Square Root

In mathematics, a square root is a fundamental concept. The square root of a number is a value that, when multiplied by itself, gives the original number. Consider the formula y² = x. We can say that the square root of x is a value y such that y × y = x. Just the same, we can figure out the square root of any number. Any positive real number has one positive and one negative square root. For instance, the square root of 9 is ±3, where 3 × 3 = 9 also (-3) × (-3) = 9. 

To find square root without calculator, you can refer the square roots table below: 

History of Square Roots

The concept of square roots dates back thousands of years, with Babylonians and Egyptians developing early methods to approximate their values. By 500 BCE, the Pythagoreans used square roots in geometry. Later, with the rise of algebra, their use expanded.

In 1450 AD, Regiomontanus introduced the symbol ‘R’, and in 1525, Christoph Rudolf gave us the modern √ sign. Today, square roots play a vital role in real-life calculations involving distance, area, and force.

The square root has several properties that make it both unique and easy to understand. Let us uncover these properties:

• A perfect square number has a perfect square root. 16 is a perfect square, since it is equal to 4 × 4. Additionally, ±4 is the square root of 16. 
   
• If the last digit of a number is an even number of zeros, it can have a square root. Take a look at this, the square root of 100 is √100 = 10 (10 × 10 = 100).
   
• The values of two square roots can be multiplied. For example, √3 × √2 = √6. 
   
• If we multiply two identical square roots, we get the radicand (the number inside the radical), such as √3 × √3 = 3. 
• If we multiply a negative square root by itself, it does not give a negative result. Instead, it yields a positive number. For instance, (-3) × (-3) = 9. 
   
• Numbers ending with 2, 3, 7, or 8 do not have a perfect square root. There is no whole number whose square equals 32, 53, 77, or 98. 
   
• Numbers ending with 1, 4, 5, 6, or 9 in the unit place have the perfect square roots. For example, ±5 is the square root of 25.
   

The square root of a number can be classified into two types: perfect squares and non-perfect squares. Let us now understand more about the classifications, and more about negative square roots. 

Perfect Squares

A perfect square is said to be an integer that is the square of another integer. In simple terms, it is a number resulting from squaring another number. As an example, 5 × 5 results 25. Here, 25 is a perfect square and it is the product of 5. Also, 5 is an integer.

 

Some examples of perfect squares are: 

\(1 = 1 × 1 = 12\)

\(4 = 2 × 2 = 22\)

\(9 = 3 × 3 = 32\)

\(16 = 4 × 4 = 42\)

\(25 = 5 × 5 = 52\)

\(36 = 6 × 6 = 62\)

These examples clearly illustrate the concept of a perfect square. A perfect square is the result that is obtained by multiplying an integer by itself.

Non-Perfect Squares

A non-perfect square root is not an integer. It will be an irrational number instead. It cannot be expressed as a simple fraction. They are numbers that do not result from multiplying an integer by itself. Its decimals never stop.
 

For example, the square root of 2 is \(√2 ≈ 1.414\). 

The square root of 2, which is 1.414 is an irrational number. Non-perfect squares have square roots whose values are not whole numbers. A few examples of non-perfect squares are:

\(√2 ≈ 1.414\)
\(√3 ≈   1.732\)
\(√5 ≈ 2.236\)
\(√8 ≈ 2.828\)

In advanced mathematics, non-perfect squares are widely used, where approximate values are required. 
 

Negative Square Roots
 

A negative square root is the negative value of a number that, when squared, gives the same result as the positive square root. For any positive number, there are actually two square roots, one positive and one negative. 
Mathematically, if √n is the positive square root of n, then -√n is its negative square root. 

For example, √9 = 3, where it is the positive square root.
–√9 = –3, here the square root is negative. 
Both 3 and –3, when squared, give 9.

Some examples of negative square roots are: 
–√2 = 4
–√4 = 16
–√5 = 25
–√7 = 49

There are different methods we use to calculate square roots of a given number. Some of the methods we use are given below:

Prime Factorization Method: 

While using the prime factorization method for square root calculation, a number is represented as the product of its prime factors. Remember that this method is only applicable to perfect square numbers. To find the square root of a perfect square, the prime factorization is the most commonly used method. We need to follow the steps mentioned below to calculate the square root of a number:

Step 1: Break down a number into its prime factors. Start with the smallest prime number, which is 2. 

Step 2: Group pairs of factors, where both factors in each pair are the same. 

Step 3: Take one factor from each pair. 

Step 4: Multiply the factors. The square root of a given number is the product of their factors.

For instance, take a look at this example of the square root of 144:

\(144 = 2 × 2 × 2 × 2 × 3 × 3\)

\((2 × 2) × (2 × 2)  × (3 × 3)\)

\((2 × 2 × 3)\)

\(144 = 122\)

\(√144 = 12\)

The square root of 144 is \( ±12\).

