to get a sense for numbers, that we encounter in everyday life, to understand the concept behind it very deeply. Take for instance the following line

1. World’s gold stock is about 170,000 metric tons.

All of us can understand the meaning of this line. But only very few of us can see what is behind 170,000 metric tons. Warren Buffet explanation for this from 2011 Annual Report.

Today the world’s gold stock is about 170,000 metric tons. If all of this gold were melded together, it would form a cube of about 68 feet per side. (Picture it fitting comfortably within a baseball infield.) At $1,750 per ounce – gold’s price as I write this – its value would be $9.6 trillion.

1 ton = 32000 ounce
170,000 metric ton = 5,440,000,000 ounces
Total value in $ = 5,440,000,000 ounces * $ 1750
Total value in $ = 9.52 trillion

There cannot be a better example of Number Sense. After reading this I learned the real value of having number sense.

2. U.S National Debt and Citizens

In the book ‘Innumeracy: Mathematical Illiteracy and Its Consequences – John Allen Paulos’ gives an explanation about understanding big numbers. Take for instance the US Public Debt is around $16.8 trillion. How would I get a feel for this? The estimated population of United States is 314 million. If every citizen of the United States pays $ 53,409($16.8 trillion/314 million) then we will not have any debt.

3. RSA Encryption and Universe Life Time

Two large prime numbers each 200 digits long when multiplied together generates a 400 digits long number. This is one part of the public key which is used for encrypting the data. To decrypt the message back you need to know the two large prime numbers which is kept private. If a hacker wants to decrypt the message he needs to find out the 2 prime numbers using the public key. How hard is this? The only way to do that is to check for each number up to the square root of the product. This is of the order 10 ^ 200.

N = P * Q
  = 10 ^ 200 * 10 ^ 200
  = 10 ^ (200 + 200)
  = 10 ^ 400

Check for the factor in (10 ^ 400/2) = 10 ^ 200

How long will this take for 1 computer to find this out.

Life time of the universe = 10 ^ 18 seconds
Imagine the computer does = 1 million calculations per second
Total calculations for the universe life time = (10 ^ 18) * 10 ^ 6
                                              = 10 ^ 24

For 10 ^ 200 it would have to run = (10 ^ 200)/(10 ^ 24)
                                  = 10 ^ 176 times the life of universe

Hence it is not easy for the hacker to decrypt the message. Even if you get a super computer we can simply increase the number of digits for P and Q.

4. CPU Clock and Speed of Light

Some time back I read in Udacity forums about how to think about CPU Clock cycles in terms of speed of light.

Clock speed of CPU = 2.7 Ghz(does 2.7 * 10 ^ 9 cycles in one second)
1 cycle is executed = 1/(2.7 * (10 ^ 9)) 
                    = 1/2.7 nanoseconds

Speed of Light = 29,979,245,800 centimeter / s
In 1 nanosecond light travels = 29.9 cms
In 1/2.7 nano seconds light travels = 29.9/2.7 = 11 cms 

For 1 CPU cycle light would have travelled 75% of a 6′ inch scale.

5. Stereo and Starbucks

In Economics it is taught that people are rational and they always think of Opportunity Costs when making purchases. But the reality is very different. In the course ‘A Beginner’s Guide to Irrational Behavior’ by Dan Ariely he explains the following. During the purchase of a Stereo system 2 choices were given.

  1. Sony    $700
  2. Pioneer $1,000 (Pioneer was of higher quality and hence it is priced higher)

People had hard time in choosing between the two and they ended up buying Pioneer. Since $300 is abstract they could not think about what can be done with that. But when given cues about what $300 can purchase for example 100 Mocha’s from Starbucks they decided to buy Sony. Relating number to concrete items can save a lot of money since it enables us to think in terms of Opportunity Costs.

What should I do

Numbers without context is useless. You need to put them in proper context and then they will start to make sense. In fact try to create contexts across disciplines.

One thought on “Number Sense

  1. Pingback: Public Key Cryptography | Seeking Wisdom

Comments are closed.