Number terms in Mathematics

 There are several types of number terms in mathematics, each with unique properties. Here’s a list of common types:

1. Even and Odd Numbers

• Even Number: A number divisible by 2 (e.g., 2, 4, 6, 8).
• Odd Number: A number not divisible by 2 (e.g., 1, 3, 5, 7).

2. Prime and Composite Numbers

• Prime Number: A number greater than 1 with only two factors: 1 and itself (e.g., 2, 3, 5, 7, 11).
• Composite Number: A number with more than two factors (e.g., 4, 6, 8, 9).

3. Perfect Number

• A number that is equal to the sum of its proper divisors (excluding itself).
  • Example: 6 → Divisors (1, 2, 3) → $1 + 2 + 3 = 6$

4. Armstrong Number (Narcissistic Number)

• A number where the sum of its digits raised to the power of the number of digits equals the number itself.
  • Example: 153 → $1^3 + 5^3 + 3^3 = 153$

5. Factorial of a Number (n!)

• The product of all positive integers from 1 to n.
  • Example: 5! = 5 × 4 × 3 × 2 × 1 = 120

6. Palindrome Number

• A number that remains the same when its digits are reversed.
  • Example: 121, 1331, 45654

7. Perfect Square Number

• A number that is the square of an integer.
  • Example: 16 → $4^2$, 25 → $5^2$

8. Perfect Cube Number

• A number that is the cube of an integer.
  • Example: 27 → $3^3$, 64 → $4^3$

9. Fibonacci Number

• A number in the sequence where each term is the sum of the previous two terms.
  • Example: 0, 1, 1, 2, 3, 5, 8, 13, 21...

10. Harshad Number (Niven Number)

• A number divisible by the sum of its digits.
  • Example: 18 → $1 + 8 = 9$, $18 \div 9 = 2$ (Divisible)

11. Automorphic Number

• A number whose square ends with the same digits as the number.
  • Example: 25 → $25^2 = 625$ (Ends with 25)

12. Strong Number

• A number where the sum of the factorials of its digits equals the number itself.
  • Example: 145 → $1! + 4! + 5! = 1 + 24 + 120 = 145$

13. Happy Number

• A number that eventually becomes 1 when replaced by the sum of the squares of its digits repeatedly.
  • Example: 19 → $1^2 + 9^2 = 82$, $8^2 + 2^2 = 68$, $6^2 + 8^2 = 100$, $1^2 + 0^2 + 0^2 = 1$

14. Ugly Number

• A number whose prime factors are only 2, 3, and 5.
  • Example: 8, 12, 25

15. Kaprekar Number

• A number where the sum of the parts of its square equals the number.
  • Example: 45 → $45^2 = 2025$ → $20 + 25 = 45$

In the grand tapestry of numbers, each thread weaves a story of its own, echoing the timeless rhythms of the universe. Beyond the familiar even, odd, prime, and perfect numbers, there exists a realm of numerical wonders, each bearing unique properties and tales. Let us journey through this landscape, honoring the traditions of mathematics and the legacy of those who unveiled these mysteries.

16. Kaprekar Number

Named after the Indian mathematician D. R. Kaprekar, a Kaprekar number is a non-negative integer, the representation of whose square can be split into two parts that add up to the original number. For example, 45 is a Kaprekar number because 45² = 2025, and 20 + 25 = 45. citeturn0search10

17. Vampire Number

A number that can be factored into two integers, each containing half the number of digits of the original number, and these factors together contain all the digits of the original number, in any order, counting multiplicity. For example, 1260 is a vampire number because 1260 = 21 × 60, and together, 21 and 60 contain the digits 1, 2, 6, and 0. citeturn0search1

18. Narcissistic Number

Also known as a pluperfect digital invariant (PPDI), an Armstrong number, or a pluperfect number, it is a number that is the sum of its own digits each raised to the power of the number of digits. For example, 153 is a narcissistic number because 1³ + 5³ + 3³ = 153. citeturn0search1

19. Undulating Number

A number that has digits in an alternating pattern, such as 121, 232, or 5656. These numbers exhibit a wave-like digit pattern, reminiscent of the undulating motion. citeturn0search3

20. Pandigital Number

A number that contains each digit from 0 to 9 at least once. For example, 1023456789 is a pandigital number.

21. Cyclic Number

A number in which cyclic permutations of the digits are successive multiples of the number. The most famous example is 142857:

• 142857 × 1 = 142857
• 142857 × 2 = 285714
• 142857 × 3 = 428571
• 142857 × 4 = 571428
• 142857 × 5 = 714285
• 142857 × 6 = 857142

Each product is a cyclic permutation of 142857.

22. Sociable Numbers

A generalization of amicable numbers, sociable numbers form a cycle such that each number is the sum of the proper divisors of the previous number, eventually returning to the starting number. For example, 12496, 14288, 15472, 14536, and 14264 form a sociable 5-cycle. 

23. Weird Number

A number that is abundant (the sum of its proper divisors is greater than the number itself) but not semiperfect (no subset of its proper divisors sums to the number itself). The smallest weird number is 70.

24. Liouville Number

A type of transcendental number that can be approximated "too well" by rational numbers. These numbers are named after the French mathematician Joseph Liouville, who first proved their existence. 

25. Colombian Number

Also known as a self number, it is a number that cannot be generated by any other number through the process of adding the sum of its digits to the original number. For example, 20 is a self number because no number added to the sum of its digits results in 20. 

In the vast expanse of mathematics, these numbers stand as testaments to the intricate beauty and endless curiosity that numbers inspire. Each type, with its distinct characteristics, invites us to delve deeper into the mysteries of arithmetic, honoring the legacy of mathematicians who have illuminated these paths.

These are some of the most well-known types of numbers. Let me know if you need further details or examples! 🚀