# The Housekeeper and the Professor

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The Housekeeper and the Professor

“The Math”

Slide2What kind of math is this?

This does not sound like the things we learned in school…The professor studied a branch of mathematics called…

Slide3Number 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.

Slide4History 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.

Slide572

70+260+12To a computer in binary code it is 1001000 or (6 dozen donuts) (3 cases of water for today’s talk)

= 6336

Slide6

Alternative Method of Subtraction

1

3

Slide7Have I seen this already in math class?

Primes A

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).

Slide8The primes

Slide9Used 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.

Slide10

Cryptography

One way primes play a major role in our everyday lives is with cryptography. They are used in secure transactions including ATM transactions Credit Card purchases

Secure e-mail

Slide11Let’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).

Slide12437 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.

Slide1338

7

=114,415,582,592

114,415,582,592 mod

437

is equivalent to 57

57 is the encrypted message!

Slide1457

283

=

81768146863498022338255639240563119665791075627862538799988651475734718859323751885572984428719002934856700633200983280485385620283507818764947593610953828020067155484078437653125585210700763161411110846430900969733128286030685899648754314369417528321362592298408077012799967269151675453412261915696311938657948949672511368260809533929683970004795042604166413601260060130911443566095563008993183206577106153855620032227752857189207192861162639953043584922980409574449551446610074416205506731745193

57

283

mod

437

is equivalent to 38

38 is last part of the safe combination!

Slide15How 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??

Slide16In 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!

Slide17Note:

Ronald Rivest, Adi Shamir, and Leonard

A

dleman 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.

Slide18The imaginary number

i

Slide19i

– The imaginary number

Knowing about imaginary numbers helps us understand real numbers better.

Electricity application

Fractals

Applications: model of lungs, antenna, fractal art

Slide20Sum 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

Slide21If you were to describe math with one word…

Relationships

Slide22
## The Housekeeper and the Professor

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