About

A symmetric integer, formally called a palindromic number, reads identically from left to right and right to left. The property is defined over the decimal digit sequence of a non-negative integer n. For a number with digits d₁d₂…d_k, symmetry holds when d_i = d_k−i+1 for all i. Negative integers are never palindromic by convention because the minus sign has no mirrored counterpart. Single-digit numbers (0 - 9) are trivially symmetric. Misidentifying palindromic numbers matters in competitive programming, cryptographic seed validation, and certain checksum algorithms where digit reversal invariance is a required property.

This tool validates symmetry by reversing the digit string and performing an exact comparison. It handles arbitrary-length integers limited only by JavaScript's string processing. Note: leading zeros are not valid integer representations, so inputs like 0120 are treated as 120 after normalization. The check runs in O(k) time where k is the digit count.

Formulas

An integer n with decimal digit representation d₁d₂…d_k is symmetric (palindromic) if and only if:

d_i = d_{k − i + 1} for all i ∈ {1, 2, …, k2}

Equivalently, the reversed digit string rev(n) must satisfy:

rev(n) = n &implies; symmetric

Where d_i is the i-th digit from the left, k is the total number of digits, and rev denotes the string reversal function. For negative integers, the minus sign is not a digit, so n < 0 &implies; FALSE. The algorithm complexity is O(k) time and O(k) space.

Reference Data

Integer	Digits	Reversed	Symmetric?	Notes
0	1	0	Yes	Trivial single digit
7	1	7	Yes	All single digits are symmetric
11	2	11	Yes	Smallest 2-digit palindrome
12	2	21	No	-
121	3	121	Yes	Classic example
123	3	321	No	-
1001	4	1001	Yes	Even-length palindrome
1234	4	4321	No	-
12321	5	12321	Yes	Odd-length, center pivot at 3
1000001	7	1000001	Yes	Sparse palindrome with inner zeros
123454321	9	123454321	Yes	Ascending-descending pattern
1234567890	10	0987654321	No	Reversed form has leading zero
−121	3 + sign	121−	No	Negative integers are never symmetric
9	1	9	Yes	Largest single-digit palindrome
99	2	99	Yes	Repdigit - always palindromic
10	2	01	No	Trailing zero breaks symmetry
1111111	7	1111111	Yes	Repunit - always palindromic
12344321	8	12344321	Yes	Even-length mirror
100	3	001	No	Powers of 10 (except 1) are never symmetric
98789	5	98789	Yes	-

Frequently Asked Questions

No. By mathematical convention, the minus sign is not a digit. Reversing −121 produces 121−, which is not a valid number representation. Therefore all negative integers fail the symmetry check and return FALSE.

Leading zeros are stripped during normalization. An input like 00121 is treated as 121, which is then checked for symmetry. The number 0 itself is a valid single-digit palindrome.

The tool operates on string representations, not numeric types. This means it supports integers with hundreds or thousands of digits without overflow. JavaScript's Number type caps at 2⁵³ − 1, but string reversal has no such limit.

A number ending in 0 has d_k = 0, but its first digit d₁ cannot be 0 (since leading zeros are not valid). Therefore d₁ ≠ d_k, and symmetry fails immediately. The sole exception is the number 0 itself.

Yes. The count of palindromic numbers with exactly k digits is 9 × 10^{⌊(k−1)/2⌋}. For example, there are 9 single-digit palindromes (1 - 9), 9 two-digit ones (11, 22, …, 99), and 90 three-digit ones (101 - 999).

This tool checks symmetry in base-10 (decimal) only. A number can be palindromic in one base but not another. For example, 9 in decimal is 1001 in binary, which is palindromic in both. However, 10 in decimal is 1010 in binary - not palindromic in either.