Long Division Method:
 

To find the square root of an imperfect number, we can apply the long division method. In this method, large numbers will be broken down into small parts.  The several steps of this method are:

Step 1: Break a number into pairs of two digits from right to left. 

Step 2: Find the greatest number whose square is smaller or equal to the first digit or pair.

Step 3: Now we can subtract the square of that number from the pair. After that, we can drop the next pair of numbers. 

Step 4: Double the number you found in step 2, and you get the new divisor.

 
Step 5: When we reach the required level, we can stop the steps. 

For example, we can find the square root of 20:
 

Step 1: We have one pair of digits for 20. 

Step 2: 4 is the largest number whose square is smaller than or equal to 20, because \((4 × 4 = 16)\). The first digit of the square root of 20 is ±4. 

Step 3: Subtract 16 from 20. 
\(20 - 16 = 4\)

Now, we have to drag down the next pair. But here, there is no other pair. So add a zero and make it 400. 
 

Step 4: Double the digit 4 and we get 8 as the result. 
 

Step 5: Next, we find the largest number, whose square is less than or equal to 400, when we multiply by the new number and add to 8. Here, 5 is the next digit. 
\(8 × 5 = 40\)
\(40 × 5 = 200\)
 

Step 6: Subtract 200 from 400 which gives 200. 
Now, we know the approximate square root of 20 is ±4.5. 

Using a Calculator:
 

This is a simple and interesting way to calculate the square roots of any given number. Finding the symbol of square root √ in the calculator is the foremost thing you should remember. Just enter your number, then press the square root symbol √. The calculator will display the square root.

Let us dive into the world of square roots and understand how it works. It has countless applications in everyday life. Square roots help students to solve academic and real-world problems. Some key benefits of mastering square roots by students are:
 

• Root for advanced math: Square roots are key in algebra, geometry, and calculus. They’re used to solving quadratic equations, and understand exponents and logarithms.
  
   
• Real-world applications: Used in architecture, physics, engineering, and finance, square roots help calculate distances, forces, energy, and compound interest.
  
   
• Builds problem-solving skills: Working with square roots improves logical thinking and analytical reasoning, helping students tackle complex math challenges confidently.
  
   
• Strengthens math confidence: Mastering square roots boosts math confidence, making students comfortable with advanced topics and real-life problem-solving.
  
   
• Foundation for future learning: A strong grasp of square roots prepares students for higher concepts in trigonometry, algebra, and data analysis, which rely on root operations.

Some tips and tricks will help students to understand more about the square roots. Also, it makes the calculations easier. After applying these tips and tricks, kids can solve difficult mathematical problems and equations. 

Learn the basics: Students should first master the fundamentals of square roots. It is a value that, when we multiply by itself, we get the original number. 
 

Remember perfect squares: Another useful trick is to by heart the perfect squares of numbers from 1 to 15. For example, we know that, 22 = 4 and 32 = 9. Memorizing these will help students quickly identify perfect squares during calculation.
 

Identify the nearest perfect squares: For non-perfect squares, estimate the nearest perfect squares. For instance, to identify the √20, find the closest perfect squares, they are √16(4) and √25(5). 
 

Understand the prime factorization: To simplify the square roots of any given number, break the number into its prime factors. It will help the students to solve problems more easily and faster. 
 

Apply division method: If the given number is large, kids can use the division method to find the square root. By breaking a large number into smaller parts, the calculations become much easier.

Use real-life examples: Parents and teachers can help students to link square roots to real-world contexts like measuring areas, distances, or field diagonals to make learning practical and engaging.

Encourage estimation practice: Ask students to estimate square roots of non-perfect squares without using calculators, it sharpens number sense and reasoning.

Visual learning through grids: Parents and teachers can use square grids or graph paper to visually represent squares and their roots. Seeing patterns helps students grasp the concept better.

Practice with fun activities: Create quick challenges, flashcards, or “find the root” games during class or at home to make memorization enjoyable.

Connect with technology: Use interactive math apps or online tools to demonstrate square root calculations visually. This helps both visual and auditory learners understand faster.

In real life, we use square roots not only in mathematics but also in solving practical problems related to science, technology, and everyday life. Here are a few real-world applications:


Geometry and area: Square numbers are used to find the area of squares and rectangles.


Construction: Builders use square numbers when designing square tiles for floors.


Sports fields and courts: Square numbers help in making designing square shaped fields or courts.


Computer graphics: Pictures in screens are often arranged in square grids that help in calculating resolutions.


Patterns and Puzzles: Square numbers appear in magic squares, chessboards, and other recreational puzzles.