Some numbers and constants

Integers (Z)

ℤ

Rational numbers (Q)

Can be expressed in p/q form where p,q ∈ ℤ

Denoted with ℚ.

Note: 22/7 is rational as it is in p/q form.

Irrational numbers

Cannot be expressed in p/q form where p,q ∈ ℤ

Note: π is irrational as it cannot be expressed in p/q form.

Real numbers (R)

ℝ = rational numbers ∪ irrational numbers

Can be denoted by a point in the number line.

Complex numbers (C)

Numbers with imaginary components.

Denoted with ℂ.

Has the form

a + bi

where a,b ∈ ℝ and i² = -1

Perfect number

Transfinite number

Transfinite ordinals

ω: Lowest transfinite ordinal number.

Algebraic number

Can be expressed as the root of a non-zero polynomial equation in one variable with rational coefficients.

Therefore,

π is not algebraic.

But √2 is algebraic.

All integers (ℤ) and rational numbers (ℚ) are algebraic numbers.

Transcendental number

A number that is not an algebraic number.

Eg: π, ℯ.

Composite number

A positive integer (ie, natural number) that can be represented as the product of two smaller positive integers.

ie, a positive integer with at least one divisor other than itself and 1.

Every natural number is either a composite number or a prime or 1.

Composite Non-composite
4 1 (1)
6 2 (prime)
8 3 (prime)
9 5 (prime)
10 7 (prime)
12 11 (prime)

Mersenne numbers

Any number (need not be prime) of the form 2ⁿ-1 for some n ∈ ℤ.

ie, Mersenne primes ⊂ Mersenne numbers.

Mersenne primes

A prime of the form 2ⁿ-1 for some n ∈ ℤ

Named after Marin Mersenne, a French mathematician who studied it.

Twin primes

Prime numbers that differ by 2.

Examples:

As we go for larger numbers, twin primes get more sparse though.

American mathematician Thomas Nicely was working with twin primes when he discovered the infamous flaw in an early Pentium processor in 1994.

It is believed that there are infinitely many twin primes (Twin prime conjecture).

Lucas numbers

Very similar to Fibonacci numbers, but with different starting points.

    Lucas: 2  1  3  4  7  11 18 29 47 76 123
Fibonacci: 1  1  2  3  5  8  13 21 34 55 89

As is the case with Fibonacci numbers, ratio of adjacent Lucas numbers approach the Golden ratio (1.618033988749…).

| 1/2    |                0.5 |
| 3/1    |                  3 |
| 4/3    | 1.3333333333333333 |
| 7/4    |               1.75 |
| 11/7   | 1.5714285714285714 |
| 18/11  | 1.6363636363636365 |
| 29/18  | 1.6111111111111112 |
| 47/29  | 1.6206896551724137 |
| 76/47  | 1.6170212765957446 |
| 123/76 |  1.618421052631579 |

Sum of Fibonacci numbers which are two terms apart would give the Lucas number corresponding to the position between those two Fibonacci numbers.

As in

Fibonacci:   0  1  1  2  3  5  8  13 21 34                                      
             |     |        |     |                  
             |     |        |     |                  
             +--+--+        +--+--+                 
                |              |                   
                v              v                  
    Lucas:   2  1  3  4  7  11 18 29 47

Todo