The year 2025 has many mathematical properties that make us believe it is going to be the perfect year. In this article, we analyze them all.
Perfect Square
The most well-known property is that 2025 is a perfect square since 45 squared equals 2025.
The next year that will have this property is 2116, as 46 squared equals 2116.
Product of Two Perfect Squares
Derived from the first property and using the properties of exponents, specifically this one:
We can deduce that:
Therefore, 2025 can also be expressed as the sum of the squares of 9 and 5.
Sum of Cubes
The sum of the cubes of all the numbers from 0 to 9 results in 2025:
This property only occurs this year, whereas similar properties in other years have been:
- Sum of squares from 0 to 9, which is 285, 1740 years ago.
- Sum of the fourth powers from 0 to 9, which is 15333, in 13308 years.
So, both of these properties are quite far from us.
However, the next two curiosities we will discuss are somewhat related to this…
Square of the Sum of All Digits
We can express 2025 as the square of the sum of all digits from 0 to 9, thus:
First, we calculate the sum of the digits from 0 to 9:
And we find that:
Thus, this property is also proven.
Sum of the Multiplication Tables from 1 to 9
Another curiosity of this year is that if we sum the multiplication tables (yes, the ones we learned in school) from 1 to 9, the total is 2025.
The multiplication table for 1: 1·1, 2·1, 3·1, 4·1… The table for 2: 2·1, 2·2, 2·3…
The sum of all the multiplication tables would be expressed something like this:
This way, we can factor out, since all numbers are multiplied by the same nine numbers: 1, 2, 3, …, 9.
Thus, we would have:
Using the famous Gauss sum, we calculate the sum of the numbers from 1 to 9:
So we have:
And if we recall the first property:
Therefore, QED (Quod erat demonstrandum, “what was to be demonstrated”), which indicates that the initial claim is now proven.
Conclusions
Well, these are just five mathematical properties, but this year can be perfect; we just need to make it happen ourselves and not wait for someone to tell us. Let’s live it!
References
title: Why 2025 is Going to Be the Perfect Year author: dario-otero altImage: 3D metallic letters of 2025. description: 2025 has many mathematical properties that make us believe it is going to be the perfect year. In this article, we analyze them all, don’t miss it! date: 2025-01-26 tags: - Math Magic - Curious Numbers
The year 2025 has many mathematical properties that make us believe it is going to be the perfect year. In this article, we analyze them all.
Perfect Square
The most well-known property is that 2025 is a perfect square since 45 squared equals 2025.
The next year that will have this property is 2116, as 46 squared equals 2116.
Product of Two Perfect Squares
Derived from the first property and using the properties of exponents, specifically this one:
We can deduce that:
Therefore, 2025 can also be expressed as the sum of the squares of 9 and 5.
Sum of Cubes
The sum of the cubes of all the numbers from 0 to 9 results in 2025:
This property only occurs this year, whereas similar properties in other years have been:
- Sum of squares from 0 to 9, which is 285, 1740 years ago.
- Sum of the fourth powers from 0 to 9, which is 15333, in 13308 years.
So, both of these properties are quite far from us.
However, the next two curiosities we will discuss are somewhat related to this…
Square of the Sum of All Digits
We can express 2025 as the square of the sum of all digits from 0 to 9, thus:
First, we calculate the sum of the digits from 0 to 9:
And we find that:
Thus, this property is also proven.
Sum of the Multiplication Tables from 1 to 9
Another curiosity of this year is that if we sum the multiplication tables (yes, the ones we learned in school) from 1 to 9, the total is 2025.
The multiplication table for 1: 1·1, 2·1, 3·1, 4·1… The table for 2: 2·1, 2·2, 2·3…
The sum of all the multiplication tables would be expressed something like this:
This way, we can factor out, since all numbers are multiplied by the same nine numbers: 1, 2, 3, …, 9.
Thus, we would have:
Using the famous Gauss sum, we calculate the sum of the numbers from 1 to 9:
So we have:
And if we recall the first property:
Therefore, QED (Quod erat demonstrandum, “what was to be demonstrated”), which indicates that the initial claim is now proven.
Conclusions
Well, these are just five mathematical properties, but this year can be perfect; we just need to make it happen ourselves and not wait for someone to tell us. Let’s live it!
