Relationships of Sets of Numbers

 

C

Complex Numbers

(formed by taking the set of real numbers and including the square roots of negative numbers)

R

Real Numbers

(The set of rational numbers along with numbers that can not be expressed as the ratio of two integers, the irrational numbers.)

These include positive square roots, e and pi.

Q Rational Numbers

(Taking the set of integers and including division of integers by non-zero elements.)

Z

Integers

(..., -2,-1,0,1,2,3...)

(Taking the set of natural numbers and including their inverses.)

N

Natural Numbers

(0,1,2,3,4...)

Z+

Positive Integers

(1,2,3,4,...)

 Zn

n Natural Numbers from 0 to n-1

(0,1,2,3,4..n-1)

derived from

a ~ b (mod n)

The set Zn  is used in modular arithmetic. For addition, the numbers are added and the remainder is recorded. For multiplication, the numbers are multiplied and their remainder is recorded. One of the interesting properties is that when n is prime, there is a unique inverse for each element in and each element appears once per row and once per column. This last property makes prime modular arithmetic useful in areas such as encryption and deriving pseudo-random numbers as well as in checksums. [Dates written as 1/5/98 with 98 as the 'year' are actually mod  Z100 functions: 1998~ 98 mod 100.]

Z5 + 0 1 2 3 4   Z5, * 0 1 2 3 4
0 0 1 2 3 4 0 0 0 0 0 0
1 1 2 3 4 0 1 0 1 2 3 4
2 2 3 4 0 1 2 0 2 4 1 3
3 3 4 0 1 2 3 0 3 1 4 2
4 4 0 1 2 3 4 0 4 3 2 1

Contrast with the non-prime Z4 (2*1 and 2*3 both equal 2, hence division by 2 is not unique.)
2*1 = 2*3, yet if we divide by 2, we get 1=3. 

NOTE, 1 and 3 have inverses since 1 is relatively prime to 4 as is 3.

Also note that for the 1 and 3 columns there are 4 unique numbers.

Z4,+ 0 1 2 3 Z4, * 0 1 2 3
0 0 1 2 3 0 0 0 0 0
1 1 2 3 0 1 0 1 2 3
2 2 3 0 1 2 0 2 0 2
3 3 0 1 2 3 0 3 2 1

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