Squares and Square Roots — Study Notes

1. Introduction

If a number is multiplied by itself, the product obtained is called the square of that number. For example, 5 x 5 = 25, so 25 is the square of 5. Squares of numbers appear all around us - in the tiles of a floor, the area of a square field, and in many patterns in nature. Understanding squares helps us calculate areas quickly and solve many practical problems.

In this chapter we will learn about square numbers and their properties, how to find squares of numbers using patterns, how to find square roots using different methods such as prime factorisation and long division, and how to use the Pythagorean triplets. The concept of square roots is used throughout mathematics, so it is very important to master this chapter thoroughly.

2. Square Numbers and Their Properties

The square of a natural number n is written as n^2. For example, 1, 4, 9, 16, 25, 36, ... are the squares of 1, 2, 3, 4, 5, 6, ... respectively. Such numbers are called perfect squares.

Important Properties of Square Numbers

1. A number ending in 2, 3, 7 or 8 is never a perfect square.
2. A square number never ends in an odd number of zeroes. For example, 100, 400, 900 end in an even number of zeroes.
3. The square of an even number is always even, and the square of an odd number is always odd.
4. The square of a number ends in 0, 1, 4, 5, 6 or 9.
5. Between the squares of n and (n + 1), there are 2n non-square numbers.

Squares of Numbers Ending in 5

A quick trick: for a number ending in 5, the square is found by multiplying the number before the 5 by one more than it, and writing 25 at the end. For example, 85^2: 8 x 9 = 72, so 85^2 = 7225.

3. Pythagorean Triplets

A set of three numbers a, b and c is called a Pythagorean triplet if: $$a^2 + b^2 = c^2$$

For example, 3, 4 and 5 form a Pythagorean triplet because 3^2 + 4^2 = 9 + 16 = 25 = 5^2. Other examples include 5, 12, 13 and 8, 15, 17.

If n is an odd number, then n, (n^2 - 1)/2 and (n^2 + 1)/2 form a Pythagorean triplet.

4. Square Roots

The square root of a number x is that number which when multiplied by itself gives x. It is denoted by the symbol sqrt(x), called the radical sign. For example, sqrt(25) = 5 because 5 x 5 = 25.

Finding Square Root by Prime Factorisation

To find the square root of a perfect square by prime factorisation: 1. Express the number as the product of its prime factors. 2. Group the factors in pairs. 3. Take one factor from each pair and multiply them.

Example: sqrt(324) = sqrt(2 x 2 x 3 x 3 x 3 x 3) = 2 x 3 x 3 = 18.

Finding Square Root by Long Division Method

The long division method is used for larger numbers and for numbers that are not perfect squares. It gives the square root digit by digit. This method is essential when we need to find the square root of numbers like 2304 or 7921 accurately.

5. Square Roots of Decimals and Other Applications

Square Root of a Decimal Number

The square root of a decimal number is found by the long division method, placing the decimal point appropriately. Alternatively, we can convert the decimal to a fraction with a perfect square denominator.

Example: sqrt(6.25) = sqrt(625/100) = 25/10 = 2.5.

Applications

• Finding the side of a square when its area is given: side = sqrt(area).
• Solving problems about the diagonal of a square and Pythagorean triplets.

Quick Revision Tables

Table 1: Squares and Square Roots of First 12 Numbers

Table 2: Quick Methods

Mind Map

graph TD
    A["Squares and Square Roots"] --> B["Square: n x n = n^2"]
    A --> C["Perfect squares"]
    C --> D["End in 0, 1, 4, 5, 6, 9"]
    C --> E["Never end in 2, 3, 7, 8"]
    A --> F["Pythagorean triplets"]
    F --> G["a^2 + b^2 = c^2, e.g. 3, 4, 5"]
    A --> H["Square root: sqrt(x)"]
    H --> I["Prime factorisation method"]
    H --> J["Long division method"]
    H --> K["Square root of decimals"]

Important Diagrams (SVG)

Diagram 1: Square Numbers as Squares of Grids

Diagram 2: Square Root by Long Division

Common Mistakes

1. Students think that a number ending in 5 is always a perfect square. Actually only numbers like 25, 225 (ending in 25) can be perfect squares; 35 is not a perfect square.
2. Students forget that perfect squares can never end in 2, 3, 7 or 8, and may waste time trying to find their square roots.
3. While finding square roots by prime factorisation, students forget to group the prime factors in pairs before taking one factor from each pair.
4. In the long division method, students make a mistake in the very first step by not pairing the digits correctly from the right (for whole numbers).
5. Students confuse the square of a number with the square root. For example, they may write sqrt(81) = 81 instead of sqrt(81) = 9.
6. When using the Pythagorean triplet formula, students forget that n must be odd for n, (n^2 - 1)/2, (n^2 + 1)/2 to be a triplet.
7. Students forget to place the decimal point correctly while finding the square root of a decimal number, giving answers that are ten times too large or too small.

Exam Tips

1. Memorise squares up to 30 and square roots up to 20; this speeds up calculations greatly.
2. For a number ending in 5, use the shortcut (n(n+1) with 25 appended) to save time.
3. In prime factorisation, always write the number as a product of primes first, then pair them.
4. Verify your square root by squaring the answer; the result must equal the original number.
5. For Pythagorean triplet questions, always check a^2 + b^2 = c^2 explicitly.
6. Practice the long division method on at least five different numbers, since it is a guaranteed exam question.
7. Remember to write sqrt sign correctly and to give the answer in the same form asked (whole number or decimal).

Conclusion

Squares and square roots are fundamental ideas in mathematics. In this chapter we learnt what square numbers are and their important properties, including the last-digit rules for perfect squares. We studied the shortcut for squaring numbers ending in 5, and the Pythagorean triplets which connect squares to right-angled triangles. We then learnt two important methods of finding square roots - the prime factorisation method and the long division method - along with the square roots of decimals. These concepts are widely used in mensuration, algebra, trigonometry and even in everyday problems like finding the side of a square plot. A firm understanding of this chapter will help students greatly in all future mathematics courses.