Palindrome Number Calculator
Check if a number is a palindrome by verifying if it reads the same forwards and backwards in different number bases.
Enter any non-negative integer to check if it is a palindrome
Check if the number is a palindrome in different number bases
A palindrome number is a number that remains the same when its digits are reversed. The concept depends on the base (radix) in which the number is represented.
Definition:
A number n is a palindrome in base b if its representation d₁d₂...dₖ equals dₖ...d₂d₁
where d₁, d₂, ..., dₖ are the digits of n in base b.
Algorithm:
- Convert the number to the selected base
- Reverse the digit string
- Compare the original and reversed strings
- If they match, the number is a palindrome in that base
Example (Decimal): Is 121 a palindrome in base 10?
- 121 in base 10 is "121"
- Reversed: "121"
- "121" = "121", so YES, it is a palindrome!
Example (Binary): Is 9 a palindrome in base 2?
- 9 in base 2 is "1001"
- Reversed: "1001"
- "1001" = "1001", so YES, it is a palindrome!
Example (Not a Palindrome): Is 123 a palindrome in base 10?
- 123 in base 10 is "123"
- Reversed: "321"
- "123" ≠ "321", so NO, it is not a palindrome
Interesting Multi-Base Palindromes:
- 585 is a palindrome in both base 10 (585) and base 2 (1001001001)
- 313 is a palindrome in base 10 (313) and base 8 (471... wait, not a palindrome!)
- All single-digit numbers (0-9) are palindromes in any base
The 196 Problem: An unsolved problem in mathematics asks whether the "reverse and add" process (repeatedly reversing a number and adding it to itself) always eventually produces a palindrome. The number 196 has been tested through billions of iterations without finding a palindrome, but no proof exists that it never will.
What is a palindrome number?
A palindrome number is a number that reads the same forwards and backwards when written in a particular base. For example, 121 is a palindrome in base 10 because it reads the same from left to right and right to left. Similarly, 9 (which is 1001 in binary) is a palindrome in base 2.
Can a number be a palindrome in one base but not another?
Yes! A number can be a palindrome in one base but not in another. For example, 21 in decimal is not a palindrome (21 reversed is 12), but in binary it is 10101, which is a palindrome. This makes checking palindromes in different bases an interesting mathematical exploration.
What are some examples of palindrome numbers in decimal?
Single-digit numbers (0-9) are all palindromes. Multi-digit palindromes include: 11, 22, 33, ..., 99 (two-digit), 101, 111, 121, 131, ..., 191, 202, ... (three-digit), and larger ones like 1001, 1111, 1221, 12321, 123321, etc.
Are there infinitely many palindrome numbers?
Yes, there are infinitely many palindrome numbers in any base. You can always construct larger palindromes by mirroring digits. For example, in base 10: 1, 11, 111, 1111, ... or 121, 12321, 1234321, ... The density of palindromes decreases as numbers get larger, but they never run out.
What is the mathematical significance of palindrome numbers?
Palindrome numbers appear in various areas of mathematics and computer science. They are studied in number theory, used in programming challenges, and have connections to other mathematical concepts. The 196 algorithm (repeatedly reversing and adding) is a famous unsolved problem related to palindromes.