The Housekeeper and the Professor“The Math”

What kind of math is this?• This does not sound like the things

we learned in school…• The professor studied a branch of

mathematics called…

Number Theory• Number theory is the branch of pure

mathematics concerned with the properties of numbers in general, and integers in particular.

• The professor studied elementary number theory. Integers are studied without use of techniques from other mathematical fields.

History of number theory• Most area of mathematics go from

inception to maturity within at most a century. But in number theory there are questions that were formulated more than 2000 years ago.

• Long considered a rather esoteric branch of mathematics, number theory has in recent years grown in practical importance through its use in areas such as coding theory, cryptography, and statistical mechanics.

72• 70+2• 60+12• To a computer in binary code it is 1001000 or

• (6 dozen donuts)• (3 cases of water for today’s talk)• = 6336

•

258

5332

2257232

22

2323

)10()20()40()81()160()320()641(

126243

646400)880)(880(8872

Alternative Method of Subtraction

448227

1

3

Have I seen this already in math class?

• PrimesA prime number is a whole number greater than 1, whose only two whole-number factors are 1 and itself. 

-They are the building blocks of our natural numbers (either numbers are primes or they are the product of primes).

The primes

– Used in identifying Least Common Denominators (LCD) which is needed for solving equations or adding fractions

– Simplifying square roots

There are several advanced uses as well.

Cryptography • One way primes play a major role in our

everyday lives is with cryptography. – They are used in secure transactions including

ATM transactionsCredit Card purchasesSecure e-mail

Let’s try encrypting a message!

I am a spy that needs to crack a safe. In order to open the safe, I need to receive a message from a double agent. I desperately need to know from him the last number to the combination of a safe.

I decide to use RSA encryption to request my information. I can announce to the “world” two public key codes that need to be used to encrypt or disguise the message (I will keep the decoder key to myself).

437 and 7

• These are my public key codes. Remember… I am keeping the decoder 283 to myself!

• The last number to the safe combination is 38. This is the message the double agent wishes to send me.

387=114,415,582,592114,415,582,592 mod 437

is equivalent to 57

57 is the encrypted message!

57283

=81768146863498022338255639240563119665791075627862538799988651475734718859323751885572984428719002934856700633200983280485385620283507818764947593610953828020067155484078437653125585210700763161411110846430900969733128286030685899648754314369417528321362592298408077012799967269151675453412261915696311938657948949672511368260809533929683970004795042604166413601260060130911443566095563008993183206577106153855620032227752857189207192861162639953043584922980409574449551446610074416205506731745193

57283 mod 437is equivalent to 38

38 is last part of the safe combination!

How could this work?

• The first part of the process is multiplying two prime numbers.

• Look at one of our public keys 437. This is the result of multiplying two prime numbers?

• Can you figure out which ones??

• In reality the two primes that are used are 200 digits or more. This makes it very hard for others to figure out our original primes (factors) and break our code. If anyone knew your original 2 primes or the decoder key…your encryption is useless.

• Computers are great at multiplying numbers but not so fast at finding factors!

Note:

Ronald Rivest, Adi Shamir, and Leonard Adleman discovered the public key coding scheme.

RSA encryption has become widely accepted commercially. In July of 1996, RSA Data Security, the company formed to promote and sell RSA solutions, was sold for about $400 million to Security Dynamics.

The imaginary number

i1

i – The imaginary number

• Knowing about imaginary numbers helps us understand real numbers better.

• Electricity application• Fractals

– Applications: model of lungs, antenna, fractal art

Sum of the numbers 1 - 100• Yet another way!

1+2+3+ … +100 =1+100 +2+99 + … + 50 +51

commutative law

= (1+100) +(2+99) +… +(50+51) associative law

=101 +101+ … +101= 50 x 101 multiplication is repeated addition

=5050

If you were to describe math with one word…

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