Sets of Numbers
1.1 A Prime number is a number that has exactly two factors: 1 and itself.
The prime numbers are the atoms of our universe of whole numbers: every positive whole number can be expressed in a unique way by multiplying together prime numbers
Example 50 = 2 * 5 * 5
Significance of Prime Numbers in Cryptography
Several computer schemes are based on the fact that finding prime factors of a large number takes impossibly large amount of computer time, whereas multiplying prime numbers is relatively easy. The best known algorithm is RSA (Rivest-Shamir-Adleman) public key encryption.
For example, a code maker chooses two large primes p and q of about 400 decimal digits each. Procedures for finding such primes require very little computer time. The primes are kept secret but their products n =p, an 800-digit number is made public. For this reason, these are often called public key code. Messages are coded by a method that requires only the knowledge of n. But to decode, both factors p and q must be know. With the most efficient computer techniquies devised, factoring an 800-digit number require billios of years on a single computer. For this reason, the codes are considered unbreakable, at least wit the current state of knowledge on factoring large numbers
2. Complex Numbers
Polar Representatoin of a Complex Number
Formula Demoivre
This theorem was published in 1707 by Abraham De Moivre, a French mathematician working in London.
Nth Root of a Complex Number
where k = 0, 1, 2, 3, 4
Example find all cube roots of 8
Note there are three roots on a circle of radius 2 centered at the origin. They are 120° apart
There are three representation of a complex number z = x +iy
(a) as a point P (x,y) in the xy plane
(b) as a vector OP from the origin to the point P
(c) as a vector that is of the same length and same direction as OP