Counting Non-Palindromic Four-Digit Numbers Using Python and Mathematics

Palindromic numbers are fascinating mathematical curiosities. A palindrome number reads the same backward as forward, such as 83438. However, not all four-digit numbers are palindromic, and in this article, we explore the intriguing world of four-digit non-palindromic numbers.

Understanding Palindromic and Non-Palindromic Four-Digit Numbers

A four-digit palindromic number has the form ABBA, where A and B are digits. Specifically:

A can be any digit from 1 to 9 (9 options). B can be any digit from 0 to 9 (10 options).

Therefore, the total number of four-digit palindromic numbers is:

9 times 10  90

Total Four-Digit Numbers and Non-Palindromic Numbers

The smallest four-digit number is 1000 and the largest is 9999. So, the total number of four-digit numbers is:

9999 - 1000   1  9000

By subtracting the number of palindromic numbers from the total number of four-digit numbers, we can find the number of non-palindromic numbers:

9000 - 90  8910

To illustrate this, let's write a simple Python program that counts the number of four-digit non-palindromic numbers.

Python Program for Counting Non-Palindromic Four-Digit Numbers

The following Python code snippet can be used to find the number of non-palindromic four-digit numbers:

print(len([i for i in range(1000, 10000) if str(i) ! str(i)[::-1]]))

This program iterates through all the four-digit numbers (from 1000 to 9999), checks if a number is not a palindrome, and counts the number of such numbers. The result is 8910.

Mathematical Verification

We can also verify this by a more direct mathematical approach. The total number of four-digit numbers without leading zeros is 9000 (from 1000 to 9999). The number of palindromic four-digit numbers is 90 as calculated earlier. Therefore, the count of non-palindromic numbers is:

9000 - 90  8910

Conclusion

By using both Python and mathematical methods, we have determined that there are 8910 four-digit numbers that are not palindromic. This result can be confirmed by the Python program and the straightforward calculation explained above.

Understanding the nature and characteristics of palindromic and non-palindromic numbers not only enhances mathematical proficiency but also deepens our appreciation for the beauty of numbers.