ℤ
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.
Cannot be expressed in p/q form where p,q ∈ ℤ
Note: π is irrational as it cannot be expressed in p/q form.
ℝ = rational numbers ∪ irrational numbers
Can be denoted by a point in the number line.
Numbers with imaginary components.
Denoted with ℂ.
Has the form
a + bi
where a,b ∈ ℝ and i² = -1
Denoted by the Hebrew letter aleph (א).
A number larger than all finite numbers.
Not necessarily infinity.
Transfinite cardinals
Transfinite ordinals
ω: Lowest transfinite ordinal 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.
A number that is not an algebraic number.
Eg: π, ℯ.
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) |
Any number (need not be prime) of the form 2ⁿ-1 for some n ∈ ℤ.
ie, Mersenne primes ⊂ Mersenne numbers.
A prime of the form 2ⁿ-1 for some n ∈ ℤ
Named after Marin Mersenne, a French mathematician who studied it.
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).
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