Simply Maths

Square numbers are a fascinating topic that has intrigued scholars, mathematicians, and enthusiasts for centuries. As this article shows, these numbers reveal some surprising and interesting facts when explored more deeply. (Stick around for the end - I'll show you how to quickly work out the squares of numbers up to 99!)

Square numbers, also known as **perfect squares**, are formed by multiplying an integer by itself. In mathematical notation, if 'n' is an integer then...

For example, 1, 4, 9, 16, and 25 are all square numbers because they can be expressed as

Square numbers can be categorized into two groups: odd and even. Odd square numbers end in **1**, **5**, or **9**, while even square numbers end in **0**, **4**, or **6**. This pattern is a result of the multiplication of odd or even integers.

A surprising fact about square numbers is that they can be represented as the sum of consecutive odd numbers. For example, 9 (3-squared) can be expressed as 1 + 3 + 5, 16 (4-squared) as 1 + 3 + 5 + 7, and 25 (5-squared) as 1 + 3 + 5 + 7 + 9. This property is related to geometric figures, where square numbers represent perfect squares, and the sum of consecutive odd numbers corresponds to the sides of those squares.

Stefan Friedrich Birkner, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons

Square numbers are closely related to **Pythagorean triples**, which are sets of three positive integers (a, b, c) that satisfy the Pythagorean theorem:

Square numbers often appear as the squares of the side lengths in these triples, making them essential in geometry and trigonometry.

Examples include ...

(5, 12, 13): a = 5, b = 12, c = 13

(8, 15, 17): a = 8, b = 15, c = 17

Square roots are the inverse operation of squaring. If you take the square root of a square number, you get the original integer. For example, the square root of 25 is 5.

Square numbers have numerous practical applications in various fields. They are used in geometry for calculating areas and side lengths, in physics for analyzing motion and energy, and in computer science for data encryption and error checking (e.g., the use of checksums).

A **digital root** is the repeated sum of the digits of a number until a single-digit number is obtained. For square numbers, the digital root is often **1, 4, 7, or 9**. This property can be used in various number games and puzzles:

36 = 3 + 6 = **9** 9 is the digital root of 36

25 = 2 + 5 = **7** 7 is the digital root of 25

16 = 1 + 6 = **7** 7 is the digital root of 16

Euler's Sum of Powers Conjecture

A surprising and unsolved problem in mathematics is Euler's Sum of Powers Conjecture, which states that there are no four positive integers a, b, c, and d, such that:

This conjecture involves different types of powers, including square numbers, and remains an open question.

**Quickly finding the square of two-digit numbers**

I'll use the square of 18 as an example:

**Square the 2 numbers**, 1 and 8 writing each as a 2-digit number:

01 and 64

**Write in line**:

0164

Multiply** all numbers (including the square)**: 1 x 8 x 2 = 16.

**Place this number under the middle numbers of the previous calculation and add them together.**

0164

__ 16 __

324

So the square of 18 is 324

Here are some more examples:

**Challenge:**

See if you can develop a method to find the square of 3-digit numbers

Square numbers are more than just a mathematical concept; they reveal intriguing patterns, connections to geometry, and practical applications - well worth an investigation!

Take a look at the following...

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