At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. From that moment until I was thirty-eight, mathematics was my chief interest and my chief source of happiness.
Bertrand Russell
Welcome to the blog Math1089 – Mathematics for All.
As 2024 bids us farewell and 2025 begins anew, may this year bring you:
12 Months of Happiness
52 Weeks of Fun
365 Days of Success
8760 Hours of Good Health
525600 Minutes of Good Luck
31536000 Seconds of Joy
and so much more! Wishing you a bright and prosperous New Year!
As we step into 2025, let’s delve into the mathematical charm of this fascinating number. Join us as we uncover even more wonders of 2025 – a year brimming with numerical splendor and inspiration. This post is the second in our series on the number 2025, building on the insights shared in our first blog. Here, we present 40 additional math insights about 2025. You can find the link to the first blog post below!
🔵 1. 2025 can be represented as 11111101001 in the binary number system.
🔵 2. 2025 can be represented as MMXXV in Roman numerals.
🔵 3. 2025 can be represented as the product of two squares:
2025 = 52 × 92
🔵 4. 2025 can also be expressed as the product of three squares:
2025 = 32 × 32 × 52
🔵 5. Adding 2025 to its reverse (5202) results in a palindrome:
2025 + 5202 = 7227.
🔵 6. 2025 is the 45th square number.
🔵 7. If the starting digit of 2025 is increased by 1, it remains a square :
2025 → (2 + 1)025 → 3025 = 552.
🔵 8. If each digit of 2025 is increased by 1, it also remains a square:
2025 → (2 + 1)(0 +1)(2 + 1)(5 + 1) → 3136 = 562.
🔵 9. 2025 is the sum of 45 consecutive numbers from 23 to 67:
23 + 24 + 25 + ∙∙∙ + 66 + 67 = 2025.
🔵 10. We are lucky to have three Thursdays in the year 2025 that fall on the 13th of the month.
🔵 11. The number 2025 is the sum of the first 45 odd numbers:
2025 = 1 + 3 + 5 + ∙∙∙ + 89 = 452.
🔵 12. The total number of divisors of 2025 are: 1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, and 2025. These total 15 in number, and 15 is also one of the factors of 2025. Therefore, 2025 is divisible by the total number of its divisors.
🔵 13. 2025 is a Harshad number because it is divisible by the sum of its digits:
The term harshad comes from the Sanskrit words, harṣa (joy) + da (give), meaning joy – giver, and the name was given by Mr. D. R. Kaprekar, an Indian school teacher.
🔵 14. If we interchange the pairs of digits in 2025, we obtain 2520, which can be expressed as the product of two reversible numbers in two different ways. In fact, 2520 = 210 × 012 = 120 × 021.
🔵 15. 2025 as a definite integration:
🔵 16. 2025 can also be represented as
- (10 × 9 × 8 × 7 ÷ 6 ÷ 5 × 4 + 3) × (2 + 1)
- (10 + (9 + 8 × 7) × 6) × 5 + 4 × 3 × 2 + 1
- 10 × (9 + 8 × 7 – 6 + 5! + 4!) – 3 × 2 + 1
- 10 × 9 × (8 + 7 + 6) + 5! + 4 × 3 + 2 + 1
- (10 + 9) × (8 + 7 + 6) × 5 + 4! + 3 + 2 + 1
🔵 17. 2025 in base 15 is 900, which is also a perfect square.
🔵 18. The sum of entries (in blue, below) in a 9 × 9 multiplication table is 2025.
🔵 19. Another representation of 2025 using basic mathematical operations:
Observe that the same digits appear in the same order.
🔵 20. 2025 can be represented as the sum of squares in various ways, as shown below:
■ As the sum of three squares: 2025 = 52 + 202 + 402
■ As the sum of seven squares: 2025 = 22 + 42 + 62 + 72 + 82 + 162 + 242 + 322.
■ As the sum of twelve squares: 2025 = 32 + 42 + 52 + 72 + 82 + 92 + 112 + 152 + 162 + 172 + 192 + 232.
■ As the sum of fifteen squares: 2025 = 12 + 32 + 52 + 62 + 72 + 92 + 102 + 112 + 122 + 132 + 142 + 152 + 162 + 172 + 182.
🔵 21. Representation of 2025 using the binary digits 0 and 1 is
2025 = 1001(1+ 1) + 11 + 11 + 1
🔵 22. 2025 as the difference of two squares:
2025 = 2052 – 2002 = 752 – 602.
🔵 23. Another fun fact: 452 = 2025 = 1012 + 1013 and 452 + 10122 = 10132.
🔵 24. 2025 can be expressed as the square of the product of two terms: the first term is the sum of the squares, and the second term is the sum of the cubes:
2025 = {(12 + 22)(13 + 23)}2.
🔵 25. 2025 is a gapful number since it is divisible by the number (25) formed by its first and last digits, as well as the number formed by last two digits (25).
🔵 26. 2025 can also be represented as the sum of a fourth power and a sixth power, as
2025 = 64 + 36.
🔵 27. Using only the number 2, we can write
🔵 28. Another representation of 2025 using its own digits and appearing in the same order:
🔵 29. Representation of 2025 as the square of the sum of squares:
2025 = (22 + 32 + 42 + 42)2
🔵 30. Below is a magic square of order 4, with a magic sum of 14847, and containing the digits 0, 2, and 5:
🔵 31. There are 2025 essentially distinct ways to colour the sides of a rectangle using at most 9 colours.
🔵 32. Using combinations, 2025 can be represented as
2025 = 10C2 × 10C2.
🔵 33. There are exactly 2025 numbers between 1 and 9999 where the last digit is strictly greater than all the other digits (if any). Examples of such numbers include: 1, 2, 3, 4, … , 8869, 8879, 8889.
🔵 34. 2025 as a continued fraction: We have the following identity
Repeated application of this identity yields
As a result, we can write
🔵 35. 2025 can also be represented in terms of π, e, and various mathematical operations.
🔵 36. Most simplest power representation using all the numbers 0, 1, 2, … , 2025 is
2025 = 10 + 20 + ∙∙∙ + 20250
🔵 37. Consider the following setup: two pairs of equal tangent circles are arranged in such a way that they are tangent to both an external semicircle and an oblique chord. In this configuration, the smallest integer radius of the semicircle that allows the radii of the inscribed circles to be integer values (specifically, 400 and 648) is 2025. [Sangaku Problem]
🔵 38. Consider writing the number 1 once, the number 2 twice, the number 3 three times, and so on up to the number 45, repeated forty-five times, forming a sequence like this: 12233344445555…454545. The total number of digits in this sequence is 2025. This is the only such instance where the total number of digits is 2025 for any number greater than 1.
🔵 39. Consider the following square matrix:
This matrix consists of rows that are cyclic permutations of the first four composite numbers 4, 6, 8, and 9. The determinant of this matrix is given by:
🔵 40. 2025 through the lens of the golden ratio:
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Awesome
Thank you so much 👍 Please share 🙏
For me this 2025 number was curiously
The sum of cubes from 1 to 9..
Thought I might make a Tshirt about the math
Now after viewing this site..I’ll need an infinite shirt
Thank you so much 🙏