1. Introduction to Prime Numbers
Prime numbers are a special set of numbers in mathematics. They are the building blocks of all other numbers. Just like atoms build all matter, prime numbers build all other numbers through multiplication.
Understanding prime numbers helps in topics like factors, fractions, algebra, and even computer security.
2. Definition of Prime Numbers
A prime number is a number greater than 1 that has exactly two factors:
- 1
- the number itself
It cannot be divided evenly by any other number.
Examples:
- 2 → factors: 1, 2 → prime
- 5 → factors: 1, 5 → prime
- 11 → factors: 1, 11 → prime
3. Why 1 Is NOT a Prime Number
Prime numbers must have two distinct factors.
The number 1 has only one factor: 1.
Therefore, it is not considered a prime number.
4. Smallest Prime Number
The smallest prime number is 2.
It is also the only even prime number, because all other even numbers can be divided by 2.
5. Identifying Prime Numbers
To check if a number is prime, we try dividing it by smaller numbers.
5.1. Method 1: Trial Division
Check if the number is divisible by any number other than 1 and itself.
Example: Is 13 prime?
- Try dividing by 2 → no
- Try dividing by 3 → no
- Try dividing by 4 → no
Since no smaller number divides 13, it is prime.
5.2. Method 2: Square Root Method
You only need to check divisibility up to \(\sqrt{n}\).
Example: To check 29, calculate \(\sqrt{29} \approx 5.3\). So check only 2, 3, 5.
6. List of Prime Numbers
Here are the first few prime numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
7. Properties of Prime Numbers
Prime numbers have several important properties.
7.1. Unique Prime Factorization
Every whole number greater than 1 can be written as a product of prime numbers. This is called the Fundamental Theorem of Arithmetic.
Examples:
- 12 = 2 × 2 × 3
- 45 = 3 × 3 × 5
7.2. Infinitely Many Primes
There is no largest prime number. Mathematicians have proven that primes go on forever.
7.3. 2 Is the Only Even Prime
All other even numbers can be divided by 2, so they cannot be prime.
8. Prime Numbers in Real Life
Prime numbers are used in many modern technologies.
8.1. Cryptography (Security)
Prime numbers are used to protect data, passwords, and online banking through encryption.
8.2. Computer Algorithms
Search engines, programming, and hashing use primes for efficient processing.
8.3. Mathematical Modeling
Prime numbers help in studying patterns and sequences.
9. Difference Between Prime and Composite Numbers
Prime numbers have only 2 factors, but composite numbers have more than 2 factors.
9.1. Comparison Table
| Prime Numbers | Composite Numbers |
|---|---|
| Exactly 2 factors | More than 2 factors |
| Examples: 2, 3, 5, 7 | Examples: 4, 6, 9, 12 |