5.4. Finding the Square of a Number

Squares of small numbers like 3, 4, 5, 6, 7, ... etc. are easy to find. But can we find the square of 23 so quickly? The answer is not so easy and we may need to multiply 23 by 23.

Calculating the square of smaller numbers, such as 3, 4, 5, or 6, can be done easily because we memorize their squares. However, for larger numbers like 23, it can be tedious to rely on simple multiplication. This shows that although squaring is easy for small numbers, it becomes cumbersome for larger numbers.

Imagine if you regularly bake cookies. You have memorized the recipe for a dozen cookies (small number), but if someone asks you to bake 23 dozen, you might need to consult your notes or a calculator for accurate measurements.

There is a way to find this without having to multiply 23 × 23. We know 23 = 20 + 3. Therefore 23² = (20 + 3)² = 20(20 + 3) + 3(20 + 3) = 20² + 20 × 3 + 3 × 20 + 3² = 400 + 60 + 60 + 9 = 529.

To calculate the square of 23, we break it down into a sum, (20 + 3). This utilizes the algebraic identity for squaring a number: (a + b)² = a² + 2ab + b². By simplifying this, we arrive at the answer without direct multiplication, showing a pattern and structure in numbers that helps facilitate calculations.

Think of a recipe where you have to double ingredients. Instead of measuring out each ingredient twice, you can simply calculate 2 times the amount of a single serving, making it both easy and more efficient.

Example 1: Find the square of the following numbers without actual multiplication. (i) 39 (ii) 42. Solution: (i) 39² = (30 + 9)² = 30(30 + 9) + 9(30 + 9) = 30² + 30 × 9 + 9 × 30 + 9² = 900 + 270 + 270 + 81 = 1521. (ii) 42² = (40 + 2)² = 40(40 + 2) + 2(40 + 2) = 40² + 40 × 2 + 2 × 40 + 2² = 1600 + 80 + 80 + 4 = 1764.

In this section, we calculate the squares of two numbers, 39 and 42, using the addition method we learned earlier. For 39, we broke it down as (30 + 9) and for 42, we used (40 + 2). This method allows us to avoid direct multiplication by using previously solved squares and the distributive property of multiplication.

Imagine organizing your wardrobe. Instead of buying clothes in bulk and estimating sizes, you can take careful measurements and find that a specific combination gives you the fit you need, just like combining smaller number segments makes squaring easier and more accurate.

5.4.1 Other patterns in squares. Consider the following pattern: 25² = 625 = (2 × 3) hundreds + 25.

This chunk discusses recognizing patterns in numbers, particularly when the unit digit is 5. It shows that the square of any number ending in 5 can be found using a specific pattern: (a × (a + 1)) hundreds + 25, which greatly simplifies calculations.

Think of how certain flower arrangements can follow specific patterns based on the number of stems. Recognizing this pattern helps you create beautiful bouquets efficiently without having to calculate each combination from scratch.

5.4.2 Pythagorean triplets. Consider the following (3² + 4² = 5²). The collection of numbers 3, 4, and 5 is known as a Pythagorean triplet.

A Pythagorean triplet consists of three positive integers a, b, and c, such that a² + b² = c². This chunk presents basic examples like (3, 4, 5) and illustrates their relevance in finding relationships between squares of numbers. This concept is especially significant in geometry, as it helps in calculations regarding right triangles.

Consider a ladder leaning against a wall. The length of the ladder, the distance from the wall, and the height up the wall forms a right triangle, similar to how the Pythagorean triplet provides a method to determine these distances.

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